Dottorato di Ricerca in Matematica XIII Ciclo

Università degli Studi di Milano, Politecnico di Milano, Università degli Studi di Pavia, Università Cattolica “Sacro

Cuore” di Brescia, Università degli studi di Trieste

Tesi di Dottorato

Generalized self similar fractals and invariant energies

Giorgio Follo

Relatore: Prof. Stefano Mortola Dipartimento di Matematica “F. Brioschi”

Politecnico di Milano Coordinatore: Prof. Bernhard Ruf

Dipartimento di Matematica “F. Enriques” Universitá degli Studi di Milano

Giorgio Follo Dipartimento di Matematica “F.Enriques” Universita’ degli Studi di Milano via Saldini, 50 20133 Milano — Italia

follo@socrates.mat.unimi.it

Contents

1 Fractals as limits of iterations 5 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Notation and preliminary results . . . . . . . . . . . . . . . . 5

1.2.1 Hausdorff measures and Hausdorff dimension . . . . . . 6 1.2.2 d-sets in ℝ^{N} . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Box-counting dimensions . . . . . . . . . . . . . . . . . 9 1.2.4 Entropy numbers . . . . . . . . . . . . . . . . . . . . . 11 1.2.5 The Hausdorff metric . . . . . . . . . . . . . . . . . . . 12 1.2.6 Sequences of indices and product spaces . . . . . . . . 14

1.3 Limit sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1 Basic notation . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.2 Existence and uniqueness . . . . . . . . . . . . . . . . 17 1.3.3 Some properties of K . . . . . . . . . . . . . . . . . . 20 1.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.4 A measure on K . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 Similitudes on ℝ^{N} . . . . . . . . . . . . . . . . . . . . . . . . 30

1.5.1 An example concerning d-sets . . . . . . . . . . . . . . 31 1.5.2 An example concerning h-sets . . . . . . . . . . . . . . 38

2 A conjecture of E. De Giorgi 41 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2 Γ-convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 Definition of capacity dimension . . . . . . . . . . . . . . . . . 51 2.4 Sets with non-empty interior . . . . . . . . . . . . . . . . . . . 54 2.5 Totally disconnected compact sets . . . . . . . . . . . . . . . . 61 2.6 Smooth curves in ℝ^{2} . . . . . . . . . . . . . . . . . . . . . . . 66 2.7 Relations with the immersion of K in ℝ^{N+M} . . . . . . . . . 78 2.8 An attempt about fractal sets . . . . . . . . . . . . . . . . . . 84 3 Invariant energy for the Koch curve 111

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.2 Γ-convergence on the closed unit interval . . . . . . . . . . . . 111 3.3 Γ-convergence on the Koch curve . . . . . . . . . . . . . . . . 116

3.3.1 Difference functionals . . . . . . . . . . . . . . . . . . . 123 3.3.2 Integral functionals . . . . . . . . . . . . . . . . . . . . 124

4 Appendix 129 4.1 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.2 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Introduction

Self similar fractals are well known. In a famous paper J. E. Hutchinson proved the existence of a self similar fractal for every finite family of similitudes; moreover he computed its Hausdorff dimension d and proved that it is a d-set (Theorem 1.19). See [19], chapter 2, section 1 or [23], chapter 1, section 3 for the definition of d-set.

More generally, if 𝔉 = {f_{1}, f_{2}, ..., f_{M}} is a finite family of contractions on a complete metric space (X, d_{X}) , then there exists a unique non-empty closed and bounded set K in X , such that

K = 𝔉(K) = ⋃_{i=1}^{M} \overline{f_{i}(K)} .

Moreover K is compact and for every non-empty closed and bounded set C ⊆ X , the sequence (𝔉^{n}(C)) , where the exponent means iteration, converges to K in the Hausdorff metric.

This theorem is proved by using the fixed point theorem; indeed the function 𝔉 is a contraction on the family of non-empty closed and bounded subsets of X , with respect to the Hausdorff metric (it is a standard fact that the family of closed and bounded subsets of a complete metric space endowed with the Hausdorff metric is a complete metric space).

We generalize this theorem. To do it we used a generalized version of the fixed point theorem.

If we have a sequence (g_{n}) of contractions on a complete metric space Y , with the properties that the sequence of their Lipschitz constant does not tend to 1 too rapidly and that there exists a bounded set Q such that g_{n}(Q) ⊆ Q for every n ∈ ℕ^{∗} , then there exists a unique point y ∈ Y such that the sequence ((g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(y_{0})) converges to y for every starting point y_{0} ∈ Y . Using this lemma we proved that, given a sequence (𝔉_{n}) of finite families of contractions on X satisfying two conditions that are similar to the ones required for (g_{n}) , there exists a unique non-empty closed and bounded set K ⊆ X such that the sequence ((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C)) converges to K in the Hausdorff metric; moreover K is compact.

Let X be complete and separable and let l = {l_{1}, l_{2}, ..., l_{M}} be a finite set of real numbers in ]0, 1[ , with ∑_{i=1}^{M} l_{i} = 1 ; in [17] it is proved that there exists a unique Borel-regular probability outer measure μ that is invariant with respect to (𝔉, l) , in the sense that

μ(A) = ∑_{i=1}^{M} l_{i}μ(f_{i}^{−1}(A)) for A ⊆ X .  (1)

Moreover K is the support of μ .

We generalized this theorem in a similar way as before; we considered a sequence (l_{n}) of finite sets of real numbers in ]0, 1[ , with ♯l_{n} = ♯𝔉_{n} = m_{n} and ∑_{i=1}^{m_{n}} l_{i}^{(n)} = 1 for every n ∈ ℕ^{∗} . Of course we can not write an expression like (1), but we can iterate its second part in the following way: let ν be a Borel regular probability outer measure and let (ν_{n}) be the sequence of measures defined by

ν_{n}(A) = ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}1}(∏_{k=1}^{n} l_{i_{k}}^{(k)})ν((f_{i_{1}}^{(1)} ∘ f_{i_{2}}^{(2)} ∘ ⋯ ∘ f_{i_{n}}^{(n)})^{−1}(A)) for A ⊆ X ;

then there exists a (unique) Borel-regular probability outer measure μ such that ν_{n} weakly converges to μ and μ does not depend on ν . Moreover K is the support of μ .

This kind of fractals gives many counterexamples to known results on self similar fractals.

For example it may happen that K is not a d-set even if all 𝔉_{n} are finite families of similitudes satisfying the Open Set Condition with the same open set and having the same similarity dimension d_{n} = d . Moreover the Hausdorff dimension d may be different from the box dimension.

A paper on this topic will appear on Georgian Mathematical Journal [16]. In [3] M. Bricchi uses fractals of this type to prove that the class of h-

sets really generalize the one of (d, ψ)-sets (that generalize the one of d-sets) introduced by D. Edmunds and H. Triebel in [9] and [10].

In the second chapter we study a method to construct invariant quadratic forms on self similar sets in ℝ^{N} , by using the notion of Γ-convergence.

An invariant form is a quadratic form f on the continuous functions, caracterized by the property that

f(u) = ρ ∑_{i=1}^{M} f(u ∘ f_{i}) ,  (2)

for a constant ρ > 0 ; here f_{1}, f_{2}, ..., f_{M} are the similitudes generating the self similar set K . These forms have been used, for example, by U. Mosco (see [21]).

The method we studied to construct these quadratic forms is due to E. De Giorgi; we briefly describe it. A form

f_{r}^{(α)}(u) = \{ \begin{matrix}\frac{1}{r^{N−α}} ∫_{K_{r}} ∥∇u(x)∥^{2}dx & if u ∈ H^{1}(K_{r}^{∘}) ⋂ C(\overline{Ω}) \\ +∞ & otherwise \end{matrix}

is associated to a compact set K ⊆ ℝ^{N} , for every r ∈]0, 1[ and for every α ∈ [0, N] . The form f_{r}^{(α)} is defined on the space (C(\overline{Ω}), ∥·∥_{L^{∞}(Ω)}) , where Ω ⊆ ℝ^{N} is an open bounded set containing its closed 1-neighborhood.

Then the possibly invariant form f^{(α)} is obtained as the Γ-limit (if it exists) of f_{r}^{(α)} as r → 0 . If this limit doesn’t exists we take the upper and lower Γ-limits f_{+}^{(α)} and f_{−}^{(α)} .

Moreover a ”capacity dimension” of K is defined as dim_{C} K = inf{α ∈ ]0, N] | f^{(α)}(u) = 0 ∀u ∈ C(\overline{Ω})\} .

We calculated the representation of f^{(α)} for some compact sets and noted that when K is self similar then f^{(dim_{C} K)} is invariant. At this moment we don’t know how to treat the general case.

We found an example in which K is self similar but f_{+}^{(α)} ≠ f_{−}^{(α)} for α equal to the Hausdorff dimension of K ; however in this example f_{+}^{(α)} = cf_{−}^{(α)}, where c is the rate between the upper and lower α-dimensional Minkowski content of K .

We compared the different notions of dimension of sets. For regular sets the capacity dimension equals the topological one.

It may happen that dim_{C} K is grater than the Hausdorff dimension of K , but we proved it is always less than or equal to the upper box dimension of K .

If K is totally disconnected, then dim_{C} K = 0 . However, if K = A × B is the cartesian product between a totally disconnected set A and a connected set B , then the capacity dimension of K could be grater than the one of B . For example, if C is the Cantor set in ℝ and K = C ×[0, 1] , then dim_{C} C = 0 and dim_{C} K = 1+log_{3} 2 , that is the Hausdorff dimension of K .

In the third chapter we study two sequences of functionals on the Koch curve. Both are defined by recurrence; the rule E_{n+1}(u) = 4 ∑_{i=1}^{4} E_{n}(u ∘ f_{i}) is iterated, where f_{1}, f_{2}, f_{3}, f_{4} are the four similitudes on ℝ^{2} , generating the Koch curve.

The first sequence starts from

E′_{0}(u) = (u(1, 0) − u(0, 0))^{2}

and then it is a sequence of difference functionals, while the second one starts from

E′′_{0}(u) = \{ \begin{matrix}∫_{0}^{1} (\frac{d}{dt}u(γ_{0}(t)))^{2} dt & if u ∘ γ_{0} ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} ,

where γ_{0}(t) = (t, 0) is the horizontal unit segment.

Both sequences Γ-converges to the functional

E(u) = \{ \begin{matrix}∫_{0}^{1} (\frac{d}{dt}u(γ(t)))^{2} dt & if u ∘ γ ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} ,  (3)

where γ is a particular homeomorphism between [0, 1] and the Koch curve.

It is easy to see that E is invariant in the sense of (2), with ρ = 4 . An other method to construct invariant forms on the Koch curve (without a representation) is described in [5], chapter 2.

We conjecture that E = f^{(α)} for some α ∈]0, log_{3} 4] ; but γ is not L^{1}-rectifiable and this fact makes it very difficult to compare the line integral in (3) with a double integral on the r-neighborhood of K .

Chapter 1

Fractals as limits of iterations

1.1 Introduction

In this chapter we define some kind of fractal sets and related measures.

The fractals and the measure we study can be regarded as a generalization of the ones constructed by J. E. Hutchinson in [17].

Relations with the notion of box dimension are studied. This notion of dimension will be used in the second chapter.

Some definitions and standard results on measure theory are recalled.

1.2 Notation and preliminary results

In this chapter (X, d_{X}) will always be a complete metric space. Additional requirements for X will be specified when necessary.

ℕ = {0, 1, 2, ...} is the set of all non negative integer numbers and ℕ^{∗} = ℕ \ {0} .

By P(X) we indicate the family of all subsets of X .

The closed and open balls in X will be indicated by the symbols B_{X}(x_{0}, r) and D_{X}(x_{0}, r) :

B_{X}(x_{0}, r) = {x ∈ X | d_{X}(x, x_{0}) ≤ r} ,
D_{X}(x_{0}, r) = {x ∈ X | d_{X}(x, x_{0}) < r} .

The diameter of a subset A of X is indicated by |A| :

|A| = sup_{x,y∈A} d_{X}(x, y)

and its number of elements is indicated by ♯A. If r > 0 , A_{r} is the r-neighborhood of A , that is

A_{r} = ⋃_{x∈A} B_{X}(x, r).

If X = ℝ^{N} then we will use the Euclidean metric: d_{ℝ^{N}}(x, y) = ∥x−y∥ =

\sqrt{∑_{i=1}^{N}(x_{i} − y_{i})^{2}} , where x = (x_{1}, x_{2}, ..., x_{N}) and y = (y_{1}, y_{2}, ..., y_{N}) . If A ⊆ ℝ^{N} , L^{N}(A) is the Lebesgue measure of A .

1.2.1 Hausdorff measures and Hausdorff dimension

In this subsection X is a complete separable metric space.

Definition. - Let A be a family of subsets of X and ψ : A → [0, +∞] be a function. Assume that the following two conditions hold:

1. for every δ > 0 there exists a sequence (A_{i}) of elements of A such that X = ⋃_{i=1}^{+∞} A_{i} and |A_{i}| ≤ δ ∀i ∈ ℕ^{∗} ,

2. for every δ > 0 there exists A ∈ A such that ψ(A) ≤ δ and |A| ≤ δ.

Then, for every δ ∈]0, +∞] and for every A ⊆ X we define

H_{δ}^{ψ}(B) = inf{∑_{i=1}^{+∞} ψ(A_{i}) | A_{i} ∈ A , |A_{i}| ≤ δ ∀i ∈ ℕ^{∗} , B ⊆ ⋃_{i=1}^{+∞} A_{i}}

and

H^{ψ}(B) = lim_{δ→0} H_{δ}^{ψ}(A) .

Remark 1. - For every δ > 0 H_{δ}^{ψ} is an outer measure.

Let B ⊆ X and δ > 0 . If ε ∈]0, δ[ then H_{δ}^{ψ}(B) ≤ H_{ε}^{ψ}(B) and so H^{ψ} is well defined and

H^{ψ}(B) = sup_{δ>0} H_{δ}^{ψ}(B) .

Theorem 1.1. - Let A and ψ be as in the previous definition. then H^{ψ} is a Borel measure on X . If all the elements of A are Borel sets then H^{ψ} is Borel regular. PROOF. - See [20], chapter 4, Theorem 4.2.

Definition. - Let A = P(X) , d ∈ [0, +∞[ and ψ : A → [0, +∞]

ψ(B) = \{ \begin{matrix}0 & if B = ∅ \\ 1 & if B ≠ ∅ , d = 0 \\ |B|^{d} & otherwise \end{matrix} .

In this case H^{ψ} is called the d-dimensional Hausdorff measure. We will write H_{δ}^{d} and H^{d} instead of H_{δ}^{ψ} and H^{ψ} .

We briefly recall some facts about Hausdorff measure; see [20] chapter 4 or [11] chapter 2 for their proofs.

For every A ⊆ X H^{0}(A) = ♯A .

If Γ ⊆ ℝ^{N} is a rectifiable curve then H^{1}(Γ) is equal to its length. There exists a constant c(N) ∈ ℝ such that H^{N} = c(N)L^{N} on ℝ^{N}. In the definition of H_{δ}^{d} one can take A to be the family of all open or

closed or convex (if X = ℝ^{N} ) subsets of X and the new definition will be equivalent to the old one.

If B ⊆ ℝ^{N} is H^{d}-measurable and H^{d}(B) < +∞ then the measure μ defined by μ(A) = H^{d}(A ⋂ B) is a Radon measure.

Theorem 1.2. - Let s, t ∈ [0, +∞[ with s < t and let A ⊆ X . If H^{s}(A) < +∞ then H^{t}(A) = 0 . PROOF. - See [20], chapter 4, Theorem 4.7.

Let A ⊆ X and s_{1}, s_{2} ∈ [0, +∞[ with s_{1} < s_{2} . If H^{s_{1}}(A) = +∞ and H^{s_{2}}(A) = 0 then there exists a unique d ∈ [s_{1}, s_{2}] such that H^{t}(A) = +∞ ∀t ∈ [0, d[ and H^{t}(A) = 0 ∀t ∈]d, +∞] .

Definition. - Let A ⊆ X , A ≠ ∅ . We call Hausdorff dimension of A the number

dim A = sup{d ∈ [0, +∞[ | H^{d}(A) > 0}.

Moreover dim ∅ = 0 .

Remark 2. - Let A ⊆ X . Then

dim A = sup{d ∈ [0, +∞[ | H^{d}(A) = +∞} =
inf{d ∈ [0, +∞[ | H^{d}(A) < +∞} = inf{d ∈ [0, +∞[ | H^{d}(A) = 0}

provided that these sets are not empty.

Lemma 1.1. - Let E ⊆ ℝ^{N} and let d > 0 . If f : E → ℝ^{N} is a mapping and c > 0 is a constant such that ∥f(x)−f(y)∥ ≤ c∥x−y∥ for every x, y ∈ E, then H^{d}(f(E)) ≤ c^{d}H^{d}(E) . PROOF. - See [13], Chapter 2, Proposition 2.2.

1.2.2 d-sets in ℝ^{N}

Definition. - Let Γ be a closed non-empty subset of ℝ^{N} and let d ∈]0, N]. A positive Borel outer measure μ with support Γ is called a d-measure on Γ if there exist c_{1}, c_{2} ∈]0, +∞[ such that for every x_{0} ∈ Γ and for every r ∈]0, 1]

c_{1}r^{d} ≤ μ(B_{ℝ^{N}}(x_{0}, r)) ≤ c_{2}r^{d}  (1.1)

holds.

Remark 3. - One can replace the condition r ∈]0, 1] in the above definition with the condition r ∈]0, r_{0}] .

To prove this, let r_{0} ∈]1, +∞[ and let k ∈ ℕ^{∗} be so that 2^{k−1} < r_{0} ≤ 2^{k}; we have 2^{−k} < r2^{−k} ≤ r_{0}2^{−k} ≤ 1 ∀r ∈]1, r_{0}] .

Now, let n_{k} ∈ ℕ^{∗} and y_{1}, y_{2}, ..., y_{n_{k}} ∈ B_{ℝ^{N}}(0, r_{0}) be so that B_{ℝ^{N}}(0, r_{0}) ⊆ ⋃_{i=1}^{n_{k}} B_{ℝ^{N}}(y_{i}, 2^{−k−1}) ; we have for every x ∈ Γ B_{ℝ^{N}}(x, r_{0}) ⊆ ⋃_{i=1}^{n_{k}} B_{ℝ^{N}}(x + y_{i}, 2^{−k−1}) and then there exist x_{1}, x_{2}, ..., x_{n_{k}} ∈ Γ such that Γ ⋂ B_{ℝ^{N}}(x, r) ⊆ Γ ⋂ B_{ℝ^{N}}(x, r_{0}) ⊆ ⋃_{i=1}^{n_{k}}(Γ ⋂ B_{ℝ^{N}}(x_{i}, 2^{−k})) ⊆ ⋃_{i=1}^{n_{k}}(Γ ⋂ B_{ℝ^{N}}(x_{i}, r2^{−k})) .

By applying (1.1) we obtain μ(B_{ℝ^{N}}(x, r)) ≤ ∑_{i=1}^{n_{k}} μ(B_{ℝ^{N}}(x_{i}, r2^{−k})) ≤ n_{k}c_{2}2^{−kd}r^{d} and μ(B_{ℝ^{N}}(x, r)) ≥ μ(B_{ℝ^{N}}(x, r2^{−k})) ≥ c_{1}2^{−kd}r^{d} .

Remark 4. - In the second inequality of (1.1) we can allow x_{0} to be out of Γ; indeed if μ is a d-measure on Γ , then μ(B_{ℝ^{N}}(x_{0}, r)) ≤ 2^{d}c_{2}r^{d} for every x_{0} ∈ ℝ^{N} and for every r ∈]0, r_{0}/2] .

Definition. - A closed non-empty subset Γ of ℝ^{N} is called a d-set if there exists a d-measure on Γ .

Definition. - Two Borel measures μ_{1}, μ_{2} on ℝ^{N} are called equivalent if there are two constants a, b ∈]0, +∞[ such that

aμ_{1}(A) ≤ μ_{2}(A) ≤ bμ_{1}(A)

for any Borel set A ⊆ ℝ^{N} .

Theorem 1.3. - Let Γ be a d-set. Then the restriction to Γ of the ddimensional Hausdorff measure H^{d} is equivalent to any d-measure μ on Γ. PROOF. - See [19], chapter 2, Proposition 1 and Theorem 1 or [23], chapter 1, Theorem 3.4

Corollary 1.4. - If Γ ⊆ ℝ^{N} is a d-set then dim(Γ ⋂ B_{ℝ^{N}}(x_{0}, r)) = d ∀x_{0} ∈ Γ ∀r ∈]0, +∞[ .

1.2. NOTATION AND PRELIMINARY RESULTS 9

1.2.3 Box-counting dimensions

Here we recall the definition and some characterizations of the box-counting dimension of a non-empty bounded subset of ℝ^{N} .

In this section Ω will be a non-empty bounded subset of ℝ^{N} . Definition. - For all δ > 0 define

N_{δ}(Ω) = min{n ∈ ℕ^{∗} | ∃C_{1}, C_{2}, ..., C_{n} ⊆ ℝ^{N} so that

Ω ⊆ ⋃_{i=1}^{n} C_{i} and |C_{i}| ≤ δ ∀i ∈ ℕ^{∗} i ≤ n} .

The lower and upper box-counting dimensions of Ω are defined as \underline{dim_{B}}Ω = lim inf_{δ→0} \frac{log N_{δ}(Ω)}{− log δ} ,

\overline{dim_{B}}Ω = lim sup_{δ→0} \frac{log N_{δ}(Ω)}{− log δ} .

If \underline{dim_{B}}Ω = \overline{dim_{B}}Ω , their common value is called the box-counting dimension of Ω and is denoted by dim_{B}Ω . Definition. - For all δ > 0 define

N(Ω, δ) = min{n ∈ ℕ^{∗} | ∃x_{1}, x_{2}, ..., x_{n} ∈ ℝ^{N} so that Ω ⊆ ⋃_{i=1}^{n} B_{ℝ^{N}}(x_{i}, δ)} ,

M(Ω, δ) = max{n ∈ ℕ^{∗} | ∃y_{1}, y_{2}, ..., y_{n} ∈ Ω so that
∥y_{i} − y_{j}∥ ≥ 2δ ∀i, j ∈ ℕ^{∗} i < j ≤ n} .

N(Ω, δ) is commonly referred as the entropy function of Ω . Proposition 1.5. - We have

\underline{dim_{B}}Ω = lim inf_{δ→0} \frac{log N(Ω, δ)}{− log δ} = lim inf_{δ→0} \frac{log M(Ω, δ)}{− log δ} ,

\overline{dim_{B}}Ω = lim sup_{δ→0} \frac{log N(Ω, δ)}{− log δ} = lim sup_{δ→0} \frac{log M(Ω, δ)}{− log δ} .

Moreover, if (δ_{n}) is a sequence of real numbers such that lim_{n} δ_{n} = 0 and there exist a constant c ∈]0, 1[ so that

0 ≤ cδ_{n} ≤ δ_{n+1} ≤ δ_{n} ∀n ∈ ℕ^{∗} ,  (1.2)

then

\underline{dim_{B}}Ω = lim inf_{n} \frac{log N_{δ_{n}}(Ω)}{− log δ_{n}} = lim inf_{n} \frac{log N(Ω, δ_{n})}{− log δ_{n}} = lim inf_{n} \frac{log M(Ω, δ_{n})}{− log δ_{n}}

(1.3)

and

\overline{dim_{B}}Ω = lim sup_{n} \frac{log N_{δ_{n}}(Ω)}{− log δ_{n}} = lim sup_{n} \frac{log N(Ω, δ_{n})}{− log δ_{n}} =
lim sup_{n} \frac{log M(Ω, δ_{n})}{− log δ_{n}} .  (1.4)

PROOF. - See [13], § 3.1.

Proposition 1.6. - We have

\underline{dim_{B}}Ω = N − lim sup_{r→0} \frac{log L^{N}(Ω_{r})}{log r}

\overline{dim_{B}}Ω = N − lim inf_{r→0} \frac{log L^{N}(Ω_{r})}{log r}.

PROOF. - See [13], § 3.2.

Remark 5. - In [13], § 3.4 it is proved that always dim Ω ≤ \underline{dim_{B}}Ω .

Proposition 1.7. - Let A ⊆ ℝ^{N} , B ⊆ ℝ^{M} be non-empty Borel sets. Then dim(A × B) ≥ dim A + dim B .

If B is bounded, then dim(A × B) ≤ dim A + \overline{dim_{B}}B .

If also A is bounded, then \overline{dim_{B}}(A × B) ≤ \overline{dim_{B}}A + \overline{dim_{B}}B . PROOF. - See [13], chapter 7, Propositions 7.2, 7.3, 7.5.

Definition. - Let A be a non-empty bounded subset of ℝ^{N} and let s ∈ [0, N] .

The s-dimensional upper and lower Minkowski contents of A are defined as

\overline{M^{(s)}}(A) = lim sup_{r→0}(2r)^{s−N}L^{N}(A_{r})

\underline{M^{(s)}}(A) = lim inf_{r→0}(2r)^{s−N}L^{N}(A_{r}) .

Remark 6. - Let A ⊆ ℝ^{N} be non-empty and bounded. Then

\underline{dim_{B}}A = inf\{s ∈ [0, N] | \underline{M^{(s)}}(A) = 0\} = sup\{s ∈ [0, N] | \underline{M^{(s)}}(A) = +∞\};

(1.5)

\overline{dim_{B}}A = inf\{s ∈ [0, N] | \overline{M^{(s)}}(A) = 0\} = sup\{s ∈ [0, N] | \overline{M^{(s)}}(A) = +∞\} . (1.6)

Let us prove (1.5). Let s > \underline{dim_{B}}A and c ∈]0, +∞[ . By Proposition 1.6 there exists a sequence (r_{n}) in ]0, 1[ such that lim_{n} r_{n} = 0 and

N − \underline{dim_{B}}A = lim_{n} \frac{log L^{N}(A_{r_{n}})}{log r_{n}} = lim_{n} \frac{log(cL^{N}(A_{r_{n}}))}{log r_{n}} .

Then there exists n_{0} ∈ ℕ^{∗} so that N − s < (log(cL^{N}(A_{r_{n}})))/ log r_{n} for every n > n_{0} ; from which r_{n}^{s−N}L^{N}(A_{r_{n}}) < 1/c for every n > n_{0} .

It follows \underline{M^{(s)}}(A) ≤ 2^{N−s}/c for every c ∈]0, +∞[ ; that is \underline{M^{(s)}}(A) = 0. Now, let s < \underline{dim_{B}}A and c ∈]0, +∞[ . Still by Proposition 1.6 we have

N − s > lim_{n} \frac{log L^{N}(A_{r_{n}})}{log r_{n}} = lim_{n} \frac{log(cL^{N}(A_{r_{n}}))}{log r_{n}}

for every sequence (r_{n}) so that lim_{n} r_{n} = 0 and there exists the limit above.

If we fix such a sequence (r_{n}) , then there exists n_{0} ∈ ℕ^{∗} such that r_{n}^{s−N}L^{N}(A_{r_{n}}) > 1/c for every n > n_{0} .

It follows lim inf_{n} r_{n}^{s−N}L^{N}(A_{r_{n}}) ≥ 1/c for every sequence (r_{n}) converging to 0 ; that is \underline{M^{(s)}}(A) ≥ 2^{N−s}/c for every c ∈]0, +∞[ and then \underline{M^{(s)}}(A) = +∞ .

The statement (1.6) is proved in the same way.

1.2.4 Entropy numbers

Let Ω be a bounded subset of X . The n-th entropy number of Ω is defined by

ε_{n}(Ω) = inf{ε > 0 | ∃ x_{1}, x_{2}, ..., x_{n} ∈ X such that Ω ⊆ ⋃_{i=1}^{n} B_{X}(x_{i}, ε)} .

The sequence (ε_{n}(Ω)) is monotonically decreasing and tends to zero if and only if Ω is precompact.

The numbers ε_{n}(Ω) are also called the outer entropy numbers of Ω ; inner entropy numbers are also useful and they are defined by

φ_{n}(Ω) = sup{ρ ≥ 0 | ∃ y_{1}, y_{2}, ..., y_{n+1} ∈ Ω such that

d_{X}(y_{i}, y_{j}) > 2ρ ∀i, j ∈ ℕ^{∗} with i < j ≤ n + 1} .

If ♯Ω ≤ n we put φ_{n}(Ω) = 0 .

Inner and outer entropy numbers have the same asymptotic behaviour; indeed the following relation holds:

φ_{n}(Ω) ≤ ε_{n}(Ω) ≤ 2φ_{n}(Ω) ∀n ∈ ℕ^{∗} .  (1.7)

See [6], for a complete treatment.

The notion of box-counting dimension is closely related to the one of entropy numbers. Here are some simple consequences of the Proposition 1.5. See [15], section 2.4 for the proofs.

Corollary 1.8. - Let d ∈]0, N] . If there exist a, b ∈]0, +∞[ such that an^{− \frac{1}{d}} ≤ ε_{n}(Ω) ≤ bn^{− \frac{1}{d}} for all n ∈ ℕ^{∗} , then dim_{B} Ω = d .

Corollary 1.9. - If dim_{B} Ω = d > 0 , then for all ε ∈]0, d[ there exists n_{ε} ∈ ℕ^{∗} so that n^{− \frac{1}{d−ε}} ≤ ε_{n}(Ω) ≤ 2(n + 1)^{− \frac{1}{d+ε}} for n > n_{ε} .

By the following proposition we determine the asymptotic behaviour of the entropy numbers of a bounded d-set.

Proposition 1.10. - Let d ∈]0, N] and Γ ⊆ ℝ^{N} be a bounded d-set. Then

(\frac{μ(Γ)}{c_{2}})^{\frac{1}{d}} n^{− \frac{1}{d}} ≤ ε_{n}(Γ) ≤ 2 (\frac{μ(Γ)}{c_{1}})^{\frac{1}{d}} n^{− \frac{1}{d}} ∀n ∈ ℕ^{∗} ,

where μ is a d-measure on Γ and c_{1} = c_{1}(|Γ|) , c_{2} = c_{2}(|Γ|) (see Remark 3). PROOF. - See [15], section 3, Proposition 3.4.

Remark 7. - Let d ∈]0, N] and Γ ⊆ ℝ^{N} be a bounded d-set. By Proposition 1.10 and Corollary 1.8 it follows that dim_{B} Γ = dim Γ = d. This fact is also proven, for instance, in [20], chapter 5, Theorem 5.7,

without using entropy numbers.

1.2.5 The Hausdorff metric

Definition. - If x_{0} ∈ X and A ⊆ X , we define the distance between x_{0} and A by

d_{X}(x_{0}, A) = inf_{x∈A} d_{X}(x_{0}, x) .

Remark 8. - For every x_{0} ∈ X and A ⊆ X ,we have d_{X}(x_{0}, A) = d_{X}(x_{0}, \overline{A}).

Definition. - Let 𝔅 be the class of all non-empty closed bounded subsets of X .

The Hausdorff metric D on 𝔅 is defined by

D(A, B) = max{sup_{x∈A} d_{X}(x, B), sup_{y∈B} d_{X}(y, A)} .

Remark 9. - D is a metric on 𝔅 . Moreover, for every A, B ∈ 𝔅

D(A, B) = inf{ε > 0 | A ⊆ ⋃_{y∈B} D_{X}(y, ε) and B ⊆ ⋃_{x∈A} D_{X}(x, ε)} .

Lemma 1.2. - Let f : X → X be a Lipschitz function and let ρ = inf{c > 0 | d_{X}(f(x), f(y)) ≤ cd_{X}(x, y) ∀x, y ∈ X} be its Lipschitz constant. Then

D(\overline{f(A)}, \overline{f(B)}) ≤ ρD(A, B) ∀A, B ∈ 𝔅 .  (1.8)

PROOF. - By Remark 8 D(\overline{f(A)}, \overline{f(B)}) = max\{sup_{u∈f(A)} d_{X}(u, f(B)) , sup_{v∈f(B)} d_{X}(v, f(A))} .

By the Lipschitz condition on f it follows that d_{X}(f(x), f(B)) ≤ ρd_{X}(x, B) and d_{X}(f(y), f(A)) ≤ ρd_{X}(y, A) , ∀x ∈ A ∀y ∈ B and then the (1.8) ■

Lemma 1.3. - Let {A_{j} | j ∈ J} , {B_{j} | j ∈ J} be two families of elements of 𝔅 . Then

D(\overline{⋃_{j∈J} A_{j}}, \overline{⋃_{j∈J} B_{j}}) ≤ sup_{j∈J} D(A_{j}, B_{j}) ,

provided that ⋃_{j∈J} A_{j} and ⋃_{j∈J} B_{j} are bounded. PROOF. - Let c > sup_{j∈J} D(A_{j}, B_{j}) : for all j ∈ J D(A_{j}, B_{j}) < c and then

A_{j} ⊆ ⋃_{y_{j}∈B_{j}} D_{X}(y_{j}, c) ⊆ ⋃_{i∈J} ⋃_{y_{i}∈B_{i}} D_{X}(y_{i}, c) = ⋃ ; y∈⋃_{i∈J} B_{i} D_{X}(y, c) .

It follows that

⋃_{j∈J} A_{j} ⊆ ⋃ ; y∈⋃_{i∈J} B_{i} D_{X}(y, c) .

In the same way we obtain

⋃_{j∈J} B_{j} ⊆ ⋃ ; x∈⋃_{j∈J} A_{j} D_{X}(x, c) .

Then

D(\overline{⋃_{j∈J} A_{j}}, \overline{⋃_{j∈J} B_{j}}) < c ■

1.2.6 Sequences of indices and product spaces

From now on, (m_{n}) is a fixed sequence of integer numbers, with m_{n} ≥ 2 for all n ∈ ℕ^{∗} . Moreover, for all n ∈ ℕ^{∗} we fix l_{1}^{(n)}, l_{2}^{(n)}, ..., l_{m_{n}}^{(n)} ∈]0, 1[ so that ∑_{j=1}^{m_{n}} l_{j}^{(n)} = 1 .

Definition. - For n ∈ ℕ^{∗} we set I_{n} = {1, 2, ..., m_{n}} and I = ∏_{n=1}^{+∞} I_{n}. Each I_{n} is equipped with the discrete topology. On I we consider the function d_{I} : I × I → ℝ ,

d_{I}(k, h) = ∑_{j=1}^{+∞} \frac{1}{2^{j}} \frac{|k_{j} − h_{j}|}{1 + |k_{j} − h_{j}|} ,

where k = (k_{j}) , h = (h_{j}) , k_{j}, h_{j} ∈ I_{j} for all j ∈ ℕ^{∗} .

Remark 10. - It is well known that d_{I} is a metric on I ; moreover d_{I} induces the product topology on I .

It follows that (I, d_{I}) is a compact metric space and then it is complete and separable (see, for example, [7], chapters 2, 5 and 6).

Definition. - Given i_{1}, i_{2}, ..., i_{n} ∈ ℕ^{∗} , we define the natural projection π_{i_{1},i_{2},...,i_{n}} : I → ∏_{j=1}^{n} I_{i_{j}} by

π_{i_{1},i_{2},...,i_{n}}(k) = (k_{i_{1}}, k_{i_{2}}, ..., k_{i_{n}}) .

Remark 11. - For every i_{j} ∈ ℕ^{∗} , π_{i_{j}} is a continuous function by the definition of product topology. Then π_{i_{1},i_{2},...,i_{n}} is continuous.

Definition. - Let X and Y be metric spaces, μ an outer measure on X and f : X → Y a function.

The image of μ under f is defined by

f_{♯}μ(A) = μ(f^{−1}(A)) ∀A ⊆ Y .

For the proof of the following two theorems see [20], chapter 1, Theorems 1.18 and 1.19.

Theorem 1.11. - Let X and Y be separable metric spaces. If f : X → Y is continuous and μ is a Radon measure on X with compact support, then f_{♯}μ is a Radon measure. Moreover, if C ⊆ X is the support of μ , then f(C) is the support of f_{♯}μ .

Definition. - Let X and Y be separable metric spaces . A mapping f : X → Y is a Borel mapping if f^{−1}(U) is a Borel set for every open set U ⊆ Y.

Let A ⊆ X be a Borel set. A function g : A → [−∞, +∞] is a Borel function if the set {x ∈ A | f(x) < c} is a Borel set for every c ∈ ℝ .

Theorem 1.12. - Let X and Y be separable metric spaces and suppose that f : X → Y is a Borel mapping, μ is a Borel measure on X and g is a non-negative Borel function on Y . Then

∫_{Y} gdf_{♯}μ = ∫_{X}(g ∘ f)dμ .

Definition. - For every n ∈ ℕ^{∗} we define a measure τ_{n} on I_{n} by

τ_{n}(A) = ∑_{j∈A} l_{j}^{(n)} ∀A ⊆ I_{n} .

Remark 12. - For every n ∈ ℕ^{∗} , τ_{n} is a Radon measure and τ_{n}(I_{n}) = 1 .

Remark 13. - From the definition of product measure of two measures it follows that

(τ_{1} × τ_{2} × ⋯ × τ_{n})(A) = ∑_{(k_{1},k_{2},...,k_{n})∈A} ∏_{j=1}^{n} l_{k_{j}}^{(j)} ∀n ∈ ℕ^{∗} ∀A ⊆ ∏_{j=1}^{n} I_{j} .

In order to define the product measure on I , we need the following theorem.

Theorem 1.13. - Let {X_{α} | α ∈ A} be a family of compact Hausdorff spaces and let, for each α ∈ A , μ_{α} be a Radon measure on X_{α} , with μ_{α}(X_{α}) = 1.

Then there exists a unique Radon measure μ on ∏_{α∈A} X_{α} such that μ(∏_{α∈A} X_{α}) = 1 and μ_{α_{1}} × μ_{α_{2}} × ⋯ × μ_{α_{n}} = π_{α_{1},α_{2},...,α_{n}♯}μ for any distinct α_{1}, α_{2}, ..., α_{n} ∈ A . PROOF. - See [14], chapter 9, Theorem 9.19.

Remark 14. - By the previous theorem there is a unique Radon measure τ on I , such that τ(I) = 1 and τ_{i_{1}} × τ_{i_{2}} × ⋯ × τ_{i_{n}} = π_{i_{1},i_{2},...,i_{n}♯}τ for any distinct i_{1}, i_{2}, ..., i_{n} ∈ ℕ^{∗} .

1.3 Limit sets

1.3.1 Basic notation

We will use the following notation:

• for any n ∈ ℕ^{∗} i ∈ I_{n} , f_{i}^{(n)} : X → X is a contraction;

– ρ_{i}^{(n)} = inf{c > 0 | d_{X}(f_{i}^{(n)}(x), f_{i}^{(n)}(y)) ≤ cd_{X}(x, y) ∀x, y ∈ X} (ρ_{i}^{(n)} is the Lipschitz constant of f_{i}^{(n)} ),

– ρ^{(n)} = max{ρ_{1}^{(n)}, ρ_{2}^{(n)}, ..., ρ_{m_{n}}^{(n)}} ,

– ρ = sup_{n∈ℕ^{∗}} ρ^{(n)} ,

– x_{i}^{(n)} ∈ X is the fixed point of f_{i}^{(n)} ;

• for every n ∈ ℕ^{∗}

– 𝔉_{n} = {f_{1}^{(n)}, f_{2}^{(n)}, ..., f_{m_{n}}^{(n)}} ,

– for every A ⊆ X 𝔉_{n}(A) = ⋃_{i=1}^{m_{n}} \overline{f_{i}^{(n)}(A)} ,

– for every A ⊆ X (𝔉_{n} ∘ 𝔉_{n+1})(A) = 𝔉_{n}(𝔉_{n+1}(A)) ;

• for every i ∈ I , n, k ∈ ℕ^{∗} with k ≤ n :

– f_{i_{1}i_{2}...i_{n}} = f_{i_{1}}^{(1)} ∘ f_{i_{2}}^{(2)} ∘ ⋯ ∘ f_{i_{n}}^{(n)} and x_{i_{1}i_{2}...i_{n}} is its fixed point,

– f_{i_{k}i_{k+1}...i_{n}}^{(k)} = f_{i_{k}}^{(k)} ∘ f_{i_{k+1}}^{(k+1)} ∘ ⋯ ∘ f_{i_{n}}^{(n)} and x_{i_{k}i_{k+1}...i_{n}}^{(k)} is its fixed point;

• F = {x_{i}^{(n)} | n ∈ ℕ^{∗} i ∈ I_{n}} is the set of all fixed points of the contractions f_{i}^{(n)} .

1.3.2 Existence and uniqueness

Lemma 1.4. - Let (g_{n}) be a sequence of contraction maps on X each of them with Lipschitz constant ρ_{n} . Let us suppose that the following two conditions hold:

1. there exists a non-empty closed and bounded set Q ⊆ X such that g_{n}(Q) ⊆ Q for every n ∈ ℕ^{∗} ;

2. lim_{n} ∏_{k=1}^{n} ρ_{k} = 0 .

Then there exists a unique x ∈ X so that

lim_{n}(g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x_{0}) = x for every x_{0} ∈ X .

Moreover x ∈ Q . PROOF. - It is easy to prove, by induction, that for any x, y ∈ X and for any n ∈ ℕ^{∗}

d_{X}((g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x), (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(y)) ≤ (∏_{k=1}^{n} ρ_{k})d_{X}(x, y) .

Now, let x_{0} ∈ Q and ε > 0 ; by the second hypothesis there exists n_{ε} ∈ ℕ^{∗} so that ∏_{k=1}^{n} ρ_{k} < ε/|Q| ∀n ∈ ℕ^{∗} with n > n_{ε} .

Then, for all m, n ∈ ℕ^{∗} with m > n > n_{ε} we have

d_{X}((g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x_{0}), (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{m})(x_{0})) ≤

(∏_{k=1}^{n} ρ_{k})d_{X}(x_{0}, (g_{n+1} ∘ g_{n+2} ∘ ⋯ ∘ g_{m})(x_{0})) < ε .

Since X is complete there exist x ∈ X such that lim_{n}(g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x_{0}) = x .

Now we prove that x does not depend on x_{0} .

Let y_{0} ∈ X and n ∈ ℕ^{∗} :

d_{X}(x, (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(y_{0})) ≤

d_{X}(x, (g_{1} ∘g_{2} ∘⋯∘g_{n})(x_{0}))+d_{X}((g_{1} ∘g_{2} ∘⋯∘g_{n})(x_{0}), (g_{1} ∘g_{2} ∘⋯∘g_{n})(y_{0})) ≤

d_{X}(x, (g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(x_{0})) + (∏_{k=1}^{n} ρ_{k})d_{X}(x_{0}, y_{0})

and by letting n → +∞ we obtain x = lim_{n}(g_{1} ∘ g_{2} ∘ ⋯ ∘ g_{n})(y_{0}) ■

Definition. - Let x_{0} ∈ X be fixed; we define ∀n ∈ ℕ^{∗} a function p_{n} : ∏_{j=1}^{n} I_{j} → X by

p_{n}(i_{1}, i_{2}, ..., i_{n}) = f_{i_{1}i_{2}...i_{n}}(x_{0}) .

Remark 15 . - Obviously p_{n} depends on x_{0} and it is a continuous function on ∏_{j=1}^{n} I_{j} .

From now on we will suppose that the following two hypotheses are valid:

1. there exists a non-empty closed bounded set Q ⊆ X such that 𝔉_{n}(Q) ⊆ Q for all n ∈ ℕ^{∗} ;

2. lim_{n} ∏_{k=1}^{n} ρ^{(k)} = 0 .

Remark 16 . - The hypotheses 1 and 2 above are implied by the following:

3. F is bounded;

4. ρ < 1 .

Indeed it is obvious that 4 ⇒ 2; moreover, let

Q = ⋂_{n=1}^{+∞} ⋂_{i=1}^{m_{n}} B_{X} (x_{i}^{(n)}, \frac{|F |}{1 − ρ}) ;

we prove that Q satisfies 1.

Q is closed and bounded; moreover F ⊆ Q .

Let n, k ∈ ℕ^{∗} , i ∈ I_{n} and j ∈ I_{k} : we have ∀x ∈ Q

d_{X}(f_{i}^{(n)}(x), x_{j}^{(k)}) ≤ d_{X}(f_{i}^{(n)}(x), x_{i}^{(n)}) + d_{X}(x_{i}^{(n)}, x_{j}^{(k)}) ≤

d_{X}(f_{i}^{(n)}(x), f_{i}^{(n)}(x_{i}^{(n)})) + |F | ≤ ρ_{i}^{(n)}d_{X}(x, x_{i}^{(n)}) + |F | ≤ ρ\frac{|F |}{1 − ρ} + |F | = \frac{|F |}{1 − ρ}

and then

f_{i}^{(n)}(x) ∈ B_{X} (x_{j}^{(k)}, \frac{|F |}{1 − ρ}) ∀k ∈ ℕ^{∗} ∀j ∈ I_{k} .

Definition. - Let x_{0} ∈ X be fixed; we define a function p : I → X by

p(k) = lim_{n} f_{k_{1}k_{2}...k_{n}}(x_{0}) .

Remark 17. - By Lemma 1.4 the function p is well defined and does not depend on x_{0} ∈ X .

If we take x_{0} ∈ Q , then we would see that p(I) ⊆ Q . We will always suppose x_{0} ∈ Q .

Definition. - We denote the set p(I) by K .

Proposition 1.14. - The function p is uniformly continuous. PROOF. - Let ε > 0 and n_{ε} ∈ ℕ^{∗} be so that ∏_{i=1}^{n_{ε}} ρ^{(i)} < ε/|Q| . Let δ = 2^{−n_{ε}−1} : for every k, h ∈ I , d_{I}(k, h) < δ implies k_{j} = h_{j} ∀j ∈ ℕ^{∗} with j ≤ n_{ε} and then, if we suppose x_{0} ∈ Q ,

d_{X}(p(k), p(h)) =

lim_{j} d_{X}(f_{k_{1}k_{2}...k_{nε}}(f_{k_{nε+1}k_{nε+2}...k_{j}}^{(n_{ε}+1)}(x_{0})), f_{k_{1}k_{2}...k_{nε}}(f_{h_{nε+1}h_{nε+2}...h_{j}}^{(n_{ε}+1)}(x_{0}))) ≤

(∏_{i=1}^{n_{ε}} ρ^{(i)}) lim_{j} d_{X}(f_{k_{nε+1}k_{nε+2}...k_{j}}^{(n_{ε}+1)}(x_{0}), f_{h_{nε+1}h_{nε+2}...h_{j}}^{(n_{ε}+1)}(x_{0})) < \frac{ε}{|Q|}|Q| = ε ■

Corollary 1.15. - K is compact and K ⊆ Q .

Proposition 1.16. - For every C ∈ 𝔅 lim_{n} D((𝔉_{1}∘𝔉_{2}∘⋯∘𝔉_{n})(C), K) = 0. PROOF. - Let x_{0} ∈ Q , ε > 0 and n_{ε} ∈ ℕ^{∗} be so that ∏_{j=1}^{n_{ε}} ρ^{(j)} < ε/|Q|. For every n ∈ ℕ^{∗} with n > n_{ε} we have D(p_{n}(π_{1,2,...,n}(I)), K) ≤ ε .

Indeed for every k ∈ I, we have p(k) ∈ K and

d_{X}(p_{n}(π_{1,2,...,n}(k)), p(k)) ≤ (∏_{j=1}^{n} ρ^{(j)})d_{X}(x_{0}, lim_{i} f_{k_{n+1}k_{n+2}...k_{i}}^{(n+1)}(x_{0})) <

(∏_{j=1}^{n_{ε}} ρ^{(j)})|Q| < ε

from which d_{X}(p_{n}(π_{1,2,...,n}(k)), K) < ε . On the other hand, ∀x ∈ K there exists k ∈ I so that p(k) = x and then d_{X}(p_{n}(π_{1,2,...,n}(I)), x) < ε .

But

p_{n}(π_{1,2,...,n}(I)) = ⋃_{k_{1}=1}^{m_{1}} ⋃_{k_{2}=1}^{m_{2}} ... ⋃_{k_{n}=1}^{m_{n}}{f_{k_{1}k_{2}...k_{n}}(x_{0})} = (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})({x_{0}}) .

Now, let C ∈ 𝔅 : by Lemmas 1.3 and 1.2 it follows that ∀n ∈ ℕ^{∗}

D((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C), (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})({x_{0}})) ≤ (∏_{j=1}^{n} ρ^{(j)})D(C, {x_{0}}) ;

then, if n > n_{ε} ,

D((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C), K) ≤

D((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(C), (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})({x_{0}}))+

D((𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(\{x_{0}\}), K) < ε (1 + \frac{D(C, \{x_{0}\})}{|Q|}) ■

1.3.3 Some properties of K

We follow the notation of the previous paragraphs

Definition. - Given a finite family of contraction maps G = {g_{1}, g_{2}, ..., g_{m}} and a subset A of X , we say that A is invariant with respect to G if G(A) = A .

Remark 18. - If 𝔉_{n} = 𝔉 = {f_{1}, f_{2}, ..., f_{m}} for every n ∈ ℕ^{∗} then K is invariant with respect to 𝔉 .

If there exists k ∈ ℕ^{∗} such that 𝔉_{n+k} = 𝔉_{n} for all n ∈ ℕ^{∗} , then K is invariant with respect to 𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{k} .

Remark 19. - If there exists n ∈ ℕ^{∗} such that 𝔉_{n+k} = 𝔉_{n} for all k ∈ ℕ^{∗}, then there is a unique non-empty compact set H ⊆ X which is invariant with respect to 𝔉_{n} . Moreover

K = ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ... ⋃_{i_{n}=1}^{m_{n}} f_{i_{1}i_{2}...i_{n}}(H) .

Lemma 1.5. - Let A = ⋃_{j=1}^{N} A_{j} ⊆ X . If A is connected then |A| ≤ ∑_{j=1}^{N} |A_{j}| . PROOF. - We consider the case N = 2 , the general statement follows by induction.

Let A = B ⋃ C be a connected subset of X and let x, y ∈ A . If x, y ∈ B or x, y ∈ C then d_{X}(x, y) ≤ |B| + |C| .

If x ∈ B and y ∈ C , then ∀z ∈ B , ∀w ∈ C ,

d_{X}(x, y) ≤ d_{X}(x, z) + d_{X}(z, w) + d_{X}(w, y) ≤ |B| + d_{X}(z, w) + |C| .

It follows that

|A| ≤ |B| + |C| + inf{d_{X}(z, w) | z ∈ B , w ∈ C} .

Let us suppose that inf{d_{X}(z, w) | z ∈ B , w ∈ C} = ε > 0 ; then B′ = ⋃_{x∈B} D_{X}(x, ε/4) and C′ = ⋃_{y∈C} D_{X}(y, ε/4) are disjoint open sets whose union contains A ⋃ B ■

Corollary 1.17. - If lim_{n} ∏_{k=1}^{n} ∑_{i=1}^{m_{k}} ρ_{i}^{(k)} = 0 then K is totally disconnected. PROOF. - Let x, y ∈ K with x ≠ y and let n_{xy} ∈ ℕ^{∗} be such that ∏_{k=1}^{n} ∑_{i=1}^{m_{k}} ρ_{i}^{(k)} < d_{X}(x, y)/|Q| ∀n ∈ ℕ^{∗} with n > n_{xy} . Then, if n > n_{xy} , we have

K = lim_{k}(𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{k})(Q) =

(𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(lim_{k}(𝔉_{n+1} ∘ 𝔉_{n+2} ∘ ⋯ ∘ 𝔉_{k})(Q))

because, by Lemmas 1.2 and 1.3, 𝔉_{1}, 𝔉_{2}, ..., 𝔉_{n} are contraction maps with respect to the Hausdorff metric and then they are continuous. It follows that

K ⊆ (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})(Q) .

Now, let A ⊆ K be connected and such that x, y ∈ A : we have

A ⊆ K ⊆ ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ... ⋃_{i_{n}=1}^{m_{n}} \overline{f_{i_{1}i_{2}...i_{n}}(Q)}

and by Lemma 1.5

|A| ≤ ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} |f_{i_{1}i_{2}...i_{n}}(Q)| ≤

∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} ρ_{i_{k}}^{(k)})|Q| = |Q| ∏_{k=1}^{n} ∑_{i=1}^{m_{k}} ρ_{i}^{(k)} < d_{X}(x, y) ≤ |A| ■

1.3.4 Examples

Cantor sets in ℝ .

Let X be the set of real numbers with the Euclidean distance; we construct a generalized version of the Cantor set. To do this we suppose to be given a real number d ∈]0, 1[ and we construct two sequences (t_{n}) and (d_{n}) in the following way:

• t_{0} = 1 and m_{n}t_{n}^{d} = t_{n−1}^{d} ; • d_{n} = \frac{t_{n−1}−m_{n}t_{n}}{m_{n}−1} .

Then, the set K of the previous paragraph is obtained by setting

f_{i}^{(n)}(x) = \frac{t_{n}x + (i − 1)(t_{n} + d_{n})}{t_{n−1}} = m_{n}^{− \frac{1}{d}} x+\frac{i − 1}{m_{n} − 1}(1−m_{n}^{− \frac{1}{d}}) ∀n ∈ ℕ^{∗} ∀i ∈ I_{n}.

Now we show some properties of the fractal set K so obtained. In the definition of the functions p_{n} we assume x_{0} = 0 .

We will show other properties of this set in section 1.5.

Remark 20. - The functions p_{n} are given by

p_{n}(i_{1}, i_{2}, ..., i_{n}) = ∑_{j=1}^{n}(i_{j} − 1)(t_{j} + d_{j}) .  (1.9)

We prove this by induction: let n ∈ ℕ^{∗} , we have f_{i_{n}}^{(n)}(0) = \frac{1}{t_{n−1}} (i_{n} − 1)(t_{n} + d_{n}) . Let us suppose that for k ∈ ℕ^{∗} 2 ≤ k ≤ n ,

f_{i_{k}i_{k+1}...i_{n}}^{(k)}(0) = \frac{1}{t_{k−1}} ∑_{j=k}^{n}(i_{j} − 1)(t_{j} + d_{j}) ,  (1.10)

then

f_{i_{k−1}i_{k}...i_{n}}^{(k−1)}(0) = f_{i_{k−1}}^{(k−1)}(f_{i_{k}i_{k+1}...i_{n}}^{(k)}(0)) = \frac{1}{t_{k−2}} ∑_{j=k−1}^{n}(i_{j} − 1)(t_{j} + d_{j}) .

Then (1.10) holds for any k ≤ n and in particular

p_{n}(i_{1}, i_{2}, ..., i_{n}) = f_{i_{1}i_{2}...i_{n}}(0) = ∑_{j=1}^{n}(i_{j} − 1)(t_{j} + d_{j}) .

It follows that

p(i) = ∑_{j=1}^{+∞}(i_{j} − 1)(t_{j} + d_{j}) ∀i ∈ I  (1.11)

and

K = {∑_{j=1}^{+∞}(i_{j} − 1)(t_{j} + d_{j}) | i_{j} ∈ I_{j} ∀j ∈ ℕ^{∗}} .  (1.12)

Example 1. - If m_{n} = 2 ∀n ∈ ℕ^{∗} and d = log_{3} 2 then K is the classical Cantor set.

Indeed, in this case t_{n} = d_{n} = 3^{−n} , f_{1}^{(n)}(x) = x/3 and f_{2}^{(n)}(x) = (x + 2)/3 ∀n ∈ ℕ^{∗} and by Remark 18 K is invariant with respect to 𝔉 = {f_{1}^{(1)}, f_{2}^{(1)}}.

It may also be noted that (1.12) becomes

K = \{∑_{j=1}^{+∞} \frac{c_{j}}{3^{j}} | c_{j} ∈ \{0, 2\} ∀j ∈ ℕ^{∗}\} .

Sierpiński gaskets.

Let X be the plane ℝ^{2} with the Euclidean distance; a generalized version of the Sierpiński gasket may be constructed in the following way.

Let (k_{n}) be a sequence of integer numbers, with k_{n} ≥ 2 for all n ∈ ℕ^{∗}; the sequence (m_{n}) is given by

m_{n} = ∑_{j=0}^{k_{n}} j = \frac{1}{2}k_{n}(k_{n} + 1) ∀n ∈ ℕ^{∗} .

For all n ∈ ℕ^{∗} , i ∈ I_{n} the contraction f_{i}^{(n)} is given by

f_{i}^{(n)}(x, y) = (\frac{x}{k_{n}} , \frac{y}{k_{n}} ) + (a_{i}^{(n)}, b_{i}^{(n)}) ,

where

a_{i}^{(n)} = \frac{k_{n} − (h_{i}^{(n)} + 1)^{2} + 2(i − 1)}{2k_{n}} , b_{i}^{(n)} = \frac{\sqrt{3}(k_{n} − h_{i}^{(n)} − 1)}{2k_{n}}

and 0 ≤ h_{i}^{(n)} < k_{n} is so that

∑_{j=0}^{h_{i}^{(n)}} j < i ≤ ∑_{j=0}^{h_{i}^{(n)}+1} j .

As before we set K = p(I) .

Remark 21. - If k_{n} = 2 for all n ∈ ℕ^{∗} , then K is the Sierpiński gasket. Indeed, in this case, we have m_{n} = 3 for all n ∈ ℕ^{∗} and

f_{1}^{(n)}(x, y) = (\frac{x}{2}, \frac{y}{2}) + (\frac{1}{4}, \frac{\sqrt{3}}{4}) ,

f_{2}^{(n)}(x, y) = (\frac{x}{2}, \frac{y}{2}) ,

f_{3}^{(n)}(x, y) = (\frac{x}{2}, \frac{y}{2}) + (\frac{1}{2}, 0) ,  (1.13)

Remark 22. - Even for the points of the Sierpiński gasket we can give a representation by means of series.

For the sake of simplicity we only consider the case in which k_{n} = 2 ∀n ∈ ℕ^{∗}. Let us consider the functions f, g : {1, 2, 3} → ℤ ,

f(i) = \{ \begin{matrix}1 & i = 1 \\ 0 & i = 2 \\ 2 & i = 3 \end{matrix} , g(i) = \{ \begin{matrix}1 & i = 1 \\ 0 & i > 1 \end{matrix} .  (1.14)

Let x_{0} = (0, 0) , then for every n ∈ ℕ^{∗} and i_{1}, i_{2}, ..., i_{n} ∈ {1, 2, 3} we have

p_{n}(i_{1}, i_{2}, ..., i_{n}) = \frac{1}{2} ∑_{j=1}^{n} \frac{1}{2^{j}} (f(i_{j}), \sqrt{3}g(i_{j})) .  (1.15)

As in Remark 20, even (1.15) is proven by induction. Let n ∈ ℕ^{∗} , by (1.13) and (1.14) we have

f_{i_{n}}^{(n)}(0, 0) = \frac{1}{4}(f(i_{n}), \sqrt{3}g(i_{n})) .

Let us suppose that for k ∈ ℕ^{∗} k ≤ n ,

f_{i_{k+1}i_{k+2}...i_{n}}^{(k+1)}(0, 0) = \frac{1}{2} ∑_{j=k+1}^{n} \frac{1}{2^{j−k}} (f(i_{j}), \sqrt{3}g(i_{j})) ,

then

f_{i_{k}i_{k+1}...i_{n}}^{(k)}(0, 0) = \frac{1}{4} ∑_{j=k+1}^{n} \frac{1}{2^{j−k}} (f(i_{j}), \sqrt{3}g(i_{j})) + \frac{1}{4}(f(i_{k}), \sqrt{3}g(i_{k})) =

\frac{1}{2} ∑_{j=k}^{n} \frac{1}{2^{j−k+1}} (f(i_{j}), \sqrt{3}g(i_{j})) .

In particular, for k = 1 we have (1.15).

It follows that

p(i) = \frac{1}{2} ∑_{j=1}^{+∞} \frac{1}{2^{j}} (f(i_{j}), \sqrt{3}g(i_{j})) ∀i ∈ I

1.4. A MEASURE ON K 25

and

K = \{\frac{1}{2} ∑_{j=1}^{+∞} \frac{1}{2^{j}} (f(i_{j}), \sqrt{3}g(i_{j})) | i_{j} ∈ \{1, 2, 3\} ∀j ∈ ℕ^{∗}\} .

1.4 A measure on K

In this section X is a complete separable metric space.

By C_{c}(X) we mean the spaces of all real valued continuous functions on X , with compact support.

We recall (§ 1.2.6) that for every n ∈ ℕ^{∗} , l_{1}^{(n)}, l_{2}^{(n)}, ..., l_{m_{n}}^{(n)} are real numbers in ]0, 1[ so that ∑_{j=1}^{m_{n}} l_{j}^{(n)} = 1 , τ_{n} is a measure on I_{n} defined by τ_{n}(A) = ∑_{j∈A} l_{j}^{(n)} ∀A ⊆ I_{n} and τ is the unique Radon measure on I such that τ(I) = 1 and τ_{i_{1}} × τ_{i_{2}} × ⋯ × τ_{i_{n}} = π_{i_{1},i_{2},...,i_{n}♯}τ for any distinct i_{1}, i_{2}, ..., i_{n} ∈ ℕ^{∗}.

Definition. - We set μ_{K} = p_{♯}τ .

Remark 23. - K is the support of μ_{K} and μ_{K}(K) = 1 .

Definition. - Let (ν_{n}) be a sequence of Radon measures on X . We say that the sequence (ν_{n}) converges weakly to a Radon measure ν if

lim_{n} ∫_{X} fdν_{n} = ∫_{X} fdν ∀f ∈ C_{c}(X) .

We denote this fact by writing ν_{n} ⇀ ν .

Remark 24. - We recall that if ν is a Borel regular measure on X and for every x ∈ X there is r > 0 such that ν(B_{X}(x, r)) < +∞ , then ν is a Radon measure on X . For the proof see [18], chapter 5, Theorem V.5.3.

Proposition 1.18. - Let ν be a Borel regular measure on X , with bounded support and such that ν(X) = 1 . For every n ∈ ℕ^{∗} we set

ν_{n} = ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)})f_{i_{1}i_{2}...i_{n}♯}ν .

Then ν_{n} ⇀ μ_{K} . PROOF. - Let ν be a Borel regular measure on X with bounded support and such that ν(X) = 1 ; let C be the support of ν and let f ∈ C_{c}(X) . We prove that

lim_{n} ∫_{X} fdν_{n} = ∫_{X} fdμ_{K} .

Let ε > 0 and δ > 0 be such that |f(x) − f(y)| < ε ∀x, y ∈ X with d_{X}(x, y) < δ ; we consider n′_{ε} ∈ ℕ^{∗} such that

(∏_{k=1}^{n′_{ε}} ρ^{(k)}) sup_{x∈C} d_{X}(x, x_{0}) < δ .  (1.16)

Moreover, let n′′_{ε} ∈ ℕ^{∗} be such that |Q| ∏_{j=1}^{n′′_{ε}} ρ^{(j)} < δ ; for every n ∈ ℕ^{∗} with n > n′′_{ε} we have

d_{X}(p_{n}(π_{1,2,...,n}(i)), p(i)) = lim_{k} d_{X}(f_{i_{1}i_{2}...i_{n}}(x_{0}), f_{i_{1}i_{2}...i_{n}}(f_{i_{n+1}i_{n+2}...i_{k}}^{(n+1)}(x_{0}))) ≤

|Q| ∏_{j=1}^{n} ρ^{(j)} < δ ∀i ∈ I ∀n ∈ ℕ^{∗} n > n′′_{ε}

and then

|f(p_{n}(π_{1,2,...,n}(i))) − f(p(i))| < ε ∀i ∈ I ∀n ∈ ℕ^{∗} n > n′′_{ε} .  (1.17)

Then, if n > max{n′_{ε}, n′′_{ε}} , we have

|∫_{X} fdν_{n} − ∫_{X} fdμ_{K} | =

|∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)}) ∫_{X} fdf_{i_{1}i_{2}...i_{n}♯}ν − ∫_{X} fdp_{♯}τ| ≤

|∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)}) (∫_{X} f(f_{i_{1}i_{2}...i_{n}}(x))dν(x) − f(f_{i_{1}i_{2}...i_{n}}(x_{0})))| +

|∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)})f(f_{i_{1}i_{2}...i_{n}}(x_{0})) − ∫_{I} f(p(i))dτ(i)| ≤

∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)}) |∫_{X} f(f_{i_{1}i_{2}...i_{n}}(x))dν(x) − ∫_{X} f(f_{i_{1}i_{2}...i_{n}}(x_{0}))dν(x)| +

|∫_{∏_{j=1}^{n} I_{j}} f(p_{n}(i_{1}, i_{2}, ..., i_{n}))d(τ_{1} × τ_{2} × ⋯ × τ_{n}) − ∫_{I} f(p(i))dτ(i)| ≤

∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)}) ∫_{X} |f(f_{i_{1}i_{2}...i_{n}}(x)) − f(f_{i_{1}i_{2}...i_{n}}(x_{0}))|dν(x)+

|∫_{I} f(p_{n}(π_{1,2,...,n}(i)))dτ(i) − ∫_{I} f(p(i))dτ(i)| ≤

∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)}) ∫_{C} |f(f_{i_{1}i_{2}...i_{n}}(x)) − f(f_{i_{1}i_{2}...i_{n}}(x_{0}))|dν(x)+

∫_{I} |f(p_{n}(π_{1,2,...,n}(i))) − f(p(i))|dτ(i) .

By (1.16)

d_{X}(f_{i_{1}i_{2}...i_{n}}(x), f_{i_{1}i_{2}...i_{n}}(x_{0})) < δ ∀x ∈ C

and then

|f(f_{i_{1}i_{2}...i_{n}}(x)) − f(f_{i_{1}i_{2}...i_{n}}(x_{0}))| < ε ∀x ∈ C .  (1.18)

Then, by (1.18) and (1.17) it follows

|∫_{X} fdν_{n} − ∫_{X} fdμ_{K} | < ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)})εν(C) + ετ(I) = 2ε

because

∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)}) = 1 ■

The measure μ_{K} generalizes the invariant measure defined in [17], section 4.

In the following example we construct a measure supported on the classical Cantor set that is a sum of weighted Dirac deltas; that is, there exists a sequence (q_{n}) of elements of K and a sequence (a_{n}) in [0, 1] such that μ_{K}(A) = ∑_{n=1}^{+∞} a_{n}δ_{q_{n}}(A) where δ_{q_{n}}(A) = 1 if q_{n} ∈ A and δ_{q_{n}}(A) = 0 otherwise.

Example 2. - Let for every n ∈ ℕ^{∗} f_{1}^{(n)}, f_{2}^{(n)} : ℝ → ℝ f_{1}^{(n)}(x) = x/3 , f_{2}^{(n)}(x) = (x + 2)/3 so that the fractal K = p(I) is the classical Cantor set. We construct a measure supported on K by choosing for every n ∈ ℕ^{∗} l_{1}^{(n)} = 2^{−2^{−n}} and (obviously) l_{2}^{(n)} = 1 − l_{1}^{(n)} .

Then for all i ∈ I τ({i}) > 0 if and only if there exists n_{0} ∈ ℕ^{∗} so that i_{j} = 1 ∀j ∈ ℕ^{∗} j > n_{0} .

Indeed, if the above condition holds then for every n > n_{0}

(τ_{1} × τ_{2} × ⋯ × τ_{n})({π_{1,2,...,n}(i)}) = ∏_{k=1}^{n} l_{i_{k}}^{(k)} =

(∏_{k=1}^{n_{0}} l_{i_{k}}^{(k)}) ∏_{k=n_{0}+1}^{n} 2^{−2^{−k}} = (∏_{k=1}^{n_{0}} l_{i_{k}}^{(k)})2^{− ∑_{k=n_{0}+1}^{n} 2^{−k}}

and so τ({i}) = 2^{−2^{−n_{0}}} ∏_{k=1}^{n_{0}} l_{i_{k}}^{(k)} > 0 .

If there exists a sequence (j_{k}) of natural numbers such that i_{j_{k}} = 2 ∀k ∈ ℕ^{∗} , then for any n ∈ ℕ^{∗} n > j_{1} we set k_{n} = max{k ∈ ℕ^{∗} | j_{k} ≤ n} and we have

(τ_{1} × τ_{2} × ⋯ × τ_{n})(\{π_{1,2,...,n}(i)\}) ≤ ∏_{k=1}^{k_{n}} l_{i_{j_{k}}}^{(k)} ≤ 2^{− \frac{kn}{2}}

from which τ({i}) = 0 .

Now, let J_{n} = {i ∈ I | i_{j} = 1 ∀h ∈ ℕ^{∗} j > n} , J = ⋃_{n=0}^{+∞} J_{n} and H = p(J) ; we prove that τ(J) = 1 . It will follow that μ_{K}(H) = 1 and \overline{H} = K .

We prove by induction that τ(J_{n}) = 2^{−2^{−n}} .

If n = 0 than

τ(J_{0}) = lim_{m} ∏_{k=1}^{m} l_{1}^{(k)} = 2^{− ∑_{k=1}^{+∞} 2^{−k}} = 2^{−2^{0}} .

Let us suppose that for n ∈ ℕ^{∗} , τ(J_{n}) = 2^{−2^{−n}} ; then ∀i ∈ J_{n+1} \ J_{n} ∃!h ∈ J_{n} such that i_{n+1} = 2 , h_{n+1} = 1 , i_{j} = h_{j} ∀j ∈ ℕ^{∗} j ≠ n + 1 . So τ({i}) = τ({h})(l_{2}^{(n+1)}/l_{1}^{(n+1)}) = τ({h})(2^{2^{−n−1}} − 1) .

Then

τ(J_{n+1}) = τ(J_{n}) + τ(J_{n+1} \ J_{n}) = τ(J_{n})2^{2^{−n−1}} = 2^{−2^{−n}}2^{2^{−n−1}} = 2^{−2^{−(n+1)}} .

Finally μ_{K}(H) = τ(J) = lim_{n} τ(J_{n}) = 1 .

Remark 25. - Assume X is locally compact. Let ν be a Borel regular measure on X with bounded support and such that ν(X) = 1 . Let for every n, k ∈ ℕ^{∗} k ≤ n , ν_{n} be defined as in Proposition 1.18 and

ν_{k}^{(n)} = ∑_{i_{k}=1}^{m_{k}} ∑_{i_{k+1}=1}^{m_{k+1}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{h=k}^{n} l_{i_{h}}^{(h)})f_{i_{k}i_{k+1}...i_{n}♯}^{(k)}ν .

Then, for all f ∈ C_{c}(X) and n, k ∈ ℕ^{∗} 1 < k ≤ n we have

∫_{X} fdν_{k−1}^{(n)} = ∑_{i_{k−1}=1}^{m_{k−1}} ∑_{i_{k}=1}^{m_{k}} ... ∑_{i_{n}=1}^{m_{n}}( ∏_{h=k−1}^{n} l_{i_{h}}^{(h)}) ∫_{X} f ∘ f_{i_{k−1}i_{k}...i_{n}}^{(k−1)}dν =

∑_{i_{k−1}=1}^{m_{k−1}} l_{i_{k−1}}^{(k−1)} ∑_{i_{k}=1}^{m_{k}} ∑_{i_{k+1}=1}^{m_{k+1}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{h=k}^{n} l_{i_{h}}^{(h)}) ∫_{X} f ∘ f_{i_{k−1}}^{(k−1)} ∘ f_{i_{k}i_{k+1}...i_{n}}^{(k)}dν =

∑_{i_{k−1}=1}^{m_{k−1}} l_{i_{k−1}}^{(k−1)} ∫_{X} f ∘ f_{i_{k−1}}^{(k−1)}dν_{k}^{(n)} .

In particular, if k = 2 we have

∫_{X} fdν_{n} = ∑_{i=1}^{m_{1}} l_{i}^{(1)} ∫_{X} f ∘ f_{i}^{(1)}dν_{2}^{(n)} ∀n ∈ ℕ^{∗} .  (1.19)

Now we suppose that for every n ∈ ℕ^{∗} m_{n} = M, 𝔉_{n} = 𝔉 = {f_{1}, f_{2}, ..., f_{M}} (and so K is invariant with respect to 𝔉 ), l_{i}^{(n)} = l_{i} ∀i ∈ ℕ^{∗} i ≤ M . In this case ν_{k}^{(n)} = ν_{n−k+1} ∀n, k ∈ ℕ^{∗} k ≤ n and then (1.19) becomes

∫_{X} fdν_{n} = ∑_{i=1}^{M} l_{i} ∫_{X} f ∘ f_{i}dν_{n−1} ∀f ∈ C_{c}(X) .

If n → +∞ , by Proposition 1.18, we have

∫_{X} fdμ_{K} = ∑_{i=1}^{M} l_{i} ∫_{X} fdf_{i♯}μ_{K} ∀f ∈ C_{c}(X) .

Let μ = ∑_{i=1}^{M} l_{i}f_{i♯}μ_{K} . We prove that μ_{K} = μ .

Let us suppose that there exists D ⊆ X such that μ_{K}(D) < μ(D). We take 3ε ∈]0, μ(D) − μ_{K}(D)[ ; since μ, μ_{K} are Radon measures, there exist A_{1}, A_{2} ⊆ X open such that D ⊆ A_{1} ⋂ A_{2} and μ(D) ≤ μ(A_{1}) < μ(D) + ε , μ_{K}(D) ≤ μ_{K}(A_{2}) < μ_{K}(D) + ε . Let A = A_{1} ⋂ A_{2} : then 0 ≤ μ(A)−μ(D) < ε , 0 ≤ μ_{K}(A)−μ_{K}(D) < ε . It follows μ(A)−μ_{K}(A) ≥ μ(D) − μ_{K}(D) + μ_{K}(D) − μ_{K}(A) > 2ε .

Moreover there exist C_{1}, C_{2} ⊆ X compact such that μ(A)−ε < μ(C_{1}) ≤ μ(A) and μ_{K}(A)−ε < μ_{K}(C_{2}) ≤ μ_{K}(A) . Let C = C_{1} ⋃ C_{2} : C is compact and 0 ≤ μ(A) − μ(C) < ε , 0 ≤ μ_{K}(A) − μ_{K}(C) < ε .

It follows μ(C) − μ_{K}(A) = μ(A) − μ_{K}(A) + μ(C) − μ(A) > ε .

Since X is locally compact and the support of μ and μ_{K} are compact we can suppose that \overline{A} is compact.

By Proposition 4.4 there exists a continuous function f : X → [0, 1] such that f(x) = 0 if and only if x ∈ X \ A and f(x) = 1 if and only if x ∈ C. Then

∫_{X} fdμ − ∫_{X} fdμ_{K} ≥ ∫_{C} fdμ − ∫_{A} fdμ_{K} ≥ μ(C) − μ_{K}(A) > ε > 0 .

1.5 Similitudes on ℝ^{N}

In this section we suppose that X = ℝ^{N} and the contractions f_{i}^{(n)} are similitudes; that is ∥f_{i}^{(n)}(x) − f_{i}^{(n)}(y)∥ = ρ_{i}^{(n)}∥x − y∥ ∀x, y ∈ ℝ^{N} ∀n, i ∈ ℕ^{∗} i ≤ m_{n} .

Let, for n ∈ ℕ^{∗} , g_{n} : ℝ → ℝ , g_{n}(s) = ∑_{i=1}^{m_{n}}(ρ_{i}^{(n)})^{s} . Then g_{n} is continuous and strictly decreasing, g_{n}(0) = m_{n} and lim_{s→+∞} g_{n}(s) = 0 . It follows that there exists a unique s_{n} ∈]0, +∞[ such that g_{n}(s_{n}) = 1 .

Definition. - If s_{n} ∈]0, +∞[ is so that ∑_{i=1}^{m_{n}}(ρ_{i}^{(n)})^{s_{n}} = 1 , then s_{n} is called the similarity dimension of 𝔉_{n} .

Definition. - We say that 𝔉_{n} satisfies the open set condition if there exists a non-empty bounded open set O_{n} ⊆ ℝ^{N} such that

• 𝔉_{n}(O_{n}) ⊆ O_{n} ,

• f_{i}^{(n)}(O_{n}) ⋂ f_{j}^{(n)}(O_{n}) = ∅ ∀i, j ∈ ℕ^{∗} i < j ≤ m_{n} .

Let us consider the case 𝔉_{n} is constant. For the next theorem we suppose 𝔉_{n} = 𝔉 = {f_{1}, f_{2}, ..., f_{M}} ∀n ∈ ℕ^{∗} ; then ρ_{i}^{(n)} = ρ_{i} ∀i ∈ ℕ^{∗} i ≤ M and s_{n} = d . Moreover we suppose that l_{i}^{(n)} = l_{i} ∀n, i ∈ ℕ^{∗} i ≤ M .

Theorem 1.19. - Suppose that 𝔉 satisfies the open set condition with a non-empty bounded open set O . If d is the similarity dimension of 𝔉 and l_{i} = ρ_{i}^{d} ∀i ∈ ℕ^{∗} i ≤ M then μ_{K} is a d-measure on K . More precisely

(\frac{σ}{|K|})^{d} r^{d} ≤ μ_{K}(B_{ℝ^{N}}(x_{0}, r)) ≤ (\frac{1 + 2b}{σa})^{N} r^{d} ∀x_{0} ∈ K ∀r ∈]0, |K|] ;

where σ = min{ρ_{1}, ρ_{2}, ..., ρ_{M}} and a, b > 0 are so that there exist x, y ∈ ℝ^{N} such that B_{ℝ^{N}}(x, a) ⊆ O ⊆ B_{ℝ^{N}}(y, b) . In particular dim K = d . Moreover

μ_{K}(B) = \frac{H^{d}(B ⋂ K)}{H^{d}(K)} ∀B ⊆ ℝ^{N} .

1.5. SIMILITUDES ON ℝ^{N} 31

PROOF. - See [17], section 5, Theorem 5.3.

1.5.1 An example concerning d-sets

If 𝔉_{n} is not constant, it may happen that K is not a d-set.

As an example we can take the Cantor set constructed in the subsection 1.3.4 (Cantor sets in ℝ). Here K is the Cantor set constructed in subsection 1.3.4. We prove that K is a d-set if and only if the sequence (m_{n}) is bounded. However, for every n ∈ ℕ^{∗} , all the contractions of the family 𝔉_{n} are similitudes, 𝔉_{n} has similarity dimension equal to d and satisfies the open set condition with the set O =]0, 1[ . Moreover H^{d}(K) = 1 .

Remark 26. - It is easy to prove, by induction, that, for every n ∈ ℕ^{∗} ,

t_{n} = (∏_{j=1}^{n} m_{j})^{− \frac{1}{d}} and d_{n} = \frac{1 − m_{n}^{1− \frac{1}{d}}}{m_{n} − 1} (∏_{j=1}^{n−1} m_{j})^{− \frac{1}{d}} .

Remark 27. - From Remark 20 we have K ⊆ [0, 1] and 0, 1 ∈ K .

Indeed, if we set k, h ∈ I , k_{j} = 1 , h_{j} = m_{j} for every j ∈ ℕ^{∗} , then by (1.11) , p(k) = 0 , p(h) = ∑_{j=1}^{+∞}(t_{j−1} − t_{j}) = t_{0} = 1 and for all i ∈ I 0 ≤ p(i) ≤ p(h) = 1 .

It may also be noted that 0 ≤ f_{j}^{(n)}(x) ≤ 1 ∀x ∈ [0, 1] and for every n, j ∈ ℕ^{∗} , j ≤ m_{n} ; then we can set Q = [0, 1] .

Remark 28. - Given n ∈ ℕ^{∗} , the ∏_{j=1}^{n} m_{j} intervals of the form [p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n})+t_{n}] are pairwise disjoint and |p_{n}(i_{1}, i_{2}, ..., i_{n})− (p_{n}(k_{1}, k_{2}, ..., k_{n}) + t_{n})| ≥ d_{n} for any different (i_{1}, i_{2}, ..., i_{n}), (k_{1}, k_{2}, ..., k_{n}) ∈ ∏_{j=1}^{n} I_{j} .

Moreover for all (i_{1}, i_{2}, ..., i_{n+1}) ∈ ∏_{j=1}^{n+1} I_{j} the interval [p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}] contains [p_{n}(i_{1}, i_{2}, ..., i_{n+1}), p_{n}(i_{1}, i_{2}, ..., i_{n+1}) + t_{n+1}] and

K = ⋂_{n=1}^{+∞} ( ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ... ⋃_{i_{n}=1}^{m_{n}}[p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}]) .

Theorem 1.20. - We have H^{d}(K) = 1 and so dim K = d . PROOF. - See [12] chapter 1, Theorem 1.15.

Lemma 1.6. - For every n ∈ ℕ^{∗} and (i_{1}, i_{2}, ..., i_{n}) ∈ ∏_{j=1}^{n} I_{j} , we have

H^{d}(K ⋂[p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}]) = t_{n}^{d} .

PROOF. - Let n ∈ ℕ^{∗} and (i_{1}, i_{2}, ..., i_{n}) ∈ ∏_{j=1}^{n} I_{j} ; we define

f : K ⋂[p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}] → K ⋂[0, t_{n}],
f(x) = x − p_{n}(i_{1}, i_{2}, ..., i_{n}) .

It is clear that 0 ≤ f(x) ≤ t_{n} ; moreover f(x) ∈ K by (1.11) and (1.9). Then f is well defined.

The function f is one to one because it is injective and ∀y ∈ K ⋂[0, t_{n}] , y = f(y + p_{n}(i_{1}, i_{2}, ..., i_{n})) . Moreover f is an isometry and then, by Lemma 1.1, H^{d}(K ⋂[0, t_{n}]) ≤ H^{d}(K ⋂[p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}]) . By applying the same arguments to f^{−1} we obtain the opposite inequality.

Finally

1 = H^{d}(K) = H^{d} ( ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ... ⋃_{i_{n}=1}^{m_{n}}(K ⋂[p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}]))

= ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} H^{d}(K ⋂[p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}]) =

(∏_{j=1}^{n} m_{j}) H^{d}(K ⋂[0, t_{n}])

and then

H^{d}(K ⋂[p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}]) = H^{d}(K ⋂[0, t_{n}]) =

(∏_{j=1}^{n} m_{j})^{−1} = t_{n}^{d} ■

Proposition 1.21. - The set K is a d-set if and only if the sequence (m_{n}) is bounded. PROOF. - Let x_{0} ∈ K and r ∈]0, 1] ; by Remark 28, for every k ∈ ℕ^{∗} there exists (i_{1}, i_{2}, ..., i_{k}) ∈ ∏_{j=1}^{k} I_{j} such that x_{0} ∈ [p_{k}(i_{1}, i_{2}, ..., i_{k}), p_{k}(i_{1}, i_{2}, ..., i_{k})+ t_{k}] . Let

n = min{k ∈ ℕ^{∗} | ∃(i_{1}, i_{2}, ..., i_{k}) ∈ ∏_{j=1}^{k} I_{j} so that

x_{0} ∈ [p_{k}(i_{1}, i_{2}, ..., i_{k}), p_{k}(i_{1}, i_{2}, ..., i_{k}) + t_{k}] ⊆ [x_{0} − r, x_{0} + r] (n is well defined because lim_{k} t_{k} = 0 by Remark 26). We prove that

\frac{1}{m_{n}} r^{d} ≤ H^{d}(k ⋂[x_{0} − r, x_{0} + r]) ≤ 2^{1+d}m_{n}r^{d} ;  (1.20)

it will follow that K is a d-set if the sequence (m_{n}) is bounded.

By Lemma 1.6 H^{d}(K ⋂[x_{0} − r, x_{0} + r]) ≥ t_{n}^{d} = t_{n−1}^{d}/m_{n} ; if t_{n−1} ≤ r , we would have x_{0} ∈ [p_{n−1}(i_{1}, i_{2}, ..., i_{n−1}), p_{n−1}(i_{1}, i_{2}, ..., i_{n−1}) + t_{n−1}] ⊆ [x_{0} − r, x_{0} + r] and this is absurd; then the first inequality follows.

Let us prove the second inequality: we have [p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n})+ t_{n}] ⊆ [x_{0} − r, x_{0} + r] ⋂[p_{n−1}(i_{1}, i_{2}, ..., i_{n−1}), p_{n−1}(i_{1}, i_{2}, ..., i_{n−1}) + t_{n−1}] and for every j ∈ ℕ^{∗} the following implications hold:

• if 1 ≤ j < i_{n−1} − 1 then [x_{0}−r, x_{0}+r] ⋂[p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, j), p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, j)+t_{n−1}] = ∅ because otherwise we would have [p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, i_{n−1}−1), p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, i_{n−1}−1)+t_{n−1}] ⊆ [x_{0}− r, x_{0} + r] ;

• if i_{n−1} + 1 < j ≤ m_{n−1} then [x_{0}−r, x_{0}+r] ⋂[p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, j), p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, j)+t_{n−1}] = ∅ because otherwise we would have [p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, i_{n−1}+1), p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, i_{n−1}+1)+t_{n−1}] ⊆ [x_{0}− r, x_{0} + r] .

Moreover at least one between the intervals [p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, i_{n−1} − 1), p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, i_{n−1} − 1) + t_{n−1}] and [p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, i_{n−1} + 1), p_{n−1}(i_{1}, i_{2}, ..., i_{n−2}, i_{n−1} + 1) + t_{n−1}] does not intersect [x_{0} − r, x_{0} + r] because otherwise we would have [p_{n−1}(i_{1}, i_{2}, ..., i_{n−1}), p_{n−1}(i_{1}, i_{2}, ..., i_{n−1}) + t_{n−1}] ⊆ [x_{0} − r, x_{0} + r] .

Then H^{d}(K ⋂[x_{0} − r, x_{0} + r]) < 2t_{n−1}^{d} = 2m_{n}t_{n}^{d} ≤ 2^{1+d}m_{n}r^{d} .

Now we suppose that the sequence (m_{n}) is not bounded; by taking x_{0} = 0 and r_{n} = t_{n} + d_{n} for every n ∈ ℕ^{∗} , we have

\frac{H^{d}(K ⋂[x_{0} − r_{n}, x_{0} + r_{n}])}{r_{n}^{d}} = \frac{H^{d}(K ⋂[0, t_{n}])}{r_{n}^{d}} = \frac{t_{n}^{d}}{(t_{n} + d_{n})^{d}} = (\frac{m_{n} − 1}{mn^{\frac{1}{d}} − 1})^{d} .

Let (m_{n_{k}}) be a subsequence of (m_{n}) such that lim_{k} m_{n_{k}} = +∞ ; then

lim_{k} \frac{H^{d}(K ⋂[x_{0} − r_{n_{k}}, x_{0} + r_{n_{k}}])}{r_{n_{k}}^{d}} = 0

because d < 1 ■ Remark 29. - If in the above proposition we suppose t_{k} ≤ d_{k} for all k ∈ ℕ^{∗} , then (1.20) becomes

\frac{1}{m_{n}} r^{d} ≤ H^{d}(k ⋂[x_{0} − r, x_{0} + r]) ≤ 2^{d}m_{n}r^{d} .  (1.21)

We recall that by Remark 26 t_{k} ≤ d_{k} if d ≤ log_{(2m_{k}−1)} m_{k} . If d ≤ log_{3} 2, then t_{k} ≤ d_{k} independently from m_{k} .

Example 3. - For the classical Cantor set (1.21) gives

\frac{1}{2}r^{d} ≤ H^{d}(k ⋂[x_{0} − r, x_{0} + r]) ≤ 2^{1+d}r^{d} ∀x_{0} ∈ K ∀r ∈]0, 1] .

Proposition 1.22. - Let for every n, i ∈ ℕ^{∗} i ≤ m_{n} , l_{i}^{(n)} = 1/m_{n} and let μ_{K} be the measure defined in section 1.4. Then μ_{K} = H^{d}|_{K} . PROOF. - First we prove that μ_{K} = H^{d}|_{K} on the intervals of the form [p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}] . Let (i_{1}, i_{2}, ..., i_{n}) ∈ ∏_{k=1}^{n} I_{k}; then

μ_{K}([p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}]) =

τ(p^{−1}([p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}])) = τ(π_{1,2,...,n}^{−1}({(i_{1}, i_{2}, ..., i_{n})})) =

(×_{k=1}^{n}τ_{k})({(i_{1}, i_{2}, ..., i_{n})}) = ∏_{k=1}^{n} l_{i_{k}}^{(k)} = (∏_{k=1}^{n} m_{k})^{−1} = t_{n}^{d}

by Remark 26. By Lemma 1.6 the claim is proved.

Since μ_{K} and H^{d}|_{K} are Radon measure on ℝ it is sufficient to prove that they agree on open sets.

Let A ⊆ ℝ be open and such that A ⋂ K ≠ ∅ . Let x ∈ A ⋂ K; then there exists i ∈ I such that x = p(i) . It is easy to see, by Remarks 20 and 28 that x ∈ ⋂_{n=1}^{+∞}[p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n})+t_{n}] , where (i_{1}, i_{2}, ..., i_{n}) = π_{1,2,...,n}(i) .

Since A is open and lim_{n} t_{n} = 0 it follows that there exists n ∈ ℕ^{∗} such that [p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}] ⊆ A .

It follows that there exists a sequence (J_{n}) of disjoint intervals of the form J_{n} = [p_{k_{n}}(i_{1}^{(n)}, i_{2}^{(n)}, ..., i_{k_{n}}^{(n)}), p_{k_{n}}(i_{1}^{(n)}, i_{2}^{(n)}, ..., i_{k_{n}}^{(n)}) + t_{k_{n}}] such that A ⋂ K = ⋃_{n=1}^{+∞} J_{n} .

Since μ_{K}(J_{n}) = H^{d}(K ⋂ J_{n}) ∀n ∈ ℕ^{∗} , we have μ_{K}(A) = H^{d}(K ⋂ A) ■

Corollary 1.23. - If the sequence (m_{n}) is bounded and l_{j}^{(n)} = 1/m_{n} ∀n ∈ ℕ^{∗} ∀j ∈ I_{n} then μ_{K} is a d-measure on K .

Now we estimate the entropy numbers of K under the assumption that t_{k} ≤ d_{k} ∀k ∈ ℕ^{∗} .

To avoid tedious notation we set for every k ∈ ℕ^{∗} ,

C_{k} = ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ... ⋃_{i_{n}=1}^{m_{n}}[p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}] .

By Remark 28 K = ⋂_{k=1}^{+∞} C_{k} .

Let k ∈ ℕ^{∗} and n_{k} = ∏_{j=1}^{k} m_{j} ; then ε_{n_{k}}(K) ≤ ε_{n_{k}}(C_{k}) ≤ t_{k}/2 .

Since the extreme points of the intervals of C_{k} are in K , it follows that

ε_{n_{k}}(K) = \frac{1}{2}t_{k} = \frac{1}{2}n_{k}^{− \frac{1}{d}} .  (1.22)

Let h ∈ ℕ^{∗} be a divisor of m_{k+1} : we compute ε_{hn_{k}}(K) .

The set C_{k+1} is a disjoint union of n_{k+1} closed intervals with amplitude t_{k+1} and mutual distance grater than or equal to t_{k+1} . Since hn_{k} divides n_{k+1} = m_{k+1}n_{k} , we can cover all the m_{k+1} closed intervals of C_{k+1} that are included into a single interval of C_{k} with h closed intervals of the form

[p_{k+1}(i_{1}, i_{2}, ..., i_{k}, \frac{lm_{k+1}}{h} + 1), p_{k+1}(i_{1}, i_{2}, ..., i_{k}, \frac{(l + 1)m_{k+1}}{h}) + t_{k+1}]

0 ≤ l < h

and as before, the extreme points of these intervals are in K ; so

ε_{hn_{k}}(K) = \frac{1}{2} (\frac{m_{k+1}}{h} t_{k+1} + (\frac{m_{k+1}}{h} − 1) d_{k+1})

and by Remark 26 we have

ε_{hn_{k}}(K) = \frac{(h − 1)m_{k+1}^{1− \frac{1}{d}} + m_{k+1} − h}{2h^{1− \frac{1}{d}} (m_{k+1} − 1)}(hn_{k})^{− \frac{1}{d}} .  (1.23)

If h is not a divisor of m_{k+1} we have ε_{hn_{k}}(K) ≤ t_{k}/(2h) and then

ε_{hn_{k}}(K) ≤ \frac{1}{2}h^{\frac{1}{d} −1}(hn_{k})^{− \frac{1}{d}} .  (1.24)

Let ⌊m_{k+1}/h⌋ = max{n ∈ ℕ^{∗} | n ≤ m_{k+1}/h} : then we may not cover K by using hn_{k} intervals of length (t_{k+1} +d_{k+1})⌊m_{k+1}/h⌋ because at least one of the intervals of C_{k+1} would not be covered; so it must be

ε_{hn_{k}}(K) > \frac{1}{2}(t_{k+1} + d_{k+1}) ⌊\frac{m_{k+1}}{h} ⌋ ,

that is

ε_{hn_{k}}(K) > \frac{h^{\frac{1}{d}} (1 − m_{k+1}^{− \frac{1}{d}})}{2(m_{k+1} − 1)} ⌊\frac{m_{k+1}}{h} ⌋ (hn_{k})^{− \frac{1}{d}} .  (1.25)

Finally, if l ∈ ℕ^{∗} l < n_{k} , then

ε_{hn_{k}+l}(K) = ε_{hn_{k}}(K)  (1.26)

because the additional l intervals can not be equally distributed between the connected components of C_{k} .

Example 4. - Let K be the classical Cantor set: by (1.22) and (1.26) we have

ε_{2^{k}+l}(K) = 2^{− \frac{k}{d} −1} = \frac{1}{2}3^{−k} .

It follows that the entropy numbers of the classical Cantor set have the following asymptotic behaviour:

\frac{1}{2}n^{− \frac{1}{d}} ≤ ε_{n}(K) < \frac{3}{2}n^{− \frac{1}{d}} ∀n ∈ ℕ^{∗} .

The constants 1/2 and 3/2 are sharp. Indeed lim_{n} ε_{2^{n+1}−1}(K)(2^{n+1} − 1)^{1/d} = 3/2 and ε_{2^{n}}(K)(2^{n})^{1/d} = 1/2 ∀n ∈ ℕ^{∗} .

Remark 30. - If the sequence (m_{n}) is bounded, K is a d-set by Proposition 1.21 and then we can apply Proposition 1.10 to see that there exist a, b ∈ ]0, +∞[ such that

an^{− \frac{1}{d}} ≤ ε_{n}(K) ≤ bn^{− \frac{1}{d}} ∀n ∈ ℕ^{∗} .

Moreover, by Corollary 1.8, the box dimension of K is d .

Remark 31. - Let us suppose that the sequence (m_{n}) is not bounded and t_{k} ≤ d_{k} ∀k ∈ ℕ^{∗} . Let (m_{n_{k}}) be a subsequence of (m_{n}) such that lim_{k} m_{n_{k}} = +∞ ; we set p_{k} = ∏_{j=1}^{n_{k}−1} m_{j} for every k ∈ ℕ^{∗} ; then, by (1.22) we have

ε_{p_{k}}(K) = \frac{1}{2}p_{k}^{− \frac{1}{d}} .

If m_{n_{k}} is even, then, by setting 2h_{k} = m_{n_{k}} , we have from (1.23)

ε_{h_{k}p_{k}}(K)(h_{k}p_{k})^{\frac{1}{d}} = \frac{2^{1− \frac{1}{d}} (h_{k} − 1) + hk^{\frac{1}{d}}}{2(2h_{k} − 1)} .  (1.27)

If m_{n_{k}} is odd, then, we set 2h_{k} = m_{n_{k}} + 1 and by (1.25) it follows

ε_{h_{k}p_{k}}(K)(h_{k}p_{k})^{\frac{1}{d}} > \frac{hk^{\frac{1}{d}} (1 − m_{n_{k}}^{− \frac{1}{d}})}{2(m_{n_{k}} − 1)} ⌊\frac{m_{n_{k}}}{h_{k}} ⌋ ,

but ⌊m_{n_{k}}/h_{k}⌋ = ⌊2 − h_{k}^{−1}⌋ = 1 ; then

ε_{h_{k}p_{k}}(K)(h_{k}p_{k})^{\frac{1}{d}} > \frac{hk^{\frac{1}{d}} (1 − (2h_{k} − 1)^{− \frac{1}{d}})}{4(h_{k} − 1)} .  (1.28)

Since lim_{k} h_{k} = +∞ , it follows from (1.27) and (1.28) that

lim sup_{n} ε_{n}(K)n^{\frac{1}{d}} = +∞ .

If the sequence (m_{n}) is not bounded it may happen that \overline{dim_{B}}K > d = dim K .

Example 5. - Let d ∈]0, 1[ and ε ∈]0, 1 − d[ . Suppose that the sequence (m_{n}) satisfies the following conditions: m_{1} = 2 and for every n ∈ ℕ^{∗} m_{n+1} is even and

m_{n+1} > 2^{\frac{3d+3ε+1}{1−d−ε}} qn+1^{\frac{ε}{d(1−d−ε)}} where q_{n+1} = ∏_{k=1}^{n} m_{k} .

Let us suppose that dim_{B} K = d . By Corollary 1.9 there exists n_{ε} ∈ ℕ^{∗} such that

ε_{n}(K) ≤ 2(n + 1)^{− \frac{1}{d+ε}} < 2n^{− \frac{1}{d+ε}} ∀n ∈ ℕ^{∗} n > n_{ε} .

By (1.27), for all n ∈ ℕ^{∗} with n > n_{ε} ,

2 > ε_{m_{n}q_{n}/2}(K) (\frac{1}{2}m_{n}q_{n})^{\frac{1}{d+ε}} =

\frac{2^{1− \frac{1}{d}} (m_{n}/2 − 1) + (m_{n}/2)^{\frac{1}{d}}}{2(m_{n} − 1)} (\frac{1}{2}m_{n}q_{n})^{\frac{1}{d+ε} − \frac{1}{d}} >

\frac{2^{− \frac{1}{d}} mn^{\frac{1}{d}} 2^{− \frac{1}{d+ε} + \frac{1}{d}} m_{n}^{\frac{1}{d+ε} − \frac{1}{d}}}{2m_{n}} q_{n}^{− \frac{ε}{d(d+ε)}} = 2^{−1− \frac{1}{d+ε}} m_{n}^{\frac{1}{d+ε} −1}q_{n}^{− \frac{ε}{d(d+ε)}} >

2^{\frac{3d+3ε+1}{d+ε} − \frac{d+ε+1}{d+ε}} qn^{\frac{ε}{d(d+ε)}} q_{n}^{− \frac{ε}{d(d+ε)}} = 2^{\frac{2d+2ε}{d+ε}} = 4 .

1.5.2 An example concerning h-sets

Definition. - For N ∈ ℕ^{∗} we say that a function h : [0, +∞[→ [0, +∞[ belongs to ℌ_{N} if it is continuous,increasing and there exist q ∈ ℕ^{∗} q ≥ 2 and a sequence (λ_{n}) in ]0, +∞[ so that:

1. there exist a, b ∈]0, +∞[ such that

aq^{−Nn} ≤ h(∏_{k=1}^{n} λ_{k}) ≤ bq^{−Nn} ∀n ∈ ℕ^{∗} ,

2. 0 < inf_{n} λ_{n} ≤ sup_{n} λ_{n} < 1/q .

Example 6. - For every d ∈]0, N[ the function h(r) = r^{d} belongs to ℌ_{N}. Indeed the conditions 1) and 2) of the previous definition are satisfied by h with q = 2 and λ_{n} = q^{−N/d} ∀n ∈ ℕ^{∗} .

The function h(r) = r^{N} is not in ℌ_{N} . Indeed, let q ∈ ℕ^{∗} and (λ_{n}) be a sequence in ]0, 1/q[ satisfying condition 2) of the previous definition; then there exists ε > 0 such that λ_{n} < q^{−1} − ε ∀n ∈ ℕ^{∗} . It follows q^{n} ∏_{k=1}^{n} λ_{k} ≤ (1 − qε)^{n} ∀n ∈ ℕ^{∗} and then lim_{n} q^{Nn}h(∏_{k=1}^{n} λ_{k}) = 0 .

Definition. - Let Γ be a compact non-empty subset of ℝ^{N} and h ∈ ℌ_{N}. We call Γ an h-set if there exists a Radon measure μ on ℝ^{N} whose support is Γ and two constants c_{1}, c_{2} ∈]0, +∞[ such that

c_{1}h(r) ≤ μ(B_{ℝ^{N}}(x, r)) ≤ c_{2}h(r) ∀x ∈ Γ ∀r ∈]0, 1[ .

If Γ is an h-set with respect to a Radon measure μ , we say that μ is an h-measure on Γ .

Remark 32. - By Example 6 every d-set in ℝ^{N} with 0 < d < N is an h-set with (for example) h(r) = r^{d} .

Let h ∈ ℌ_{N} and let q ∈ ℕ^{∗} q ≥ 2 , (λ_{n}) be related to h as in the definition of ℌ_{N} .

Let {a_{1}, a_{2}, ..., a_{q^{N}} } = {(i_{1}/q, i_{2}/q, ..., i_{N}/q) ∈ ℝ^{N} | i_{j} ∈ ℕ i_{j} < q ∀j ∈ ℕ^{∗} j ≤ N} .

We set for every n ∈ ℕ^{∗} m_{n} = q^{N} and f_{i}^{(n)} : ℝ^{N} → ℝ^{N} f_{i}^{(n)}(x) = λ_{n}x + a_{i} .

As usual I = ∏_{n=1}^{+∞} I_{n} = {1, 2, ..., q^{N}}^{ℕ^{∗}} and K = p(I) .

We recall that in this case ρ < 1/q < 1 and x_{i}^{(n)} = \frac{1}{1−λ_{n}} a_{i} ∀n, i ∈ ℕ^{∗} i ≤ q^{N} ; since sup_{n} λ_{n} < 1 we have that F (the set of all fixed points) is bounded. Then K is well defined.

Moreover we set, for every n, i ∈ ℕ^{∗} i ≤ q^{N} l_{i}^{(n)} = 1/q^{N} and as in section 1.4, μ_{K} = p_{♯}τ .

Theorem 1.24. - The set K defined above is an h-set with respect to the same function h used in its construction.

The measure μ_{K} constructed above is an h-measure with respect to the same h .

Let A = {A ⊆ ℝ^{N} | |A| < +∞} and ψ : A → [0, +∞[ , ψ(A) = h(|A|). Then H^{ψ} is an h-measure on K and every h-measure μ on K is equivalent to H^{ψ} . PROOF. - See [3], section 6, Theorem 6.6.

Theorem 1.25. - Let h ∈ ℌ_{N} and Γ ⊆ ℝ^{N} be an h-set. Then,

dim(Γ ⋂ B_{ℝ^{N}}(x, r)) = lim inf_{t→0} \frac{log h(t)}{log t} ∀x ∈ Γ ∀r ∈]0, 1[ .

PROOF. - See [3], section 8, Theorem 8.12.

The definition of h-set is really more general than the one of d-set.

Example 7. - Let d ∈]0, N[ , s ∈]0, 1/2[ and

h(r) = r^{d}e^{−| log r|^{s}} .

In [3], section 7, Proposition 7.5 it is proved that h ∈ ℌ_{N} . Then we can construct a set K ⊆ ℝ^{N} that is an h-set with respect to this function. Moreover there exist c_{1}, c_{2} ∈]0, +∞[ so that

c_{1}h(r) ≤ H^{ψ}(K ⋂ B_{ℝ^{N}}(x, r)) ≤ c_{2}h(r) ∀x ∈ K ∀r ∈]0, 1[ ,

where ψ is as in Theorem 1.24.

Let us suppose that K is a d′-set with d′ ∈]0, N] . Since

lim_{r→0} \frac{log h(r)}{log r} = d ,

by Theorem 1.25, dim K = d and then d′ = d .

Let x ∈ K and r ∈]0, 1[ . Let δ > 0 ; by the definition of H_{δ}^{d} and Remark 1 there exists a sequence (A_{i}) of subsets of ℝ^{N} such that |A_{i}| ≤ δ ∀i ∈ ℕ^{∗} , K ⋂ B_{ℝ^{N}}(x, r) ⊆ ⋃_{i=1}^{+∞} A_{i} and

∑_{i=1}^{+∞} |A_{i}|^{d} < H_{δ}^{d}(K ⋂ B_{ℝ^{N}}(x, r)) + 1 ≤ H^{d}(K ⋂ B_{ℝ^{N}}(x, r)) + 1 .

By the definition of H_{δ}^{ψ} ,

H_{δ}^{ψ}(K ⋂ B_{ℝ^{N}}(x, r)) ≤ ∑_{i=1}^{+∞} h(|A_{i}|) ≤ e^{−| log δ|^{s}} ∑_{i=1}^{+∞} |A_{i}|^{d} <

e^{−| log δ|^{s}}(H^{d}(K ⋂ B_{ℝ^{N}}(x, r)) + 1) .

If δ → 0 , we have H^{ψ}(K ⋂ B_{ℝ^{N}}(x, r)) = 0 which proves that K is not an h-set, in opposition to our hypothesis.

A conjecture of E. De Giorgi

2.1 Introduction

We define a notion of dimension for compact subsets of ℝ^{N} . The method we use is similar to the one commonly used to define the Minkowski dimension (see [20], section 5.5), but the Minkowski contents \underline{M^{(α)}}, \overline{M^{(α)}} are replaced by the functionals f_{−}^{(α)}, f_{+}^{(α)} defined on the continuous functions on K . By definition they are the lower and upper limit as r → 0 of the functional f_{r}^{(α)} defined on the continuous functions on the closed r-neighborhood of K . The limit just mentioned is taken in the sense of Γ-convergence that is defined in the second section, while in the third one we define the dimension of K .

2.2 Γ-convergence

In this section we define the notion of Γ-convergence of functions defined on a topological space X and recall some properties of Γ-limits.

Let x ∈ X ; we denote by U(x) the set of all open neighborhood of x . By ℝ^{∗} we mean the extended real line [−∞, +∞] .

Definition. - Let D ⊆ ℝ and f : D × X → ℝ^{∗} be a function. If r_{0} ∈ ℝ^{∗} is an accumulation point of D , the Γ-lower limit and the Γ-upper limit of f as r → r_{0} are defined by

Γ- lim inf_{r→r_{0}} f(r, x) = sup_{U∈U(x)} lim inf_{r→r_{0}} inf_{y∈U} f(r, y) = sup_{U∈U(x)} sup_{V ∈U(r_{0})} inf_{r∈V} inf_{y∈U} f(r, y) ,

Γ- lim sup_{r→r_{0}} f(r, x) = sup_{U∈U(x)} lim sup_{r→r_{0}} inf_{y∈U} f(r, y) = sup_{U∈U(x)} inf_{V ∈U(r_{0})} sup_{r∈V} inf_{y∈U} f(r, y) .

If there exists a function g : X → ℝ^{∗} such that Γ- lim inf_{r→r_{0}} f(r, x) = Γ- lim sup_{r→r_{0}} f(r, x) = g(x) ∀x ∈ X , then we write g(x) = Γ- lim_{r→r_{0}} f(r, x) and we say that f(r, ·) Γ-converges to g as r → r_{0} or that g is the Γ-limit of f(r, ·) as r → r_{0} .

The next proposition compare the Γ-convergence with the pointwise one.

Proposition 2.1. - Let D ⊆ ℝ and f : D × X → ℝ^{∗} be a function and r_{0} ∈ ℝ^{∗} be an accumulation point of D . Then, for all x ∈ X ,

Γ- lim inf_{r→r_{0}} f(r, x) ≤ lim inf_{r→r_{0}} f(r, x) ,

Γ- lim sup_{r→r_{0}} f(r, x) ≤ lim sup_{r→r_{0}} f(r, x) .

In particular, if f(r, ·) Γ-converges to g and converges pointwise to h, then g(x) ≤ h(x) ∀x ∈ X . PROOF. - Let x ∈ X : for every U ∈ U(x) and for every r ∈ D we have inf_{y∈U} f(r, y) ≤ f(r, x) , hence

lim inf_{r→r_{0}} inf_{y∈U} f(r, y) ≤ lim inf_{r→r_{0}} f(r, x)

and

lim sup_{r→r_{0}} inf_{y∈U} f(r, y) ≤ lim sup_{r→r_{0}} f(r, x) .

The conclusion follows easily ■

We will give examples in which the two notions of convergence give different limits.

Definition. - Let f : X → ℝ^{∗} be a function and x ∈ X . We say that f is lower (upper) semicontinuous at x if f(x) = −∞ (f(x) = +∞) or if for every t ∈] − ∞, f(x)[ (t ∈]f(x), +∞[) there exists U ∈ U(x) such that f(y) > t (f(y) < t) ∀y ∈ U .

We say that f is lower (upper) semicontinuous on X if it is lower (upper) semicontinuous at every point x ∈ X.

Remark 1. - A function f : X → ℝ^{∗} is continuous if and only if it is upper and lower semicontinuous.

Remark 2. - A function f : X → ℝ^{∗} is lower (upper) semicontinuous at a point x ∈ X if and only if

f(x) = sup_{U∈U(x)} inf_{y∈U} f(y) (f(x) = inf_{U∈U(x)} sup_{y∈U} f(y)) .

If f is lower (upper) semicontinuous at a point x ∈ X , then

f(x) ≤ lim inf_{n} f(x_{n}) (f(x) ≥ lim sup_{n} f(x_{n}))

for every sequence (x_{n}) converging to x in X .

Proposition 2.2. - Suppose that X satisfies the first axiom of countability. Let f : X → ℝ^{∗} be a function and let x ∈ X . The following facts are equivalent:

1. f is lower semicontinuous at x ;

2. f(x) ≤ lim inf_{n} f(x_{n}) for every sequence (x_{n}) converging to x in X ;

3. f(x) ≤ lim_{n} f(x_{n}) for every sequence (x_{n}) converging to x in X , such that lim_{n} f(x_{n}) exists and is finite.

PROOF. - See [8], chapter 1, Proposition 1.3.

Definition. - A function f : X → ℝ^{∗} satisfying condition (2) of the previous proposition is said to be sequentially lower semicontinuous.

Remark 3. - If a function f : X → ℝ^{∗} is lower semicontinuous, then it is sequentially lower semicontinuous.

If X satisfies the first axiom of countability then f is lower semicontinuous if and only if it is sequentially lower semicontinuous.

Definition. - Let D ⊆ ℝ and f : D ×X → ℝ^{∗} be a function. We say that the functions f(r, ·) : X → ℝ^{∗} are equi-lower (equi-upper) semicontinuous at a point x ∈ X if for every ε > 0 there exists U ∈ U(x) so that f(r, y) ≥ f(r, x) − ε (f(r, y) ≤ f(r, x) + ε) ∀y ∈ U ∀r ∈ D .

We say that the functions f(r, ·) : X → ℝ^{∗} are equi-lower (equi-upper) semicontinuous on X if they are equi-lower (equi-upper) semicontinuous at every point x ∈ X .

We say that the functions f(r, ·) : X → ℝ^{∗} are equi-continuous at a point x ∈ X if for every ε > 0 there exists U ∈ U(x) so that f(r, y) ∈]f(r, x) − ε, f(r, x) + ε[ ∀y ∈ U ∀r ∈ D .

As before we say that the functions f(r, ·) : X → ℝ^{∗} are equi-continuous on X if they are equi-continuous at each point of X .

Proposition 2.3. - Let D ⊆ ℝ and f : D × X → ℝ^{∗} be a function and r_{0} ∈ ℝ^{∗} be an accumulation point of D . Assume that the functions f(r, ·) : X → ℝ^{∗} are equi-lower semicontinuous on X . Then, for all x ∈ X

Γ- lim inf_{r→r_{0}} f(r, x) = lim inf_{r→r_{0}} f(r, x) ,
Γ- lim sup_{r→r_{0}} f(r, x) = lim sup_{r→r_{0}} f(r, x) .

In particular f(r, ·) Γ-converges to g : X → ℝ^{∗} if and only if it converges pointwise to g in X . PROOF. - Let x ∈ X : for every ε > 0 there exists U ∈ U(x) so that f(r, x) − ε ≤ inf_{y∈U} f(r, y) for every r ∈ D . This implies

lim inf_{r→r_{0}} f(r, x) − ε ≤ lim inf_{r→r_{0}} inf_{y∈U} f(r, y) ≤ sup_{U∈U(x)} lim inf_{r→r_{0}} inf_{y∈U} f(r, y)

If ε → 0 then we obtain lim inf_{r→r_{0}} f(r, x) ≤ Γ- lim inf_{r→r_{0}} f(r, x) . The opposite inequality is given by Proposition 2.1.

The proof of the second equality is analogous ■

Definition. - Let f : X → ℝ^{∗} be a function. We call lower semicontinuous envelope of f the function defined by

\overline{f}(x) = sup\{g(x) | g : X → ℝ^{∗} is lower semicontinuous and
g(y) ≤ f(y) ∀y ∈ X} .

Remark 4. - Let h : X → ℝ^{∗} be a lower semicontinuous function such that h(y) ≤ f(y) ∀y ∈ X ; then for x ∈ X

h(x) = sup_{U∈U(x)} inf_{y∈U} h(y) ≤ sup_{U∈U(x)} inf_{y∈U} sup{g(y) | g : X → ℝ^{∗}

is lower semicontinuous and g(z) ≤ f(z) ∀z ∈ X\} = sup_{U∈U(x)} inf_{y∈U} \overline{f}(y) .

It follows that

\overline{f}(x) ≤ sup_{U∈U(x)} inf_{y∈U} \overline{f}(y) .

Since the opposite inequality holds for every function, it follows that \overline{f} is lower semicontinuous at x .

By definition \overline{f}(x) ≤ f(x) and \overline{f}(x) ≥ g(x) ∀x ∈ X and for every lower semicontinuous function g : X → ℝ^{∗} such that g(y) ≤ f(y) ∀y ∈ X. Therefore \overline{f} is the greatest lower semicontinuous function majorized by f. In particular f = \overline{f} if and only if f is lower semicontinuous.

Proposition 2.4. - Let f : X → ℝ^{∗} be a function. Then

\overline{f}(x) = sup_{U∈U(x)} inf_{y∈U} f(y) ∀x ∈ X .

PROOF. - See [8], chapter 3, Proposition 3.3.

Proposition 2.5. - Suppose that X satisfies the first axiom of countability. Let f : X → ℝ^{∗} be a function and let x ∈ X . Then \overline{f} is characterized by the following properties:

1. for every sequence (x_{n}) converging to x in X it is

\overline{f}(x) ≤ lim inf_{n} f(x_{n}) ;

2. there exists a sequence (x_{n}) converging to x in X such that

\overline{f}(x) ≥ lim sup_{n} f(x_{n}) .

PROOF. - See [8], chapter 3, Proposition 3.6.

Usually the notion of Γ-convergence is defined for sequences of functions, that is when D = ℕ^{∗} and r_{0} = +∞ .

In this case we write f_{n}(x) instead of f(n, x) .

Proposition 2.6. - Let (f_{n}) be a sequence of functions from X to ℝ^{∗} . If f_{n}(x) ≤ f_{n+1}(x) ∀x ∈ X ∀n ∈ ℕ^{∗} , then (f_{n}) Γ-converges and

Γ- lim_{n} f_{n}(x) = lim_{n} \overline{f}_{n}(x) = sup_{n∈ℕ^{∗}} f_{n}(x) ∀x ∈ X .

PROOF. - See [8], chapter 5, Proposition 5.4.

Proposition 2.7. - Let (f_{n}) be a sequence of functions from X to ℝ^{∗} and let f : X → ℝ^{∗} be a function. If for every x ∈ X f_{n}(x) ≥ f_{n+1}(x) ∀n ∈ ℕ^{∗} and lim_{n} f_{n}(x) = f(x) , then (f_{n}) Γ-converges and

Γ- lim_{n} f_{n}(x) = \overline{f}(x) ∀x ∈ X .

PROOF. - See [8], chapter 5, Proposition 5.7.

Proposition 2.8. - Let (f_{n}) be a sequence of functions from X to ℝ^{∗} and g_{1}, g_{2}, g : X → ℝ^{∗} be functions. Assume that X satisfies the first axiom of countability. Then, for every x ∈ X , the following statements hold:

1.

g_{1}(x) = Γ- lim inf_{n} f_{n}(x)

if and only if the following two conditions are satisfied:

(a) g_{1}(x) ≤ lim inf_{n} f_{n}(x_{n}) for every sequence (x_{n}) converging to x in X ,

(b) there exists a sequence (x_{n}) converging to x in X such that g_{1}(x) = lim inf_{n} f_{n}(x_{n}) ;

2.

g_{2}(x) = Γ- lim sup_{n} f_{n}(x)

if and only if the following two conditions are satisfied:

(a) g_{2}(x) ≤ lim sup_{n} f_{n}(x_{n}) for every sequence (x_{n}) converging to x in X ,

(b) there exists a sequence (x_{n}) converging to x in X such that g_{2}(x) = lim sup_{n} f_{n}(x_{n}) ;

3.

g(x) = Γ- lim_{n} f_{n}(x)

if and only if the following two conditions are satisfied:

(a) g(x) ≤ lim inf_{n} f_{n}(x_{n}) for every sequence (x_{n}) converging to x in X ,

(b) there exists a sequence (x_{n}) converging to x in X such that g(x) = lim_{n} f_{n}(x_{n}) .

PROOF. - See [8], chapter 8, Proposition 8.1.

The following proposition links the Γ-convergence of sequences of functions with the Γ-convergence of nets of functions.

Proposition 2.9. - Let D ⊆ ℝ , f : D × X → ℝ^{∗} be a function, r_{0} ∈ ℝ^{∗} be an accumulation point of D and g_{1}, g_{2} : X → ℝ^{∗} be functions. Assume that X satisfies the first axiom of countability. Then, for every x ∈ X the following statements hold:

1.

g_{1}(x) = Γ- lim inf_{r→r_{0}} f(r, x)

if and only if the following two conditions are satisfied:

(a) g_{1}(x) ≤ lim inf_{n} f(r_{n}, x_{n}) for every sequence (r_{n}) converging to r_{0} in D and for every sequence (x_{n}) converging to x in X ,

(b) there exists a sequence (r_{n}) converging to r_{0} in D and a sequence (x_{n}) converging to x in X such that g_{1}(x) ≥ lim inf_{n} f(r_{n}, x_{n}) ;

2.

g_{2}(x) = Γ- lim sup_{r→r_{0}} f(r, x)

if and only if the following two conditions are satisfied:

(a) there exists a sequence (r_{n}) converging to r_{0} in D such that g_{2}(x) ≤ lim sup_{n} f(r_{n}, x_{n}) for every sequence (x_{n}) converging to x in X ,

(b) for every sequence (r_{n}) converging to r_{0} in D there exists a sequence (x_{n}) converging to x in X such that g_{2}(x) ≥ lim sup_{n} f(r_{n}, x_{n}) .

PROOF. - See [4], section 3, Proposition 3.3.

Corollary 2.10. - Let D ⊆ ℝ , f : D ×X → ℝ^{∗} be a function, r_{0} ∈ ℝ^{∗} be an accumulation point of D and g_{1}, g_{2}, g : X → ℝ^{∗} be functions. Assume that X satisfies the first axiom of countability. Then, for every x ∈ X the following statements hold:

1.

g_{1}(x) = Γ- lim inf_{r→r_{0}} f(r, x)

if and only if the following two conditions are satisfied:

(a) g_{1}(x) ≤ Γ- lim inf_{n} f(r_{n}, x) for every sequence (r_{n}) converging to r_{0} in D ,

(b) there exists a sequence (r_{n}) converging to r_{0} in D such that g_{1}(x) = Γ- lim_{n} f(r_{n}, x) ; 2.

g_{2}(x) = Γ- lim sup_{r→r_{0}} f(r, x)

if and only if the following two conditions are satisfied:

(a) g_{2}(x) ≥ Γ- lim sup_{n} f(r_{n}, x) for every sequence (r_{n}) converging to r_{0} in D ,

(b) there exists a sequence (r_{n}) converging to r_{0} in D such that g_{2}(x) = Γ- lim_{n} f(r_{n}, x) .

3.

g(x) = Γ- lim_{r→r_{0}} f(r_{n}, x)

if and only if g(x) = Γ- lim_{n} f(r_{n}, x) for every sequence (r_{n}) in D converging to r_{0} in D .

Moreover, in these statements, one can restrict the choice of (r_{n}) to the strictly monotone sequences. PROOF. - We prove 1). Let x ∈ X and g_{1}(x) = Γ- lim inf_{r→r_{0}} f(r, x) . Take a sequence (r_{n}) converging to r_{0} in D . Then, given U ∈ U(x) , for ε > 0 , there exists n_{ε} ∈ ℕ^{∗} such that |r_{n} − r_{0}| < ε for every n > n_{ε} and then

inf ; r∈]r_{0}−ε,r_{0}+ε[⋂ D inf_{y∈U} f(r, y) ≤ inf_{y∈U} f(r_{n}, y)

for every n > n_{ε} ; then

inf ; r∈]r_{0}−ε,r_{0}+ε[⋂ D inf_{y∈U} f(r, y) ≤ inf_{k>n_{ε}} inf_{y∈U} f(r_{k}, y) ≤ sup_{n∈ℕ^{∗}} inf_{k≥n} inf_{y∈U} f(r_{k}, y)

for every ε > 0; which implies

sup_{ε>0} inf ; r∈]r_{0}−ε,r_{0}+ε[⋂ D inf_{y∈U} f(r, y) ≤ sup_{n∈ℕ^{∗}} inf_{k≥n} inf_{y∈U} f(r_{k}, y)

for all U ∈ U(x) and then g_{1}(x) ≤ Γ- lim inf_{n} f(r_{n}, x) .

By Proposition 2.9 1) b) there exists a sequence r_{n} converging to r_{0} in D and a sequence y_{n} converging to x in X so that g_{1}(x) ≥ lim inf_{n} f(r_{n}, y_{n}) and by Proposition 2.9 1) a) g_{1}(x) = lim inf_{n} f(r_{n}, y_{n}) and so there exists a subsequence (r_{n_{k}}) of (r_{n}) and a subsequence (y_{n_{k}}) of (y_{n}) so that g_{1}(x) = lim_{k} f(r_{n_{k}}, y_{n_{k}}) . But still by Proposition 2.9 1) a) g_{1}(x) ≤ lim inf_{k} f(r_{n_{k}}, x_{k}) for every sequence (x_{k}) converging to x in X . Then, by Proposition 2.8 1) g_{1}(x) = Γ- lim inf_{k} f(r_{n_{k}}, x) .

Since there exists only one number l satisfying the two conditions 1) a) and 1) b), then this number must be g_{1}(x) and so the equivalence in 1) is proved.

Proof of 2). Let x ∈ X and g_{2}(x) = Γ- lim sup_{r→r_{0}} f(r, x) . Take a sequence (r_{n}) converging to r_{0} in D . Then, given U ∈ U(x) , we have

g_{2}(x) ≥ lim sup_{r→r_{0}} inf_{y∈U} f(r, y) .

Let ε > 0 : there exists δ > 0 such that

g_{2}(x) + ε > inf_{y∈U} f(r, y) ∀r ∈]r_{0} − δ, r_{0} + δ[⋂ D

and so there exists n_{δ} ∈ ℕ^{∗} such that, for every n > n_{δ} we have |r_{n}−r_{0}| < δ and then g_{2}(x) + ε > inf_{y∈U} f(r_{n}, y).

It follows

g_{2}(x) + ε ≥ lim sup_{n} inf_{y∈U} f(r_{n}, y) ∀ε > 0 ∀U ∈ U(x)

and then g_{2}(x) ≥ Γ- lim sup_{n} f(r_{n}, x) (note that in the proof of 1) a) and 1)

b) we have not used the fact that X satisfies the first axiom of countability). By Proposition 2.9 2) a) there exists a sequence (r_{n}) converging to r_{0} in

D such that g_{2}(x) ≤ lim sup_{n} f(r_{n}, x_{n}) for every sequence (x_{n}) converging to x in X. But, by Proposition 2.9 2) b) there exists a sequence (y_{n}) converging to x in X such that g_{2}(x) ≥ lim sup_{n} f(r_{n}, y_{n}) ; then, by Proposition 2.8 2) g_{2}(x) = Γ- lim sup_{n} f(r_{n}, x) .

To prove 3), let g(x) = Γ- lim_{r→r_{0}} f(r, x) , g_{1}(x), g_{2}(x) be as in 1) and 2) and (r_{n}) be a sequence converging to r_{0} in D . Then, by 2) and 1),

Γ- lim sup_{n} f(r_{n}, x) ≤ g_{2}(x) = g(x) = g_{1}(x) ≤ Γ- lim inf_{n} f(r_{n}, x) .

Conversely, if g(x) = Γ- lim_{n} f(r_{n}, x) for all sequences (r_{n}) converging to r_{0} in D , then, still by 1) and 2), there is a sequence (r_{n}) converging to r_{0} in D such that

g_{1}(x) = Γ- lim inf_{n} f(r_{n}, x) = g(x)

and another sequence (s_{n}) converging to r_{0} in D such that

g_{2}(x) = Γ- lim sup_{n} f(s_{n}, x) = g(x) ■

For the rest of this section we only consider the case in which D = ℕ^{∗} and r_{0} = +∞ ; we will write f_{n}(x) in place of f(n, x) .

Proposition 2.11. - Let (f_{n}) be a sequence of functions from X to ℝ^{∗} and let (f_{n_{k}}) be a subsequence of (f_{n}) . Then, for every x ∈ X

Γ- lim inf_{n} f_{n}(x) ≤ Γ- lim inf_{k} f_{n_{k}}(x)

and

Γ- lim sup_{n} f_{n}(x) ≥ Γ- lim sup_{k} f_{n_{k}}(x) .

In particular, if (f_{n}) Γ-converges to f : X → ℝ^{∗} , then (f_{n_{k}}) Γconverges to f . PROOF. - We only prove the first inequality, the second being analogous.

Let x ∈ X , U ∈ U(x) and i ∈ ℕ^{∗} . We have

inf_{h≥n_{i}} inf_{y∈U} f_{h}(y) ≤ inf_{h≥i} inf_{y∈U} f_{n_{h}}(y) ≤ sup_{k∈ℕ^{∗}} inf_{h≥k} inf_{y∈U} f_{n_{h}}(y) ;

then

sup_{n∈ℕ^{∗}} inf_{h≥n} inf_{y∈U} f_{h}(y) ≤ sup_{k∈ℕ^{∗}} inf_{h≥k} inf_{y∈U} f_{n_{h}}(y) ≤ sup_{V ∈U(x)} sup_{k∈ℕ^{∗}} inf_{h≥k} inf_{y∈U} f_{n_{h}}(y)

and the result easily follows ■

Proposition 2.12. - Let (f_{n}) and (g_{n}) be two sequences of functions from X to ℝ^{∗} . If f_{n}(x) ≤ g_{n}(x) ∀x ∈ X ∀n ∈ ℕ^{∗} , then, for every x ∈ X

Γ- lim inf_{n} f_{n}(x) ≤ Γ- lim inf_{n} g_{n}(x)

and

Γ- lim sup_{n} f_{n}(x) ≤ Γ- lim sup_{n} g_{n}(x) .

In particular, if (f_{n}) Γ-converges to f : X → ℝ^{∗} and (g_{n}) Γ-converges to g : X → ℝ^{∗} , then f(x) ≤ g(x) for every x ∈ X .

If h : X → ℝ^{∗} is a lower semicontinuous function and h(x) ≤ f_{n}(x) ∀x ∈ X ∀n ∈ ℕ^{∗} , then

h(x) ≤ Γ- lim inf_{n} f_{n}(x) ∀x ∈ X .

PROOF. - It is straightforward ■ Proposition 2.13. - Let (f_{n}) be a sequence of functions from X to ℝ^{∗}. Then the functions Γ- lim inf_{n} f_{n} and Γ- lim sup_{n} f_{n} are lower semicontinuous on X . PROOF. - See [8], chapter 6, Proposition 6.8.

Theorem 2.14. - Let (f_{n}) be a sequence of functions between X and ℝ^{∗} and let f(x) = Γ- lim inf_{n} f_{n}(x) ∀x ∈ X .

Suppose that there exists a contably compact set K ⊆ X such that

inf_{x∈X} f_{n}(x) = inf_{x∈K} f_{n}(x)

for every n ∈ ℕ^{∗} . Then f attains its minimum on X and

min_{x∈X} f(x) = lim inf_{n} inf_{x∈X} f_{n}(x) .

If, in addition, (f_{n}) Γ-converges to f in X, then f attains its minimum on X and

min_{x∈X} f(x) = lim_{n} inf_{x∈X} f_{n}(x) .

PROOF. - See [8], chapter 7, Theorem 7.4.

Proposition 2.15. - Let (f_{n}) be a sequence of functions between X and ℝ^{∗} and f : X → ℝ^{∗} be a function. Assume that X satisfies the first axiom of countability.

Then (f_{n}) Γ-converges to f if and only if for every subsequence (f_{n_{k}}) of (f_{n}) there exists a subsequence (f_{n_{k_{h}}}) of (f_{n_{k}}) that Γ-converges to f . PROOF. - See [8], chapter 8, Proposition 8.3.

Theorem 2.16. - Assume that X satisfies the second axiom of countability. Then every sequence (f_{n}) of functions from X to ℝ^{∗} has a Γ-convergent subsequence. PROOF. - See [8], chapter 8, Theorem 8.5.

2.3 Definition of capacity dimension

From now on Ω will be an open bounded subset of ℝ^{N} and K a compact subset of Ω . We recall that, for r > 0 , K_{r} is the closed r-neighborhood of K . We will consider the space C(\overline{Ω}) equipped with the norm ∥u∥_{L^{∞}(Ω)} = inf{c > 0 | |u(x)| ≤ c for L^{N}-almost every x ∈ Ω} = sup{|u(x)| | x ∈ Ω} .

Definition. - For any α ∈ [0, N] and for every r > 0 such that K_{r} ⊆ Ω we define the functionals f_{r}^{(α)}, f_{−}^{(α)}, f_{+}^{(α)} : C(\overline{Ω}) → ℝ^{∗}

f_{r}^{(α)}(u) = \{ \begin{matrix}\frac{1}{r^{N−α}} ∫_{K_{r}} ∥∇u(x)∥^{2}dx & if u ∈ H^{1}(K_{r}^{∘}) ⋂ C(\overline{Ω}) \\ +∞ & otherwise \end{matrix} ;

f_{−}^{(α)}(u) = Γ- lim inf_{r→0} f_{r}^{(α)}(u) ;

f_{+}^{(α)}(u) = Γ- lim sup_{r→0} f_{r}^{(α)}(u) .

If f_{−}^{(α)}(u) = f_{+}^{(α)}(u) then we call the common value f^{(α)}(u) .

Remark 5. - It is clear that f_{r}^{(α)}(u), f_{+}^{(α)}(u), f_{−}^{(α)}(u) ≥ 0 ∀u ∈ C(\overline{Ω}) , ∀α ∈ [0, N] , ∀r > 0 .

Remark 6. - Let 0 ≤ α < β ≤ N and let r > 0 be so that K_{r} ⊆ Ω . Then f_{r}^{(β)}(u) ≤ f_{r}^{(α)}(u) for all u ∈ C(\overline{Ω}) .

Definition. - We call lower and upper capacity dimension of K the numbers

\underline{dim_{C}}K = inf\{α ∈ [0, N] | f_{−}^{(α)}(u) = 0 ∀u ∈ C(\overline{Ω})\}
\overline{dim_{C}}K = inf\{α ∈ [0, N] | f_{+}^{(α)}(u) = 0 ∀u ∈ C(\overline{Ω})\} .

If \underline{dim_{C}}K = \overline{dim_{C}}K , their common value is called the capacity dimension of K and is denoted by dim_{C} K .

Remark 7. - In the previous definition we can replace C(\overline{Ω}) with any dense subset X ⊆ C(\overline{Ω}) .

We prove it for the upper capacity dimension. Let D = inf{α ∈ [0, N] | f^{(α)}(u) = 0 ∀u ∈ X\} , where X is a dense subset of C(\overline{Ω}) . It is clear that D ≤ \overline{dim_{C}}K ; we prove the opposite inequality.

Let us suppose to have f^{(α)}(u) = 0 ∀u ∈ X , for a given α ∈ [0, N] . Given u ∈ C(\overline{Ω}) , let (r_{n}) be a sequence of positive real numbers such

that f_{+}^{(α)}(u) = Γ- lim_{n} f_{r_{n}}^{(α)}(u) . Then f_{+}^{(α)}(u) ≤ lim inf_{n} f_{r_{n}}^{(α)}(u_{n}) for all sequences (u_{n}) in C(\overline{Ω}) such that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 .

Let (u_{n}) be a sequence in X such that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 ; for any n ∈ ℕ^{∗} f^{(α)}(u_{n}) = 0 and then there exists a sequence (v_{k}^{(n)}) in C(\overline{Ω}) such that lim_{k} ∥v_{k}^{(n)} − u_{n}∥_{L^{∞}(Ω)} = 0 and 0 = f^{(α)}(u_{n}) = lim_{k} f_{r_{k}}^{(α)}(v_{k}^{(n)}). By a standard diagonal argument it is possible to construct a strictly increasing sequence (k_{n}) in ℕ^{∗} so that lim_{n} ∥v_{k_{n}}^{(n)}−u∥_{L^{∞}(Ω)} = 0 and lim_{n} f_{r_{kn}}^{(α)}(u_{k_{n}}^{(n)}) = 0 .

For every h ∈ ℕ^{∗} we set u_{h} = v_{k_{n}}^{(n)} if k_{n} ≤ h < k_{n+1} ; then f_{+}^{(α)}(u) ≤ lim inf_{h} f_{r_{h}}^{(α)}(u_{h}) = 0 .

In particular, by Theorem 4.5 we can replace C(\overline{Ω}) with the set of polynomials on ℝ^{N} .

Proposition 2.17. - We have \overline{dim_{C}}K ≤ \overline{dim_{B}}K . PROOF. - Let α ∈]\overline{dim_{B}}K, N] ; by Remark 7 it is sufficient to prove that f^{(α)}(u) = 0 ∀u ∈ C^{1}(\overline{Ω}) .

Let u ∈ C^{1}(\overline{Ω}) ; for all r > 0 such that K_{r} ⊆ Ω we have

0 ≤ f_{r}^{(α)}(u) ≤ (∥\frac{∂u}{∂x}∥_{L^{∞}(Ω)}^{2} + ∥\frac{∂u}{∂y}∥_{L^{∞}(Ω)}^{2}) \frac{L^{N}(K_{r})}{r^{N−α}} .

Let us suppose that there exists a sequence (r_{n}) of positive real numbers such that lim_{n} r_{n} = 0 and lim_{n} L^{N}(K_{r_{n}})r_{n}^{α−N} = L > 0 ; then, by Proposition 1.6,

N − \overline{dim_{B}}K = lim inf_{r→0} \frac{log L^{N}(K_{r})}{log r} ≤ lim inf_{n} \frac{log L^{N}(K_{r_{n}})}{log r_{n}} =

lim inf_{n} \frac{log \frac{L^{N} (K_{rn})}{r_{n}^{N−α}} + log r_{n}^{N−α}}{log r_{n}} = N − α

from which α ≤ \overline{dim_{B}}K .

It follows that lim_{r→0} f_{r}^{(α)}(u) = 0 ∀u ∈ C^{1}(\overline{Ω}) and by Proposition 2.1 also the Γ-limit has to be 0 ■

Remark 8. - Let α ∈ [0, N[ and u ∈ C(\overline{Ω}) . If f_{+}^{(α)}(u) < +∞ (f_{−}^{(α)}(u) < +∞), then f_{+}^{(β)}(u) = 0 (f_{−}^{(β)}(u) = 0) ∀β ∈]α, N] .

Indeed, for every neighborhood U of u we have

inf_{r>0} sup_{0<s<r} inf_{v∈U} f_{s}^{(α)}(v) ≤ f_{+}^{(α)}(u) < +∞ .

Then there exists r > 0 such that

inf_{v∈U} f_{s}^{(α)}(v) < 1 + f_{+}^{(α)}(u) ∀s ∈]0, r[ .

Since f_{s}^{(β)}(v) = s^{β−α}f_{s}^{(α)}(v) ∀s > 0 ∀v ∈ C(\overline{Ω}) it follows that

inf_{v∈U} f_{s}^{(β)}(v) < s^{β−α}(1 + f_{+}^{(α)}(u)) < r^{β−α}(1 + f_{+}^{(α)}(u)) ∀s ∈]0, r[

and then

sup_{0<s<r} inf_{v∈U} f_{s}^{(β)}(v) ≤ r^{β−α}(1 + f_{+}^{(α)}(u)) .

Then

inf_{r>0} sup_{0<s<r} inf_{v∈U} f_{s}^{(β)}(v) = 0

for every neighborhood U of u ; from which f_{+}^{(β)}(u) = 0 (the same arguments run for f_{−}^{(β)}(u)).

In particular, if f_{+}^{(α)}(u) < +∞ (f_{−}^{(α)}(u) < +∞) ∀u ∈ X , with X dense in C(\overline{Ω}) , by Remark 7 we obtain \overline{dim_{C}}K ≤ α (\underline{dim_{C}}K ≤ α).

Remark 9. - To prove that \overline{dim_{C}}K = α ∈]0, N] (\underline{dim_{C}}K = α ∈]0, N]) it is sufficient to prove that f_{+}^{(α)}(u) < +∞ (f_{−}^{(α)}(u) < +∞) for every u in a dense subset of C(\overline{Ω}) (for example C^{1}(\overline{Ω})) and there exists u ∈ C(\overline{Ω}) such that f_{+}^{(α)}(u) > 0 (f_{−}^{(α)}(u) > 0).

Remark 10. - Let H ⊆ K be compact. Then, for every r > 0 we have H_{r} ⊆ K_{r} .

Let, for every α ∈ [0, N] and r > 0 , g_{r}^{(α)} be defined as f_{r}^{(α)} with H inside of K ; we have g_{r}^{(α)}(u) ≤ f_{r}^{(α)} for any u ∈ C(\overline{Ω}) .

It follows g_{−}^{(α)}(u) ≤ f_{−}^{(α)}(u) and g_{+}^{(α)}(u) ≤ f_{+}^{(α)}(u) for any u ∈ C(\overline{Ω}) and then \underline{dim_{C}}H ≤ \underline{dim_{C}}K , \overline{dim_{C}}H ≤ \overline{dim_{C}}K .

2.4 Sets with non-empty interior

Notation. - We set

Q = D_{ℝ^{N−1}}(0, 1)×]−1, 1[ , Q_{+} = D_{ℝ^{N−1}}(0, 1)×]0, 1[ , Q_{0} = D_{ℝ^{N−1}}(0, 1)×{0} .

Lemma 2.1. - Let p ∈ [1, +∞] and u ∈ W^{1,p}(Q_{+}) . Then the function v : Q → ℝ defined by

v(x_{1}, x_{2}, ..., x_{N}) = \{ \begin{matrix}u(x_{1}, x_{2}, ..., x_{N}) & if x_{N} > 0 \\ u(x_{1}, x_{2}, ..., −x_{N}) & if x_{N} < 0 \end{matrix}

belongs to W^{1,p}(Q) and

∥v∥_{L^{p}(Q)} ≤ 2∥u∥_{L^{p}(Q_{+})} , ∥v∥_{W^{1,p}(Q)} ≤ 2∥u∥_{W^{1,p}(Q_{+})} .

PROOF. - See [2], chapter 9, Lemma IX.2.

Definition. - An open set V ⊆ ℝ^{N} is said to be of class C^{1} if for every x ∈ ∂V there exists U ∈ U(x) and a one to one function T : Q → U so that

1. T ∈ C^{1}(\overline{Q}) ,

2. T^{−1} ∈ C^{1}(\overline{U}) ,

3. T (Q_{+}) = U ⋂ V ,

4. T (Q_{0}) = U ⋂(∂V) .

Proposition 2.18. - If K = \overline{K^{∘}} , K^{∘} is non-empty and of class C^{1} , then

f^{(N)}(u) = \{ \begin{matrix}∫_{K^{∘}} ∥∇u(x)∥^{2}dx & if u ∈ H^{1}(K^{∘}) ⋂ C(\overline{Ω}) \\ +∞ & otherwise \end{matrix} .

PROOF. - Let (r_{n}) be a sequence of real numbers such that r_{n} > r_{n+1} > 0 for every n ∈ ℕ^{∗} and lim_{n} r_{n} = 0 . Then, for every n ∈ ℕ^{∗} and u ∈ C(\overline{Ω}) , K_{r_{n+1}} ⊆ K_{r_{n}} and f_{r_{n+1}}^{(N)}(u) ≤ f_{r_{n}}^{(N)}(u) .

Let f : C(\overline{Ω}) → ℝ^{∗} be defined by f(u) = ∫_{K^{∘}} ∥∇u(x)∥^{2}dx if there exists

r_{0} > 0 such that u ∈ H^{1}(K_{r_{0}}^{∘}) ⋂ C(\overline{Ω}) and f(u) = +∞ otherwise: we prove that (f_{r_{n}}^{(N)}) converges to f pointwise.

Let u ∈ C(\overline{Ω}) . If f(u) = +∞ then f_{r_{n}}^{(N)}(u) = +∞ ∀n ∈ ℕ^{∗} .

If f(u) < +∞ then there exists r_{0} > 0 such that u ∈ H^{1}(K_{r_{0}}^{∘}) ⋂ C(\overline{Ω}) and for every n ∈ ℕ^{∗} with r_{n} < r_{0} we have

|f_{r_{n}}^{(N)}(u) − f(u)| = ∫_{K_{rn}\K^{∘}} ∥∇u(x)∥^{2}dx ≤

∫_{(∂K)_{rn}} ∥∇u(x)∥^{2}dx = ∫_{(∂K)_{r_{0}}} ∥∇u(x)∥^{2}χ_{(∂K)_{rn}}(x)dx ,

where by χ_{A} we denote the characteristic function of the set A ; that is

χ_{A}(x) = \{ \begin{matrix}1 & x ∈ A \\ 0 & x ∉ A \end{matrix} .

The function ∥∇u∥^{2} belongs to L^{1}((∂K)_{r_{0}}) and then, by the Lebesgue theorem (Theorem 4.7),

lim_{n} ∫_{(∂K)_{r_{0}}} ∥∇u(x)∥^{2}χ_{(∂K)_{rn}}(x)dx = ∫_{(∂K)_{r_{0}}} ∥∇u(x)∥^{2}χ_{∂K}(x)dx =

∫_{∂K} ∥∇u(x)∥^{2}dx = 0

because L^{N}(∂K) = 0 .

By Proposition 2.7 f^{(N)} = \overline{f} ; we prove that f^{(N)} is as we announced. We apply Proposition 2.5. First we consider the case u ∈ H^{1}(K^{∘}) ⋂ C(\overline{Ω}) . Let (u_{n}) be a sequence

in C(\overline{Ω}) such that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 and for every n ∈ ℕ^{∗} there

exists r_{n} > 0 such that u_{n} ∈ H^{1}(K_{r_{0}}^{∘}) . Let ε > 0 and for every

i ∈ ℕ^{∗} i ≤ N , φ_{i} ∈ C_{c}^{∞}(K^{∘}) be such that

∥\frac{∂u}{∂x_{i}} − φ_{i}∥_{L^{2}(K^{∘})} < \frac{ε}{\sqrt{N}} .

Moreover, let n_{ε} ∈ ℕ^{∗} be so that

∥u_{n} − u∥_{L^{∞}(Ω)}L^{N}(K) ∑_{i=1}^{N} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})} < ε

for every n > n_{ε} .

We have, for n > n_{ε} ,

f^{(N)}(u) = ∑_{i=1}^{N} |(\frac{∂u}{∂x_{i}} , \frac{∂u}{∂x_{i}} )_{L^{2}(K^{∘})}| ≤

∑_{i=1}^{N} |(\frac{∂u}{∂x_{i}} , \frac{∂u}{∂x_{i}} − φ_{i})_{L^{2}(K^{∘})}| + ∑_{i=1}^{N} |(\frac{∂u}{∂x_{i}} , φ_{i})_{L^{2}(K^{∘})}| ≤

∑_{i=1}^{N} |∫_{K^{∘}} u(x)\frac{∂φ_{i}}{∂x_{i}} (x)dx| + ∑_{i=1}^{N} ∥\frac{∂u}{∂x_{i}} ∥_{L^{2}(K^{∘})} ∥\frac{∂u}{∂x_{i}} − φ_{i}∥_{L^{2}(K^{∘})} ≤

∑_{i=1}^{N} |∫_{K^{∘}}(u(x) − u_{n}(x))\frac{∂φ_{i}}{∂x_{i}} (x)dx| + ∑_{i=1}^{N} |∫_{K^{∘}} u_{n}(x)\frac{∂φ_{i}}{∂x_{i}} (x)dx| +

\sqrt{∑_{i=1}^{N} ∥\frac{∂u}{∂x_{i}} ∥_{L^{2}(K^{∘})}^{2}}\sqrt{∑_{i=1}^{N} ∥\frac{∂u}{∂x_{i}} − φ_{i}∥_{L^{2}(K^{∘})}^{2}} <

∑_{i=1}^{N} |∫_{K^{∘}} \frac{∂u_{n}}{∂x_{i}} (x)φ_{i}(x)dx|+∥u_{n}−u∥_{L^{∞}(Ω)}L^{N}(K) ∑_{i=1}^{N} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})}+ε\sqrt{f^{(N)}(u)} <

∑_{i=1}^{N} ∥\frac{∂u_{n}}{∂x_{i}} ∥_{L^{2}(K^{∘})} ∥φ_{i}∥_{L^{2}(K^{∘})} + (1 + \sqrt{f^{(N)}(u)}) ε ≤

\sqrt{∑_{i=1}^{N} ∥\frac{∂u_{n}}{∂x_{i}} ∥_{L^{2}(K^{∘})}^{2}}\sqrt{∑_{i=1}^{N} ∥φ_{i}∥_{L^{2}(K^{∘})}^{2}} + (1 + \sqrt{f^{(N)}(u)}) ε ≤

\sqrt{f(u_{n})}\sqrt{∑_{i=1}^{N} (∥φ_{i} − \frac{∂u}{∂x_{i}} ∥_{L^{2}(K^{∘})} + ∥\frac{∂u}{∂x_{i}} ∥_{L^{2}(K^{∘})})^{2}} + (1 + \sqrt{f^{(N)}(u)}) ε <

\sqrt{f(u_{n})}\sqrt{∑_{i=1}^{N} (ε + ∥\frac{∂u}{∂x_{i}} ∥_{L^{2}(K^{∘})})^{2}} + (1 + \sqrt{f^{(N)}(u)}) ε .

Now, if n → +∞ , we have

f^{(N)}(u) ≤ \sqrt{∑_{i=1}^{N} (ε + ∥\frac{∂u}{∂x_{i}} ∥_{L^{2}(K^{∘})})^{2}} lim inf_{n} \sqrt{f(u_{n})} + (1 + \sqrt{f^{(N)}(u)}) ε

and if ε → 0 ,

f^{(N)}(u) ≤ \sqrt{f^{(N)}(u)} lim inf_{n} \sqrt{f(u_{n})} ,

from which f^{(N)}(u) ≤ lim inf_{n} f(u_{n}) .

We prove the condition 2) of Proposition 2.5.

If there exists r_{0} > 0 such that u ∈ H^{1}(K_{r_{0}}^{∘}) ⋂ C(\overline{Ω}) then the condition is satisfied with u_{n} = u ∀n ∈ ℕ^{∗} (in other words f is already lower semicontinuous at u ).

Let us suppose that u ∈ H^{1}(K^{∘}) ⋂ C(\overline{Ω}) but for every r > 0 u ∉ H^{1}(K_{r}^{∘}). To construct a sequence (u_{n}) that converges to u in L^{∞}(Ω) and such that lim_{n} f(u_{n}) = f^{(N)}(u) we first extend u out of K^{∘} by reflection (see, for example, the proof of the Theorem IX.7 in the chapter 9 of the book [2]).

Let n ∈ ℕ^{∗} : since ∂K is compact and K^{∘} is of class C^{1} , there exist k_{n} ∈ ℕ^{∗} and k_{n} open sets U_{1}^{(n)}, U_{2}^{(n)}, ..., U_{k_{n}}^{(n)} ⊆ ℝ^{N} such that ∂K ⊆ ⋃_{i=1}^{k_{n}} U_{i}^{(n)} and for every i ∈ ℕ^{∗} i ≤ k_{n} there exists T_{i}^{(n)} : Q → U_{i}^{(n)} one to one, so that

T_{i}^{(n)} ∈ C^{1}(\overline{Q}) , T_{i}^{(n)}^{−1} ∈ C^{1}(\overline{U_{i}^{(n)}}) ,

T_{i}^{(n)}(Q_{+}) = U_{i}^{(n)} ⋂ K^{∘} , T_{i}^{(n)}(Q_{0}) = U_{i}^{(n)} ⋂(∂K) .

Let x ∈ ∂K , the same arguments hold about K^{∘} ⋂ D_{ℝ^{N}}(x, r_{n}/2) ; then it is not restrictive to suppose that |U_{i}^{(n)}| ≤ r_{n} ∀i ∈ ℕ^{∗} i ≤ k_{n} .

By the lemma on the partition of the unity (Lemma 4.1) there exist φ_{0}^{(n)}, φ_{1}^{(n)}, ..., φ_{k_{n}}^{(n)} ∈ C^{∞}(ℝ^{N}) so that ∀x ∈ ℝ^{N} ∀i ∈ ℕ i ≤ k_{n} we have 0 ≤ φ_{i}^{(n)}(x) ≤ 1 and ∑_{j=0}^{k_{n}} φ_{j}^{(n)}(x) = 1 ; moreover for 1 ≤ i ≤ k_{n} the support of φ_{i}^{(n)} is contained in U_{i}^{(n)} and the support of φ_{0}^{(n)} is contained in ℝ^{N} \ (∂K) .

Since the support of φ_{0}^{(n)} does not intersect ∂K it follows that φ_{0}^{(n)}χ_{K} ∈

C_{c}^{∞}(K^{∘}) and then φ_{0}^{(n)}χ_{K}u ∈ H^{1}(Ω) .

Let i ∈ ℕ^{∗} i ≤ k_{n} : the function u ∘ T_{i}^{(n)}|_{Q_{+}} belongs to H^{1}(Q_{+}) . By Lemma 2.1 the function v_{i}^{(n)} : Q → ℝ

v_{i}^{(n)}(y_{1}, y_{2}, ..., y_{N}) = \{ \begin{matrix}u(T_{i}^{(n)}(y_{1}, y_{2}, ..., y_{N})) & y_{N} ≥ 0 \\ u(T_{i}^{(n)}(y_{1}, y_{2}, ..., −y_{N})) & y_{N} < 0 \end{matrix}

belongs to H^{1}(Q) . Then the function v_{n} : Ω → ℝ ,

v_{n}(x) = φ_{0}^{(n)}(x)χ_{K}(x)u(x) + ∑_{i=1}^{k_{n}} φ_{i}^{(n)}(x)v_{i}^{(n)}(T_{i}^{(n)}^{−1}(x))

(with the convention that φ_{i}^{(n)}(x)v_{i}^{(n)}(T_{i}^{(n)}^{−1}(x)) = 0 if x ∉ U_{i}^{(n)} ) is in H^{1}(Ω) .

Let x ∈ K : if x ∉ ⋃_{i=1}^{k_{n}} U_{i}^{(n)} then φ_{0}^{(n)}(x) = 1 and v_{n}(x) = u(x); else, if i_{1}, i_{2}, ..., i_{h_{n}} ∈ {1, 2, ..., k_{n}} are all the indeces such that x ∈ ⋂_{j=1}^{h_{n}} U_{i_{j}}^{(n)}, we

have v_{n}(x) = φ_{0}^{(n)}(x)u(x) + ∑_{j=1}^{h_{n}} φ_{i_{j}}^{(n)}(x)u(T_{i_{j}}^{(n)}(T_{i_{j}}^{(n)}^{−1}(x))) = u(x) . That is v_{n}|_{K} = u|_{K} .

Since the support of φ_{0}^{(n)} does not intersect ∂K that is compact, there exists δ_{n} ∈]0, r_{n}/2[ such that for every x ∈ ℝ^{N} \ K with φ_{0}^{(n)}(x) > 0 and for every y ∈ K we have ∥x − y∥ ≥ δ_{n} .

We prove that lim_{n} ∥v_{n} − u∥_{L^{∞}(K_{δn})} = 0 .

Let ε > 0 : since u is uniformly continuous there exists δ > 0 such that |u(x) − u(y)| < ε ∀x, y ∈ Ω such that ∥x − y∥ < δ .

Let n_{ε} ∈ ℕ^{∗} be so that r_{n} < δ for all n > n_{ε} and let x ∈ K_{δ_{n}} \ K . Let i_{1}, i_{2}, ..., i_{h_{n}} ∈ {1, 2, ..., k_{n}} be all the indeces such that x ∈ ⋂_{j=1}^{h_{n}} U_{i_{j}}^{(n)} ; we have

|v_{n}(x) − u(x)| = | ∑_{j=1}^{h_{n}} φ_{i_{j}}^{(n)}(x)v_{i_{j}}^{(n)}(T_{i_{j}}^{(n)}^{−1}(x)) − u(x)| ≤

∑_{j=1}^{h_{n}} φ_{i_{j}}^{(n)}(x)|v_{i_{j}}^{(n)}(T_{i_{j}}^{(n)}^{−1}(x)) − u(x)| .

By the way the functions v_{i_{j}}^{(n)} are constructed it follows that for every

j ∈ ℕ^{∗} j ≤ h_{n} there exists y_{i_{j}}^{(n)} ∈ U_{i_{j}}^{(n)} such that v_{i_{j}}^{(n)}(T_{i_{j}}^{(n)}^{−1}(x)) = u(y_{i_{j}}^{(n)}); then

|v_{n}(x) − u(x)| ≤ ∑_{j=1}^{h_{n}} φ_{i_{j}}^{(n)}(x)|u(y_{i_{j}}^{(n)}) − u(x)| < ε

since ∥x − y_{i_{j}}^{(n)}∥ ≤ |U_{i_{j}}^{(n)}| ≤ r_{n} < δ .

Now we construct the sequence (u_{n}) . We still use the lemma on the partition of the unity.

Let, for every n ∈ ℕ^{∗} , σ_{n} = δ_{n}/5 and s_{n} = σ_{n}/3 . The set K_{4σ_{n}}\ K^{∘}_{2σ_{n}} is compact and contained into its open s_{n}-neighborhood. Then there exist

l_{n} ∈ ℕ^{∗} and x_{1}^{(n)}, x_{2}^{(n)}, ..., x_{l_{n}}^{(n)} ∈ K_{4σ_{n}}\ K^{∘}_{2σ_{n}} such that

K_{4σ_{n}}\ K^{∘}_{2σ_{n}}⊆ ⋃_{i=1}^{h_{n}} D_{ℝ^{N}}(x_{i}^{(n)}, s_{n}) .

Again, by the lemma on the partition of the unity there exist ψ_{0}^{(n)}, ψ_{1}^{(n)}, ..., ψ_{l_{n}}^{(n)} ∈ C^{∞}(ℝ^{N}) satisfying the same conditions as before with D_{ℝ^{N}}(x_{i}^{(n)}, s_{n}) instead of U_{i}^{(n)} . We define ω_{1}^{(n)}, ω_{2}^{(n)} : Ω → ℝ , ω_{1}^{(n)}(x) = ψ_{0}^{(n)}(x)χ_{K_{3σn}}(x) , ω_{2}^{(n)}(x) = ψ_{0}^{(n)}(x)(1 − χ_{K_{3σn}}(x)) .

Then the sequence (u_{n}) is defined by

u_{n}(x) = ω_{1}^{(n)}(x)v_{n}(x) + (ω_{2}^{(n)}(x) + ∑_{i=1}^{l_{n}} ψ_{i}^{(n)}(x))u(x) .

For every n ∈ ℕ^{∗} the function u_{n} is continuous on \overline{Ω} because the support of ψ_{0}^{(n)} is contained in (ℝ^{N} \ K_{4σ_{n}}) ⋃ K_{2σ_{n}} .

Let us prove that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 . Let ε > 0 , n_{ε} ∈ ℕ^{∗} be as before and let n ∈ ℕ^{∗} n > n_{ε} . If x ∈ Ω \ K_{2σ_{n}} then u_{n}(x) = u(x) because ω_{1}^{(n)}(x) = 0 . If x ∈ K_{2σ_{n}} then

|u_{n}(x) − u(x)| ≤ |ω_{1}^{(n)}(x)v_{n}(x) − ω_{1}^{(n)}(x)u(x)|+

|ω_{1}^{(n)}(x)u(x) + (ω_{2}^{(n)}(x) + ∑_{i=1}^{l_{n}} ψ_{i}^{(n)}(x))u(x) − u(x)| =

ω_{1}^{(n)}(x)|v_{n}(x) − u(x)| + | ∑_{i=0}^{l_{n}} ψ_{i}^{(n)}(x)u(x) − u(x)| ≤ |v_{n}(x) − u(x)| .

But x ∈ K_{δ_{n}} and we have already shown that in this case |v_{n}(x)−u(x)| < ε.

For every n ∈ ℕ^{∗} u_{n} = v_{n} on K_{σ_{n}} and then u_{n} ∈ H^{1}(K^{∘}_{σ_{n}}) , from which f(u_{n}) = f^{(N)}(u_{n}) = f^{(N)}(u) because u_{n} = v_{n} = u on K . Then f^{(N)}(u) = \overline{f}(u) .

Finally we consider the case u ∈ C(\overline{Ω}) \ H^{1}(K^{∘}) .

Let (u_{n}) be a sequence in C(\overline{Ω}) such that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 ; we have to show that lim_{n} f(u_{n}) = +∞ . We suppose f(u_{n}) < +∞ ∀n ∈ ℕ^{∗} .

Let ε > 0 and φ_{1}, φ_{2}, ..., φ_{N} ∈ C_{c}^{∞}(K^{∘}) be so that ∥φ_{i}∥_{L^{2}(K^{∘})} ≤ 1 and

let n_{ε} ∈ ℕ^{∗} be so that

∥u_{n} − u∥_{L^{∞}(Ω)}L^{N}(K) ∑_{i=1}^{N} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})} < ε ∀n > n_{ε} .

For n > n_{ε} we have

|∑_{i=1}^{N} ∫_{K^{∘}} u(x)\frac{∂φ_{i}}{∂x_{i}} (x)dx| ≤

∑_{i=1}^{N} |∫_{K^{∘}}(u(x) − u_{n}(x))\frac{∂φ_{i}}{∂x_{i}} (x)dx| + ∑_{i=1}^{N} |∫_{K^{∘}} u_{n}(x)\frac{∂φ_{i}}{∂x_{i}} (x)dx| ≤

∑_{i=1}^{N} |∫_{K^{∘}} \frac{∂u_{n}}{∂x_{i}} (x)φ_{i}(x)dx| + ∥u_{n} − u∥_{L^{∞}(Ω)}L^{N}(K) ∑_{i=1}^{N} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})} <

∑_{i=1}^{N} ∥\frac{∂u_{n}}{∂x_{i}} ∥_{L^{2}(K^{∘})} ∥φ_{i}∥_{L^{2}(K^{∘})} + ε ≤

\sqrt{N ∑_{i=1}^{N} ∥\frac{∂u_{n}}{∂x_{i}} ∥_{L^{2}(K^{∘})}^{2}} + ε = \sqrt{Nf(u_{n})} + ε .

By letting n → +∞ and ε → 0 we have

|∑_{i=1}^{N} ∫_{K^{∘}} u(x)\frac{∂φ_{i}}{∂x_{i}} (x)dx| ≤ \sqrt{N lim inf_{n} f(u_{n})}  (2.1)

for every φ_{1}, φ_{2}, ..., φ_{N} ∈ C_{c}^{∞}(K^{∘}) with ∥φ_{i}∥_{L^{2}(K^{∘})} ≤ 1 ∀i ∈ ℕ^{∗} i ≤ N .

We define the linear functionals g_{i} : C_{c}^{∞}(K^{∘}) → ℝ

< g_{i}, φ >= ∫_{K^{∘}} u(x)\frac{∂φ}{∂x_{i}} (x)dx i ∈ ℕ^{∗} i ≤ N .

Let us suppose that sup{| < g_{i}, φ > | | φ ∈ C_{c}^{∞}(K^{∘}) , ∥φ∥_{L^{2}(K^{∘})} ≤ 1} =

C_{i} < +∞ ∀i ≤ N ; then g_{i} is continuous on C_{c}^{∞}(K^{∘}) with respect to the L^{2}- norm. Since g_{i} is linear it is uniformly continuous and by Theorem 4.1, there exists h_{i} : L^{2}(K^{∘}) → ℝ uniformly continuous such that h_{i}(φ) =< g_{i}, φ > for every φ ∈ C_{c}^{∞}(K^{∘}) . Since h_{i} is continuous on L^{2}(K^{∘}) , C_{C}^{∞}(K^{∘}) is dense in L^{2}(K^{∘}) and g_{i} is linear it is easy to see that h_{i} is linear. Then there

exists w_{i} ∈ L^{2}(K^{∘}) such that < h_{i}, v >= ∫_{K^{∘}} w_{i}(x)v(x)dx ∀v ∈ L^{2}(K^{∘}) and

then

∫_{K^{∘}} u(x)\frac{∂φ}{∂x_{i}} (x)dx =< g_{i}, φ >=< h_{i}, φ >= ∫_{K^{∘}} w_{i}(x)φ(x)dx

∀φ ∈ C_{c}^{∞}(K^{∘}) ∀i ∈ ℕ^{∗} i ≤ N

that is u ∈ H^{1}(K^{∘}) and \frac{∂u}{∂x_{i}} = −w_{i} ∀i ∈ ℕ^{∗} i ≤ N .

Than there exists i ∈ ℕ^{∗} such that C_{i} = +∞ and by (2.1) lim_{n} f(u_{n}) = +∞ ■

Corollary 2.19. - If K^{∘}≠ ∅ then dim_{C} K = N . PROOF. - If K^{∘} is of class C^{1} then by the previous proposition dim_{C} K = N. Otherwise dim_{C} K ≥ N because K contains a closed ball ■

2.5 Totally disconnected compact sets

We prove that if K is totally disconnected then dim_{C} K = 0 .

Lemma 2.2. - Let K ⊆ ℝ^{N} be compact. Then the following conditions are equivalent:

1. K is totally disconnected;

2. For all ε > 0 there exist r > 0 and n closed subsets C_{1}, C_{2}, ..., C_{n} of ℝ^{N} so that the following conditions are satisfied:

(a) |C_{i}| < ε ∀i ∈ ℕ^{∗} i ≤ n ,

(b) C_{i} ⋂ C_{j} = ∅ ∀i, j ∈ ℕ^{∗} i < j ≤ n ,
(c) K_{r} = ⋃_{i=1}^{n} C_{i} .

PROOF. - Let us suppose that K is totally disconnected and that 2) is false, that is there exists ε_{0} > 0 such that for all r > 0 , n ∈ ℕ^{∗} and C_{1}, C_{2}, ..., C_{n} ⊆ ℝ^{N} closed and disjoint, such that K_{r} = ⋃_{i=1}^{n} C_{i} , there exists i ∈ ℕ^{∗} i ≤ n such that |C_{i}| ≥ ε_{0} .

Let (r_{n}) be a sequence of real numbers such that r_{n} > r_{n+1} ∀n ∈ ℕ^{∗} and lim_{n} r_{n} = 0 . Given n ∈ ℕ^{∗} with r_{n} < ε_{0} , let k_{n} ∈ ℕ^{∗} and C_{1}^{(n)}, C_{2}^{(n)}, ..., C_{k_{n}}^{(n)} ⊆ ℝ^{N} be closed disjoint subsets of ℝ^{N} so that K_{r_{n}} = ⋃_{i=1}^{k_{n}} C_{i}^{(n)} ; we suppose that h_{n} ∈ ℕ^{∗} , i_{1}, i_{2}, ..., i_{h_{n}} ∈ {1, 2, ..., k_{n}} are the indeces such that |C_{i_{j}}^{(n)}| ≥ ε_{0} ∀j ∈ ℕ^{∗} j ≤ h_{n} . We claim that at least one between C_{i_{1}}^{(n)}, C_{i_{2}}^{(n)}, ..., C_{i_{hn}}^{(n)} contains a closed connected subset with diameter grater than or equal to ε_{0} .

Let us suppose that C_{i_{1}}^{(n)} is not connected; then there exist two nonempty closed disjoint sets D_{1}^{(n)}, D_{2}^{(n)} ⊆ ℝ^{N} so that C_{i_{1}}^{(n)} = D_{1}^{(n)} ⋃ D_{2}^{(n)}.

Let us suppose that there exists j ∈ {1, 2} so that |D_{j}^{(n)}| ≥ ε_{0} and D_{j}^{(n)} is not connected. Then, by proceeding in the same way we can find two non-empty closed disjoint sets D_{j1}^{(n)}, D_{j2}^{(n)} ⊆ ℝ^{N} so that D_{j}^{(n)} = D_{j1}^{(n)} ⋃ D_{j2}^{(n)}.

Think to repeat this procedure any time a set D_{j_{1}j_{2}...j_{k}}^{(n)} is not connected and |D_{j_{1}j_{2}...j_{k}}^{(n)}| ≥ ε_{0} ; we prove that this algorithm stops in a finite number of steps.

Let A_{1}^{i_{1}}, A_{2}^{i_{1}}, A_{3}^{i_{1}} be three families of sets defined as follows:

• A_{1}^{i_{1}} = {C_{i}^{(n)}^{∘} | i ∈ ℕ^{∗} i ≠ i_{1}} ;

• A_{2}^{i_{1}} = {D^{∘}_{j_{1}j_{2}...j_{k}}^{(n)} | k ∈ ℕ^{∗} , j_{1}, j_{2}, ..., j_{k} ∈ {1, 2} and D_{j_{1}j_{2}...j_{k}}^{(n)} is

connected} ⋃{D^{∘}_{j_{1}j_{2}...j_{k}}^{(n)} | k ∈ ℕ^{∗} , j_{1}, j_{2}, ..., j_{k} ∈ {1, 2} and |D_{j_{1}j_{2}...j_{k}}^{(n)}| <

ε_{0}} (A_{2}^{i_{1}} is the family of those sets D_{j_{1}j_{2}...j_{k}}^{(n)} for which the algorithm above mentioned has stopped);

• A_{3}^{i_{1}} = {(⋂_{k=1}^{+∞} D_{j_{1}j_{2}...j_{k}}^{(n)})^{∘} | (j_{h}) is a sequence in {1, 2} such that for every k ∈ ℕ^{∗} |D_{j_{1}j_{2}...j_{k}}^{(n)}| ≥ ε_{0} and D_{j_{1}j_{2}...j_{k}}^{(n)}is not connected} .

We point out that the open sets in the definition of A_{1}^{i_{1}}, A_{2}^{i_{1}}, A_{3}^{i_{1}} are not empty. Let us prove this fact for A_{3}^{i_{1}} . Given a sequence (j_{h}) as in the definition of A_{3}^{i_{1}} , we have D_{j_{1}j_{2}...j_{h+1}}^{(n)} ⊆ D_{j_{1}j_{2}...j_{h}}^{(n)} ∀h ∈ ℕ^{∗} and all these sets are compact and non-empty; then ⋂_{h=1}^{+∞} D_{j_{1}j_{2}...j_{h}}^{(n)} ≠ ∅ . Let y ∈ ⋂_{h=1}^{+∞} D_{j_{1}j_{2}...j_{h}}^{(n)} : then y ∈ K_{r_{n}} , from which ∃x ∈ K such that y ∈ B_{ℝ^{N}}(x, r_{n}) that is connected; it follows B_{ℝ^{N}}(x, r_{n}) ⊆ D_{j_{1}j_{2}...j_{h}}^{(n)} ∀h ∈ ℕ^{∗} and then ∅ ≠ D_{ℝ^{N}}(x, r_{n}) ⊆ (⋂_{k=1}^{+∞} D_{j_{1}j_{2}...j_{k}}^{(n)})^{∘} .

Now we prove that K ⊆ ⋃_{l=1}^{3} ⋃_{A∈A_{l}^{i_{1}}} A .

Let x ∈ K : there exists a unique i ∈ ℕ^{∗} i ≤ k_{n} such that x ∈ C_{i}^{(n)} . If i ≠ i_{1} then C_{i}^{(n)} ∈ A_{1}^{i_{1}} . We suppose that x ∈ C_{i_{1}}^{(n)} . If there exists h ∈ ℕ^{∗}

and j_{1}, j_{2}, ..., j_{h} ∈ {1, 2} such that x ∈ D_{j_{1}j_{2}...j_{h}}^{(n)} and D^{∘}_{j_{1}j_{2}...j_{h}}^{(n)}∈ A_{2}^{i_{1}} , then

we have B_{ℝ^{N}}(x, r_{n}) ⊆ D_{j_{1}j_{2}...j_{h}}^{(n)} and so x ∈D^{∘}_{j_{1}j_{2}...j_{h}}^{(n)}∈ A_{2}^{i_{1}} .

Otherwise there exists a sequence (j_{h}) as in the definition of A_{3}^{i_{1}}, such that x ∈ ⋂_{h=1}^{+∞} D_{j_{1}j_{2}...j_{h}}^{(n)} and as before, we can prove that x ∈ (⋂_{h=1}^{+∞} D_{j_{1}j_{2}...j_{h}}^{(n)})^{∘} .

It follows that A^{i_{1}} = ⋃_{l=1}^{3} A_{l}^{i_{1}} is an open disjoint covering of K . Moreover, let A ∈ A_{2}^{i_{1}} ⋃ A_{3}^{i_{1}} ; given y ∈ A we have y ∈ K_{r_{n}} and as before we can prove that K ⋂ A ≠ ∅ , which implies that A^{i_{1}} does not admit proper sub coverings. But ♯A^{i_{1}} < +∞ since K is compact and if A_{3}^{i_{1}} ≠ ∅, at least one between A_{2}^{i_{1}} and A_{3}^{i_{1}}would not be finite; then A_{3}^{i_{1}} = ∅ .

Now, let us suppose that A_{2}^{i_{1}} = {D^{∘}_{j_{1}j_{2}...j_{k}}^{(n)} | k ∈ ℕ^{∗} , j_{1}, j_{2}, ..., j_{k} ∈ {1, 2}

and |D_{j_{1}j_{2}...j_{k}}^{(n)}| < ε_{0}} (in other words we suppose that for every D^{∘}_{j_{1}j_{2}...j_{k}}^{(n)}∈ A_{2}^{i_{1}}

the algorithm has stopped because |D_{j_{1}j_{2}...j_{k}}^{(n)}| < ε_{0}); we repeat the procedure for C_{i_{2}}^{(n)} .

If also A_{2}^{i_{2}} is composed by sets with diameter less than ε_{0} , we consider C_{i_{3}}^{(n)} and so on.

If A_{2}^{i_{l}} is composed by sets with diameter less than ε_{0} for every l ∈

ℕ^{∗} l ≤ h_{n} , then we set C^{i_{l}} = {D_{j_{1}j_{2}...j_{k}}^{(n)} | D^{∘}_{j_{1}j_{2}...j_{k}}^{(n)}∈ A_{2}^{i_{l}}} and C_{n} =

{C_{i}^{(n)} | i ∈ ℕ^{∗} i ≤ k_{n} , |C_{i}^{(n)}| < ε_{0}} ⋃(⋃_{l=1}^{h_{n}} C^{i_{l}}) and C_{n} satisfies condition 2), but by our assumptions, this fact can not happen.

Then there exist k, l ∈ ℕ^{∗} l ≤ h_{n} and j_{1}, j_{2}, ..., j_{k} ∈ {1, 1} such that

D^{∘}_{j_{1}j_{2}...j_{k}}^{(n)}∈ A_{2}^{i_{l}} , |D_{j_{1}j_{2}...j_{k}}^{(n)}| ≥ ε_{0} and D_{j_{1}j_{2}...j_{k}}^{(n)} is connected.

It is not restrictive to suppose that there exists l_{n} ∈ ℕ^{∗} l_{n} ≤ h_{n} so that C_{i_{ln}}^{(n)} is connected and |C_{i_{ln}}^{(n)}| ≥ ε_{0} ; we also suppose that i_{l_{n}} = 1 .

Now let n ∈ ℕ^{∗} : C_{1}^{(n+1)} ⊆ K_{r_{n+1}} ⊆ K_{r_{n}} = ⋃_{i=1}^{k_{n}} C_{i}^{(n)} and since C_{i}^{(n)} ⋂ C_{j}^{(n)} = ∅ for i ≠ j , it follows that there exists i ∈ ℕ^{∗} i ≤ k_{n} such that C_{1}^{(n+1)} ⊆ C_{i}^{(n)} and we can suppose i = 1 (indeed C_{1}^{(n+1)} is connected and |C_{1}^{(n+1)}| ≥ ε_{0}).

Let K′ = ⋂_{n=1}^{+∞} C_{1}^{(n)} ; then K′ ⊆ ⋂_{n=1}^{+∞} ⋃_{i=1}^{k_{n}} C_{i}^{(n)} = ⋂_{n=1}^{+∞} K_{r_{n}} = K . We prove that ♯K′ ≥ 2 . For every n ∈ ℕ^{∗} there exist x_{n}, y_{n} ∈ C_{1}^{(n)} ⊆

C_{1}^{(1)} so that ∥x_{n} − y_{n}∥ > ε_{0}/2 . Since C_{1}^{(1)} is compact there exist two subsequences (x_{n_{k}}) and (y_{n_{k}}) and x, y ∈ ℝ^{N} such that lim_{k} x_{n_{k}} = x and lim_{k} y_{n_{k}} = y . Let k ∈ ℕ^{∗} : for every h > k x_{n_{h}} ∈ C_{1}^{(n_{h})} ⊆ C_{1}^{(n_{k})} that is closed; then x ∈ C_{1}^{(n_{k})} and so x ∈ K′ (recall that C_{1}^{(n+1)} ⊆ C_{1}^{(n)} ∀n ∈ ℕ^{∗}). The same conclusion holds for y .

Let k′_{ε_{0}}, k′′_{ε_{0}} ∈ ℕ^{∗} be such that ∥x_{n_{k}} − x∥ < ε_{0}/8 for any k > k′_{ε_{0}} and ∥y_{n_{k}} − y∥ < ε_{0}/8 for any k > k′′_{ε_{0}} . Then, for every k > max{k′_{ε_{0}}, k′′_{ε_{0}}} we have ε_{0}/2 < ∥x_{n_{k}} − y_{n_{k}}∥ ≤ ∥x_{n_{k}} − x∥ + ∥x − y∥ + ∥y − y_{n_{k}}∥ < ∥x − y∥ + ε_{0}/4 and ∥x − y∥ > ε_{0}/4 ; that is x ≠ y and x, y ∈ K′ .

Finally we prove that K′ is connected. If not, let A, B ⊆ ℝ^{N} be open and disjoint, with A ⋂ K′ ≠ ∅ , B ⋂ K′ ≠ ∅ , K′ ⊆ A ⋃ B . Let n ∈ ℕ^{∗} ; if C_{1}^{(n)} ⊆ A ⋃ B , then ∅ ≠ K′ ⋂ A ⊆ C_{1}^{(n)} ⋂ A and ∅ ≠ K′ ⋂ B ⊆ C_{1}^{(n)} ⋂ B; but C_{1}^{(n)} is connected; then there exists x_{n} ∈ C_{1}^{(n)}\(A ⋃ B) ⊆ C_{1}^{(1)} .

Since C_{1}^{(1)} is compact there exists a subsequence (x_{n_{k}}) of (x_{n}) and x ∈ ℝ^{N} such that lim_{k} x_{n_{k}} = x . For every k ∈ ℕ^{∗} C_{1}^{(n_{k})} is closed and then x ∈ C_{1}^{(n_{k})} ; it follows that x ∈ C_{1}^{(n)} ∀n ∈ ℕ^{∗} , that is x ∈ K′ .

But x_{n_{k}} ∈ ℝ^{N}\(A ⋃ B) ∀k ∈ ℕ^{∗} and this set is closed; then x ∈ K′\(A ⋃ B) .

We have proved that there is a connected subset K′ of K with at least two points; then K is not totally disconnected.

Now we assume that condition 2) holds.

Let x, y ∈ K , x ≠ y and ε ∈]0, \frac{1}{2}∥x − y∥[ ; we take r > 0 , n ∈ ℕ^{∗} , C_{1}, C_{2}, ..., C_{n} ⊆ ℝ^{N} as in condition 2).

By 2)c) there exist i, j ∈ ℕ^{∗} i ≤ n j ≤ n such that x ∈ C_{i} and y ∈ C_{j}; moreover i and j are unique by 2)b).

If i = j , then |C_{i}| ≥ ∥x − y∥ > 2ε ; so i ≠ j by 2)a).

Let δ = \frac{1}{3} min\{d_{ℝ^{N}}(C_{k}, C_{h}) | k, h ∈ ℕ^{∗} k ≤ n h ≤ n\} , where d_{ℝ^{N}}(C_{k}, C_{h}) = inf{∥u−v∥ | u ∈ C_{k} , v ∈ C_{h}} ; we set A_{k} = ⋃_{z∈C_{k}} D_{ℝ^{N}}(z, δ) ∀k ∈ ℕ^{∗} k ≤ n , A = A_{i} , B = ⋃_{k≠i} A_{k} . Then A, B are open and disjoint; moreover x ∈ A , y ∈ B and K ⊆ K_{r} = ⋃_{k=1}^{n} C_{k} ⊆ ⋃_{k=1}^{n} A_{k} = A ⋃ B ■

Proposition 2.20. - Let a :]0, +∞[→ [0, +∞[ be a function and for every r > 0 such that K_{r} ⊆ Ω , let f_{r} : C(\overline{Ω}) → ℝ^{∗}

f_{r}(u) = \{ \begin{matrix}a(r) ∫_{K_{r}} ∥∇u(x)∥^{2}dx & if u ∈ H^{1}(K_{r}^{∘}) ⋂ C(\overline{Ω}) \\ +∞ & otherwise \end{matrix}

Then Γ- lim_{r→0} f_{r}(u) = 0 for every u ∈ C(\overline{Ω}) . PROOF. - Of course Γ- lim inf_{r→0} f_{r}(u) ≥ 0 ∀u ∈ C(\overline{Ω}) .

Let u ∈ C(\overline{Ω}) and ε > 0 . Since u is uniformly continuous on Ω there exists δ > 0 such that |u(x) − u(y)| < ε for every x, y ∈ Ω such that ∥x − y∥ < δ . By Lemma 2.2 there exist r > 0 and n closed subsets C_{1}, C_{2}, ..., C_{n} of Ω so that the following conditions are satisfied:

1. |C_{i}| < δ/2 ∀i ∈ ℕ^{∗} i ≤ n ,

2. C_{i} ⋂ C_{j} = ∅ ∀i, j ∈ ℕ^{∗} i < j ≤ n ,

3. K_{r} = ⋃_{i=1}^{n} C_{i} .

Since the sets C_{i} are compact and disjoint there exists σ ∈]0, δ/22[ such that inf{∥x − y∥ | x ∈ C_{i} , y ∈ C_{j}} > 11σ for every i, j ∈ ℕ^{∗} i < j ≤ n .

For every i ∈ ℕ^{∗} i ≤ n , (C_{i})_{3σ}\ (C_{i})_{2σ}^{∘} is compact and contained into its open σ-neighborhood; then there exist k_{i} ∈ ℕ^{∗} and x_{1}^{(i)}, x_{2}^{(i)}, ..., x_{k_{i}}^{(i)} ∈

(C_{i})_{3σ}\ (C_{i})_{2σ}^{∘} such that (C_{i})_{3σ}\ (C_{i})_{2σ}^{∘}⊆ ⋃_{j=1}^{k_{i}} D_{ℝ^{N}}(x_{i}^{(j)}, σ) . By the lemma

on the partition of the unity there exist φ_{0}^{(i)}, φ_{1}^{(i)}, ..., φ_{k_{i}}^{(i)} ∈ C^{∞}(ℝ^{N}) such that for every x ∈ ℝ^{N} and for every j ∈ ℕ^{∗} j ≤ k_{i} we have 0 ≤ φ_{j}^{(i)}(x) ≤ 1 and ∑_{j=1}^{k_{i}} φ_{j}^{(i)}(x) = 1 ; moreover, for 1 ≤ i ≤ k_{i} , the support of φ_{j}^{(i)} is contained in D_{ℝ^{N}}(x_{j}^{(i)}, σ) and the support of φ_{0}^{(i)} does not intersect

(C_{i})_{3σ}\ (C_{i})_{2σ}^{∘} .

Let for every i ∈ ℕ^{∗} i ≤ n , z_{i} be a point of C_{i} , D_{i} = (C_{i})_{5σ}\ (C_{i})_{3σ}^{∘}

and D = Ω \ ⋃_{i=1}^{n}(C_{i})_{5σ} ; we set v : Ω → ℝ

v(x) = χ_{D}(x)u(x)+

∑_{i=1}^{n}(φ_{0}^{(i)}(x)χ_{(C_{i})_{3σ}}(x)u(z_{i}) + (φ_{0}^{(i)}(x)χ_{D_{i}}(x) + ∑_{j=1}^{k_{i}} φ_{j}^{(i)}(x))u(x)) .

It is easy to see that v is continuous, ∥u − v∥_{L^{∞}(Ω)} < ε and for every i ∈ ℕ^{∗} i ≤ n , v(x) = u(z_{i}) for every x ∈ C_{i} . This fact implies that v is constant on the connected components of K_{r} ; then f_{s}(u) = 0 for every s ∈]0, r] and

lim sup_{r→0} inf\{f_{r}(v) | v ∈ C(\overline{Ω}) ∥u − v∥_{L^{∞}(Ω)} < ε\} ≤ 0 .  (2.2)

Since (2.2) holds for every ε > 0 it follows Γ- lim sup_{r→0} f_{r}(u) ≤ 0 ■

Corollary 2.21. - If K is totally disconnected then dim_{C} K = 0 . PROOF. - By the previous proposition f^{(α)}(u) = 0 ∀u ∈ C(\overline{Ω}) ∀α ∈ ]0, N] ■

2.6 Smooth curves in ℝ^{2}

In this section we prove the conjecture for a simple regular curve in ℝ^{2} .

Definition. - We call curve in ℝ^{2} a function γ : [a, b] → ℝ^{2} , γ(t) = (x(t), y(t)) .

We call trace of γ the set K(γ) = γ([a, b]) .

The curve γ is said to be simple if it is injective; it is said to be regular if γ ∈ C^{1}([a, b]) (that is x, y ∈ C^{1}([a, b])) and ∥γ′(t)∥ > 0 ∀t ∈ [a, b] .

Remark 11. - If γ is a regular curve then there exist m, M ∈]0, +∞[ so that m ≤ ∥γ′(t)∥ ≤ M ∀t ∈ [a, b] .

Notation. - We call ℓ(γ) the length of K(γ) .

Remark 12. - It is well known that ℓ(γ) = ∫_{a}^{b} ∥γ′(t)∥dt .

Definition. - Let γ_{1} : [a, b] → ℝ^{2} , γ_{2} : [c, d] → ℝ^{2} be two regular curves in ℝ^{2} . We say that γ_{1}, γ_{2} are equivalent if there exists a function φ : [a, b] → [c, d] so that:

1. γ_{1}(t) = γ_{2}(φ(t)) ∀t ∈ [a, b] ;

2. φ ∈ C^{1}([a, b]) ;

3. φ(a) = c , φ(b) = d .

We will refer to φ as a regular change of parametrization relative to γ_{1} and γ_{2} .

Remark 13. - It is clear that in the previous definition φ′(t) > 0 ∀t ∈ [a, b], since γ′_{2}(φ(t))φ′(t) = γ′_{1}(t) ≠ 0 ∀t ∈ [a, b] and φ(a) = c , φ(b) = d .

Remark 14. - It is well known that the equivalence of regular curves is a reflexive, symmetric and transitive relation. In particular, if γ_{1} and γ_{2} are equivalent regular curves is ℝ^{2} and φ is a regular change of parametrization relative to γ_{1} and γ_{2} , then φ^{−1} is a regular change of parametrization relative to γ_{2} and γ_{1} .

Moreover K(γ_{1}) = K(γ_{2}) .

2.6. SMOOTH CURVES IN ℝ^{2} 67

Remark 15. - Let γ : [a, b] → ℝ^{2} be a regular curve in ℝ^{2} , ℓ = ℓ(γ) and p, q ∈ ℝ , q > 0 . Then there exists a regular curve γ_{1} : [p, p + qℓ] → ℝ^{2} equivalent to γ and such that ∥γ′_{1}(t)∥ = 1/q ∀t ∈ [p, p + qℓ] .

To prove this, let ψ : [a, b] → [p, p + qℓ] ψ(t) = p + q ∫_{a}^{t} ∥γ′(z)∥dz , then ψ ∈ C^{1}([a, b]) , ψ′(t) = q∥γ′(t)∥ > 0 ∀t ∈ [a, b] , ψ(t) = p ⟺ t = a , ψ(t) = p + qℓ ⟺ t = b .

Let φ = ψ^{−1} : φ ∈ C^{1}([p, p + ql]) , φ(p) = a , φ(p + ql) = b and φ′(t) = 1/(q∥γ′(φ(t))∥) > 0 ∀t ∈ [p, p + ql] .

The curve γ_{1} = γ∘φ is regular and equivalent to γ ; moreover ∥γ′_{1}(t)∥ = ∥φ′(t)γ′(φ(t))∥ = 1/q .

It is easy to see that if γ_{2} : [c, d] → ℝ^{2} is a regular curve equivalent to γ and η : [a, b] → [c, d] is a regular change of parametrization relative to γ and γ_{2} , then ∫_{a}^{t} ∥γ′(z)∥dz = ∫_{c}^{η(t)} ∥γ′_{2}(z)∥dz and so γ_{1} is invariant under regular changes of parametrization. In particular ℓ(γ_{2}) = ℓ(γ_{1}) = ℓ .

Finally, if γ ∈ C^{k}([a, b]) then γ_{1} ∈ C^{k}([p, p + ql]) .

Definition. - Let γ be a simple regular curve in ℝ^{2} and let us suppose that γ ∈ C^{2}([a, b]) . Let γ_{1} : [0, ℓ(γ)] → ℝ^{2} be a regular curve equivalent to γ and so that ∥γ′_{1}(t)∥ = 1 ∀t ∈ [0, ℓ(γ)] . We call curvature of γ at the point γ(t) the number c_{γ}(t) = ∥γ′′_{1}(ψ(t))∥ ∀t ∈ [a, b] , where ψ is as in the previous remark (that is γ = γ_{1} ∘ ψ).

If c_{γ}(t) ≠ 0 , the number r_{γ}(t) = 1/c_{γ}(t) is the curvature radius of γ at the point γ(t) .

We set R_{γ} = 1/∥c_{γ}∥_{L^{∞}(a,b)} if γ is not a segment, otherwise we set R_{γ} = +∞ .

The vector N_{γ}(t) = \frac{1}{∥γ′(t)∥}(−y′(t), x′(t)) is called normal unit vector to γ at the point γ(t) .

If r ∈]0, R_{γ}[ we set T_{γ} : [a, b] × [−r, r] → ℝ^{2} T_{γ}(t, s) = γ(t) + sN_{γ}(t) .

Lemma 2.3. - Let γ ∈ C^{2}([0, 1]) be a simple regular curve in ℝ^{2} such that ∥γ′(t)∥ = ℓ(γ) ∀t ∈ [0, 1] and let r ∈]0, R_{γ}[ be so that K(γ)_{r} is simply connected. Then the following facts hold:

1. the function T_{γ} is injective;

2. for all (t, s) ∈ [0, 1] × [−r, r] ,

det ∇T_{γ}(t, s) = ℓ(γ) + \frac{s}{ℓ(γ)}∥γ′′(t)∥ > 0 ;

3. for every [a, b] ⊆ [0, 1]

L^{2}(K(γ|_{[a,b]})_{r}) − πr^{2} = ∫_{−r}^{r} ∫_{a}^{b} det ∇T_{γ}(t, s)dtds =

2rℓ(γ|_{[a,b]}) = 2rℓ(γ)(b − a) ;

in particular L^{2}(K(γ)_{r}) = 2rℓ(γ) + πr^{2} ;

4. for every u : [0, 1] → ℝ L^{1}-measurable

2rℓ(γ) ∫_{0}^{1} u(t)dt = ∫_{−r}^{r} ∫_{0}^{1} u(t) det ∇T_{γ}(t, s)dtds .

PROOF. - For simplicity we set K = K(γ) and ℓ = ℓ(γ) ; moreover we suppose that γ is not a segment, because otherwise the statement is obvious.

The function T_{γ} is injective because r < R_{γ} and K_{r} is simply connected.

Now we compute det ∇T_{γ}(t, s) . Let η : [0, 1] → ℝ^{3} η(t) = (x(t), y(t), 0) and N_{η}(t) = \frac{1}{ℓ}(−y′(t), x′(t), 0) ; for every t ∈ [0, 1] , η′(t) and N′_{η}(t) are perpendicular to N_{η}(t) ; moreover η′(t), N_{η}(t), N′_{η}(t) are perpendicular to (0, 0, 1) ; then (here × stays for the outer product of vectors)

det ∇T_{γ}(t, s) = dz((η′(t)+sN′_{η}(t))×N_{η}(t)) = ∥η′(t)∥∥N_{η}(t)∥+s∥N′_{η}(t)∥∥N_{η}(t)∥ =

ℓ + s ∥\frac{d}{dt} (\frac{1}{ℓ}(−y′(t), x′(t)))∥ = ℓ + \frac{s}{ℓ}∥(−y′′(t), x′′(t))∥ = ℓ + \frac{s}{ℓ}∥γ′′(t)∥ .

To compute the curvature of γ we have ψ(t) = ∫_{0}^{t} ∥γ′(z)∥dz = ℓt and then γ_{1}(σ) = γ(ψ^{−1}(σ)) = γ(σ/ℓ) , γ′_{1}(σ) = \frac{1}{ℓ}γ′(σ/ℓ) , γ′′_{1}(σ) = \frac{1}{ℓ^{2}} γ′′(σ/ℓ) and

c_{γ}(t) = ∥γ′′_{1}(ψ(t))∥ = \frac{1}{ℓ^{2}} ∥γ′′ (\frac{1}{ℓ}ψ(t))∥ = \frac{1}{ℓ^{2}} ∥γ′′(t)∥ .  (2.3)

Since c_{γ} ≤ 1/R_{γ} , we have −\frac{R_{γ}}{ℓ} ∥γ′′(t)∥ ≥ −ℓ , from which

det ∇T_{γ}(t, s) ≥ ℓ − \frac{r}{ℓ}∥γ′′(t)∥ > ℓ − \frac{R_{γ}}{ℓ} ∥γ′′(t)∥ ≥ 0 .

Then ∀[a, b] ⊆ [0, 1] we have

L^{2}(K(γ|_{[a,b]})_{r}) − πr^{2} = L^{2}(T_{γ}([a, b] × [−r, r])) = ∫_{−r}^{r} ∫_{a}^{b} det ∇T_{γ}(t, s)dtds .

By Fubini’s theorem

L^{2}(K(γ|_{[a,b]})_{r}) − πr^{2} = ∫_{a}^{b} ∫_{−r}^{r} det ∇T_{γ}(t, s)dsdt =

2rℓ(b − a) = 2r ∫_{a}^{b} ∥γ′(t)∥dt = 2rℓ(γ|_{[a,b]}) .

Moreover, if u : [0, 1] → ℝ is L^{1}-measurable, by Tonelli’s theorem

∫_{−r}^{r} u(t) ∫_{0}^{1} det ∇T_{γ}(t, s)dtds = ∫_{0}^{1} u(t) ∫_{−r}^{r} (ℓ + \frac{s}{ℓ}∥γ′′(t)∥) dsdt =

2rℓ ∫_{0}^{1} u(t)dt ■

Proposition 2.22. - Let γ ∈ C^{3}([a, b]) be a simple regular curve in ℝ^{2} , with K(γ) ⊆ Ω .

If K = K(γ) then

f^{(1)}(u) = \{ \begin{matrix}2 ∫_{a}^{b} \frac{1}{∥γ′(t)∥} (\frac{d}{dt}u(γ(t)))^{2} dt & if u ∘ γ ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} .

PROOF. - We suppose that K(γ) is not a segment because this case will be treated in paragraph 2.7.

To simplify the notation we set K = K(γ) , ℓ = ℓ(γ) , f_{r} = f_{r}^{(1)} and f = f^{(1)} .

It is clear that the functions f_{r} depend only on K and not on its parametrization; we prove that so does f .

Let γ_{1} : [a, b] → ℝ^{2} and γ_{2} : [c, d] → ℝ^{2} be two equivalent curves and let φ : [a, b] → [c, d] be a regular change of parametrization relative to γ_{1} and γ_{2}. Then

∫_{a}^{b} \frac{1}{∥γ′_{1}(t)∥} (\frac{d}{dt}u(γ_{1}(t)))^{2} dt =

∫_{a}^{b} \frac{1}{∥φ′(t)γ′_{2}(φ(t))∥}(∇u(γ_{2}(φ(t))), φ′(t)γ′_{2}(φ(t)))_{ℝ^{2}}^{2}dt =

∫_{c}^{d} \frac{1}{∥γ′_{2}(s)∥} (\frac{d}{ds}u(γ_{2}(s)))^{2} ds

and so it is not restrictive to suppose a = 0 , b = 1 and ∥γ′(t)∥ = ℓ ∀t ∈ [0, 1] .

Let (r_{n}) be a sequence of positive real numbers such that R_{γ} > r_{n} > r_{n+1} > 0 ∀n ∈ ℕ^{∗} and lim_{n} r_{n} = 0 and let u ∈ C(\overline{Ω}) ; we prove that f(u) ≤ lim inf_{n} f_{r_{n}}(u_{n}) for every sequence (u_{n}) in C(\overline{Ω}) uniformly convergent to u .

First we suppose that u ∘ γ ∈ H^{1}(0, 1) ; and u_{n} ∈ H^{1}(K_{r_{n}}^{∘}) ∀n ∈ ℕ^{∗}. Let ε > 0 : we set the following. φ ∈ C_{c}^{∞}(]0, 1[) is such that ∥φ −

(u ∘ γ)′∥_{L^{2}(0,1)} < ε . δ > 0 is such that |u(x_{1}, y_{1}) − u(x_{2}, y_{2})| < ε/3 for all (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Ω with ∥(x_{1}, y_{1}) − (x_{2}, y_{2})∥ < δ and n′_{ε} ∈ ℕ^{∗} is so that r_{n′_{ε}} < δ . M = ∥\frac{∂}{∂t}(φ det ∇T_{γ})∥_{L^{∞}(]0,1[×]−r_{1},r_{1}[)} and n′′_{ε} ∈ ℕ^{∗} is so that ∥u_{n} − u∥_{L^{∞}(0,1)} < ε/(M + 3) ∀n ∈ ℕ^{∗} with n > n′′_{ε}.

Now, let n_{ε} = max{n′_{ε}, n′′_{ε}} and n ∈ ℕ^{∗} , n > n_{ε} : we have

f(u) = \frac{2}{ℓ} |(\frac{d(u ∘ γ)}{dt}, \frac{d(u ∘ γ)}{dt})_{L^{2}(0,1)}| ≤

\frac{2}{ℓ} |(\frac{d(u ∘ γ)}{dt}, \frac{d(u ∘ γ)}{dt} − φ)_{L^{2}(0,1)}| + \frac{2}{ℓ} |(\frac{d(u ∘ γ)}{dt}, φ)_{L^{2}(0,1)}| ≤

\frac{2}{ℓ} ∥\frac{d(u ∘ γ)}{dt}∥_{L^{2}(0,1)} ∥\frac{d(u ∘ γ)}{dt} − φ∥_{L^{2}(0,1)} + \frac{2}{ℓ} |∫_{0}^{1} (\frac{d}{dt}u(γ(t))) φ(t)dt| .

By Lemma 2.3

f(u) < \frac{1}{r_{n}ℓ^{2}} |∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} (\frac{d}{dt}u(γ(t))) φ(t) det ∇T_{γ}(t, s)dtds| + ε\sqrt{\frac{2}{ℓ}f(u)} =

\frac{1}{r_{n}ℓ^{2}} |∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} u(γ(t))\frac{∂}{∂t}(φ(t) det ∇T_{γ}(t, s))dtds| + ε\sqrt{\frac{2}{ℓ}f(u)} ≤

\frac{1}{r_{n}ℓ^{2}} |∫_{−r_{n}}^{r_{n}} ∫_{0}^{1}(u(γ(t)) − u_{n}(γ(t)))\frac{∂}{∂t}(φ(t) det ∇T_{γ}(t, s))dtds| +

\frac{1}{r_{n}ℓ^{2}} |∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} u_{n}(γ(t))\frac{∂}{∂t}(φ(t) det ∇T_{γ}(t, s))dtds| + ε\sqrt{\frac{2}{ℓ}f(u)} .

Let t ∈ [0, 1] and s ∈ [−r_{n}, r_{n}] : |s| ≤ r_{n} < δ and then

|u_{n}(γ(t)) − u_{n}(γ(t) + sN_{γ}(t))| ≤ |u_{n}(γ(t)) − u(γ(t))|+

|u(γ(t)) − u(γ(t) + sN_{γ}(t))| + |u(γ(t) + sN_{γ}(t)) − u_{n}(γ(t) + sN_{γ}(t))| <

2∥u_{n} − u∥_{L^{∞}(Ω)} + ε/3 < ε .

It follows

|\frac{1}{r_{n}ℓ^{2}} |∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} u_{n}(γ(t))\frac{∂}{∂t}(φ(t) det ∇T_{γ}(t, s))dtds| +

2.6. SMOOTH CURVES IN ℝ^{2} 71

−\frac{1}{r_{n}ℓ^{2}} |∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} u_{n}(γ(t) + sN_{γ}(t))\frac{∂}{∂t}(φ(t) det ∇T_{γ}(t, s))dtds| | < 2\frac{M}{ℓ^{2}} ε

and then

f(u) < \frac{1}{r_{n}ℓ^{2}} |∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} u_{n}(γ(t) + sN_{γ}(t))\frac{∂}{∂t}(φ(t) det ∇T_{γ}(t, s))dtds| +

(2\frac{M}{ℓ^{2}} + \sqrt{\frac{2}{ℓ}f(u)}) ε .

We set v_{n}(t, s) = u_{n}(γ(t) + sN_{γ}(t)) and by integrating by parts it follows

f(u) < \frac{1}{r_{n}ℓ^{2}} |∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} \frac{∂v_{n}}{∂t} (t, s)φ(t) det ∇T_{γ}(t, s)dtds|+(2\frac{M}{ℓ^{2}} + \sqrt{\frac{2}{ℓ}f(u)}) ε .

By the Schwarz’s inequality we have

f(u) < \frac{1}{r_{n}ℓ^{2}} ∥\frac{∂v_{n}}{∂t} \sqrt{det ∇T_{γ}}∥_{L^{2}(]0,1[×]−r_{n},r_{n}[)} ∥φ\sqrt{det ∇T_{γ}}∥_{L^{2}(]0,1[×]−r_{n},r_{n}[)} +

(2\frac{M}{ℓ^{2}} + \sqrt{\frac{2}{ℓ}f(u)}) ε .

We set g_{n}(v_{n}) = \frac{1}{r_{n}ℓ^{2}} ∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} \frac{∂v_{n}}{∂t} (t, s)^{2} det ∇T_{γ}(t, s)dtds and by Lemma 2.3 it follows

f(u) < \sqrt{g_{n}(v_{n})}\sqrt{\frac{2}{ℓ} ∫_{0}^{1} φ(t)^{2}dt} + (2\frac{M}{ℓ^{2}} + \sqrt{\frac{2}{ℓ}f(u)}) ε ≤

\sqrt{g_{n}(v_{n})}\sqrt{\frac{2}{ℓ}} (∥φ − \frac{d(u ∘ γ)}{dt}∥_{L^{2}(0,1)} + ∥\frac{d(u ∘ γ)}{dt}∥_{L^{2}(0,1)}) +

(2\frac{M}{ℓ^{2}} + \sqrt{\frac{2}{ℓ}f(u)}) ε <

\sqrt{g_{n}(v_{n})} (\sqrt{\frac{2}{ℓ}}ε + \sqrt{f(u)}) + (2\frac{M}{ℓ^{2}} + \sqrt{\frac{2}{ℓ}f(u)}) ε .

Now if n → +∞ , we have

f(u) ≤ (\sqrt{\frac{2}{ℓ}}ε + \sqrt{f(u)}) lim inf_{n} \sqrt{g_{n}(v_{n})} + (2\frac{M}{ℓ^{2}} + \sqrt{\frac{2}{ℓ}f(u)}) ε

and if ε → 0 ,

f(u) ≤ \sqrt{f(u)} lim inf_{n} \sqrt{g_{n}(v_{n})} ,

from which

f(u) ≤ lim inf_{n} g_{n}(v_{n}) .

Now we replace g_{n}(v_{n}) with f_{r_{n}}(u_{n}) . For every n ∈ ℕ^{∗}

\frac{∂v_{n}}{∂t} (t, s)^{2} = (∇u_{n}(γ(t) + sN_{γ}(t)), γ′(t) + sN′_{γ}(t))_{ℝ^{2}}^{2} ≤

∥∇u_{n}(γ(t) + sN_{γ}(t))∥^{2}∥γ′(t) + sN′_{γ}(t)∥^{2}

and by (2.3),

∥γ′(t) + sN′_{γ}(t)∥ ≤ ℓ + \frac{|s|}{ℓ}∥γ′′(t)∥ ≤ ℓ(1 + r_{n}c_{γ}(t)) ≤ ℓ (1 + \frac{r_{n}}{R_{γ}} ) .

Let ε > 0 : there exists n_{ε} ∈ ℕ^{∗} such that for every n ∈ ℕ^{∗} with n > n_{ε} we have f(u) < g_{n}(v_{n}) + ε .

Then, for every n > n_{ε} ,

f(u) < \frac{1}{r_{n}} (1 + \frac{r_{n}}{R_{γ}} )^{2} ∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} ∥∇u_{n}(γ(t)+sN_{γ}(t))∥^{2} det ∇T_{γ}(t, s)dtds+ε ≤

(1 + \frac{r_{n}}{R_{γ}} )^{2} f_{r_{n}}(u_{n}) + ε .

As before, if n → +∞ and ε → 0 , we obtain f(u) ≤ lim inf_{n} f_{r_{n}}(u_{n}) . Now we suppose u ∘ γ ∉ H^{1}(0, 1) . Let ε > 0 and φ ∈ C_{c}^{∞}(]0, 1[) be so that ∥φ∥_{L^{2}(0,1)} ≤ 1 ; as before

we set δ > 0 such that |u(x_{1}, y_{1}) − u(x_{2}, y_{2})| < ε/3 ∀(x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Ω with ∥(x_{1}, y_{1}) − (x_{2}, y_{2})∥ < δ and n′_{ε} ∈ ℕ^{∗} so that r_{n′_{ε}} < δ . M = ∥\frac{∂}{∂t}(φ det ∇T_{γ})∥_{L^{∞}(]0,1[×]−r_{1},r_{1}[)} and n′′_{ε} ∈ ℕ^{∗} is so that ∥u_{n} − u∥_{L^{∞}(0,1)} < ε/(M + 3) ∀n ∈ ℕ^{∗} with n > n′′_{ε} . Moreover n′′′_{ε} is so that ∥u_{n} − u∥ < ε/∥φ′∥_{L^{∞}(0,1)} ∀n ∈ ℕ^{∗} with n > n′′′_{ε} .

Now, let n_{ε} = max{n′_{ε}, n′′_{ε}, n′′′_{ε}} and n ∈ ℕ^{∗} , n > n_{ε} : we have

|∫_{0}^{1} u(γ(t))φ′(t)dt| ≤

|∫_{0}^{1}(u(γ(t)) − u_{n}(γ(t)))φ′(t)dt| + |∫_{0}^{1} u_{n}(γ(t))φ′(t)dt| ≤

|∫_{0}^{1} \frac{d}{dt}u_{n}(γ(t))φ(t)dt| + ∥u_{n} − u∥_{L^{∞}(0,1)}∥φ′∥_{L^{∞}(0,1)} <

\frac{1}{2r_{n}ℓ} |∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} \frac{d}{dt}u_{n}(γ(t))φ(t) det ∇T_{γ}(t, s)dtds| + ε .

By using the same arguments of the previous proof and the fact that ∥φ∥_{L^{2}(0,1)} ≤ 1 we can show that

|∫_{0}^{1} u(γ(t))φ′(t)dt| < \sqrt{\frac{ℓ}{2}g_{n}(v_{n})} + (\frac{M}{ℓ} + 1) ε ,

where g_{n} and v_{n} are defined in the same way.

Since

|\frac{∂v_{n}}{∂t} (t, s)| ≤ ∥∇u_{n}(γ(t) + sN_{γ}(t))∥ℓ (1 + \frac{r_{n}}{R_{γ}} ) ,

we have

|∫_{0}^{1} u(γ(t))φ′(t)dt| < \sqrt{\frac{ℓ}{2} (1 + \frac{r_{n}}{R_{γ}} )^{2} f_{r_{n}}(u_{n})} + (\frac{M}{ℓ} + 1) ε .

By letting n → +∞ and ε → 0 we have

|∫_{0}^{1} u(γ(t))φ′(t)dt| ≤ \sqrt{\frac{ℓ}{2} lim inf_{n} f_{r_{n}}(u_{n})}

∀φ ∈ C_{c}^{∞}(]0, 1[) with ∥φ∥_{L^{2}(0,1)} ≤ 1 .

We define a linear functional g : C_{c}^{∞}(]0, 1[) → ℝ < g, φ >= ∫_{0}^{1} u(γ(t))φ′(t)dt .

Let us suppose that sup{| ∫_{0}^{1} u(γ(t))φ′(t)dt| | φ ∈ C_{c}^{∞}(]0, 1[) ∥φ∥_{L^{2}(0,1)} ≤ 1} = C < +∞ ; then g is continuous on C_{c}^{∞}(]0, 1[) with respect to the L^{2}- norm. Since g is linear it is uniformly continuous and by Theorem 4.1, there exists h : L^{2}(0, 1) → ℝ uniformly continuous such that h(φ) =< g, φ > for every φ ∈ C_{c}^{∞}(]0, 1[) . Since h is continuous on L^{2}(]0, 1[) , C_{C}^{∞}(]0, 1[) is dense in L^{2}(]0, 1[) and g is linear it is easy to see that h is linear. Then there exists w ∈ L^{2}(0, 1) such that < h, v >= ∫_{0}^{1} w(t)v(t)dt ∀v ∈ L^{2}(0, 1) and then

∫_{0}^{1} u(γ(t))φ′(t)dt =< g, φ >=< h, φ >= ∫_{0}^{1} w(t)φ(t)dt ∀φ ∈ C_{c}^{∞}(]0, 1[) ;

that is u ∘ γ ∈ H^{1}(0, 1) and \frac{d(u∘γ)}{dt} = −w . Then C = +∞ and lim_{n} f_{r_{n}}(u_{n}) = +∞ .

To complete the proof of the proposition we have to show that there exists a sequence (u_{n}) in C(\overline{Ω}) that is uniformly convergent to u and such that lim_{n} f_{r_{n}}(u_{n}) = f(u) .

We construct this sequence in the following way. Let n ∈ ℕ^{∗} n > 2 ; for simplicity we set K_{n} = K_{r_{n}} . We define 3δ_{n} = min{r_{n−1} −

r_{n}, r_{n−2} − r_{n−1}} . The set K_{n−2}\ K_{n−1}^{∘} is compact and contained into its open δ_{n}-neighborhood ⋃_{(x,y)∈K_{n−2}\K_{n−1}^{∘}} D_{ℝ^{2}}((x, y), δ_{n}). Then there exist

k_{n} ∈ ℕ^{∗} and (x_{1}^{(n)}, y_{1}^{(n)}), (x_{2}^{(n)}, y_{2}^{(n)}), ..., (x_{k_{n}}^{(n)}, y_{k_{n}}^{(n)}) ∈ K_{n−2}\ K_{n−1}^{∘} so that

K_{n−2}\ K_{n−1}^{∘}⊆ ⋃_{i=1}^{k_{n}} D_{ℝ^{2}}((x_{i}^{(n)}, y_{i}^{(n)}), δ_{n}) .

By the lemma on the partition of unity there exist φ_{0}^{(n)}, φ_{1}^{(n)}, ..., φ_{k_{n}}^{(n)} ∈ C^{∞}(ℝ^{2}) so that ∀(x, y) ∈ ℝ^{2} ∀i ∈ ℕ i ≤ k_{n} we have 0 ≤ φ_{i}^{(n)}(x, y) ≤ 1 and ∑_{j=0}^{k_{n}} φ_{j}^{(n)}(x, y) = 1 ; moreover, for 1 ≤ i ≤ k_{n} , the support of φ_{i}^{(n)} is contained in D_{ℝ^{2}}((x_{i}^{(n)}, y_{i}^{(n)}), δ_{n}) and the support of φ_{0}^{(n)} is

contained in K_{n−1}^{∘} ⋃(ℝ^{2}\K_{n−2}) . We define ψ_{n}, ω_{n} : ℝ^{2} → ℝ ψ_{n}(x, y) =

φ_{0}^{(n)}(x, y)χ_{K_{n−2}}(x, y) , ω_{n}(x, y) = φ_{0}^{(n)}(x, y)(1 − χ_{K_{n−2}}(x, y)) .

Now, let π_{t} : ℝ^{2} → ℝ π_{t}(t, s) = t : the function u_{n} : \overline{Ω} → ℝ is defined by

u_{n}(x, y) = u(γ(π_{t}(T_{γ}^{−1}(x, y))))ψ_{n}(x, y) + (ω_{n}(x, y) + ∑_{i=1}^{k_{n}} φ_{i}^{(n)}(x, y))u(x, y)

with the convention that T_{γ}^{−1}(x, y) = (0, 0) if (x, y) ∉ K_{1} or (x, y) ∈ B_{ℝ^{2}}(γ(0), r_{1})\T_{γ}([0, 1] × [−r_{1}, r_{1}]) and T_{γ}^{−1}(x, y) = (1, 0) if (x, y) ∈ B_{ℝ^{2}}(γ(1), r_{1})\T_{γ}([0, 1] × [−r_{1}, r_{1}]) .

Then the following facts hold:

1. u_{n} ∈ C(\overline{Ω}) ∀n ∈ ℕ^{∗} n > 2 ;

2. u_{n}(x, y) = u(γ(π_{t}(T_{γ}^{−1}(x, y)))) ∀(x, y) ∈ K_{n} ∀n ∈ ℕ^{∗} n > 2 ; 3. lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 ;

4. lim_{n} f_{r_{n}}(u_{n}) = f(u) .

The first claim is obvious; indeed ψ_{n}, ω_{n} ∈ C^{∞}(\overline{Ω}) , since their support

is contained in K_{n−1}^{∘} ⋃(ℝ^{2}\K_{n−2}) .

Let n ∈ ℕ^{∗}\{1, 2} ; for every i ∈ ℕ^{∗} i ≤ k_{n} we have D_{ℝ^{2}}((x_{i}^{(n)}, y_{i}^{(n)}), δ_{n}) ⋂ K_{n} = ∅ and then ∀(x, y) ∈ K_{n} φ_{i}^{(n)}(x, y) = ω_{n}(x, y) = 0 . It follows that ψ_{n}(x, y) = 1 and the second claim is proven.

To prove the third claim, let ε > 0 and δ > 0 , n_{ε} ∈ ℕ^{∗} be so that |u(x_{1}, y_{1})−u(x_{2}, y_{2})| < ε ∀(x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Ω with ∥(x_{1}, y_{1})−(x_{2}, y_{2})∥ < δ and r_{n−2} < δ ∀n ∈ ℕ^{∗} with n > n_{ε} .

Then |u_{n}(x, y) − u(x, y)| < ε ∀(x, y) ∈ Ω ∀n > n_{ε} .

Indeed, if (x, y) ∈ Ω\K_{n−2} then u_{n}(x, y) = u(x, y) because ψ_{n}(x, y) = 0. Otherwise, let (t, s) = T_{γ}^{−1}(x, y) , we have |s| ≤ r_{n−2} < δ , from which

|u_{n}(x, y) − u(x, y)| ≤ |ψ_{n}(x, y)u(γ(π_{t}(t, s))) − ψ_{n}(x, y)u(x, y)|+

|ψ_{n}(x, y)u(x, y) + (ω_{n}(x, y) + ∑_{i=1}^{k_{n}} φ_{i}^{(n)}(x, y))u(x, y) − u(x, y)| =

ψ_{n}(x, y)|u(γ(t)) − u(T_{γ}(t, s))| + |(∑_{i=0}^{k_{n}} φ_{n}(x, y) − 1)u(x, y)| ≤

|u(γ(t)) − u(γ(t) + sN_{γ}(t))| < ε .

Now we prove the fourth claim. For n ∈ ℕ^{∗} n > 2 ,

f_{r_{n}}(u_{n}) = \frac{1}{r_{n}} ∫_{T_{γ}([0,1]×[−r_{n},r_{n}])} ∥∇u_{n}(x, y)∥^{2}dxdy

because u_{n} is constant on K_{n} \ T_{γ}([0, 1] × [−r_{n}, r_{n}]) .

We set B = max{∥γ′′(t)∥ | t ∈ [0, 1]} and v : [0, 1] × [−r_{1}, r_{1}] → ℝ v(t, s) = u(γ(t)) ; it follows that u_{n}(x, y) = v(T_{γ}^{−1}(x, y)) ∀(x, y) ∈ T_{γ}([0, 1] × [−r_{n}, r_{n}]) ∀n ∈ ℕ^{∗} n > 2 . We also suppose that r_{n} < ℓ^{2}/B .

With these assumptions we have

f_{r_{n}}(u_{n}) = \frac{1}{r_{n}} ∫_{T_{γ}([0,1]×[−r_{n},r_{n}])} ∥∇v(T_{γ}^{−1}(x, y))∇T_{γ}^{−1}(x, y)∥^{2}dxdy =

\frac{1}{r_{n}} ∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} ∥∇v(t, s)∇T_{γ}(t, s)^{−1}∥^{2} det ∇T_{γ}(t, s)dtds =

\frac{1}{r_{n}} ∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} \frac{1}{det ∇T_{γ}(t, s)} (\frac{d}{dt}u(γ(t)))^{2} dtds .

Then

|f_{r_{n}}(u_{n}) − f(u)| =

|\frac{1}{r_{n}} ∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} \frac{1}{ℓ + \frac{s}{ℓ}∥γ′′(t)∥} (\frac{d}{dt}u(γ(t)))^{2} dtds − \frac{1}{2r_{n}} ∫_{−r_{n}}^{r_{n}} f(u)ds| =

|\frac{1}{r_{n}} ∫_{−r_{n}}^{r_{n}} ∫_{0}^{1} (\frac{1}{ℓ + \frac{s}{ℓ}∥γ′′(t)∥} − \frac{1}{ℓ}) (\frac{d}{dt}u(γ(t)))^{2} dtds| ≤

∫_{0}^{1} |\frac{1}{r_{n}} ∫_{−r_{n}}^{r_{n}} (\frac{1}{ℓ + \frac{s}{ℓ}∥γ′′(t)∥} − \frac{1}{ℓ}) ds| (\frac{d}{dt}u(γ(t)))^{2} dt .

Now,

|\frac{1}{r_{n}} ∫_{−r_{n}}^{r_{n}} (\frac{1}{ℓ + \frac{s}{ℓ}∥γ′′(t)∥} − \frac{1}{ℓ}) ds| ≤

\frac{∥γ′′(t)∥}{ℓr_{n}} (|∫_{−r_{n}}^{0} \frac{s}{ℓ^{2} + s∥γ′′(t)∥}ds| + |∫_{0}^{r_{n}} \frac{s}{ℓ^{2} + s∥γ′′(t)∥}ds|) <

\frac{∥γ′′(t)∥}{ℓr_{n}} (∫_{0}^{r_{n}} \frac{s}{ℓ^{2} − r_{n}∥γ′′(t)∥}ds + ∫_{0}^{r_{n}} \frac{s}{ℓ^{2}} ds) =

\frac{∥γ′′(t)∥}{2ℓ} (\frac{1}{ℓ^{2} − r_{n}∥γ′′(t)∥} + \frac{1}{ℓ^{2}} ) r_{n} ≤ \frac{B}{2ℓ} (\frac{1}{ℓ^{2} − Br_{n}} + \frac{1}{ℓ^{2}} ) r_{n} .

Then

|f_{r_{n}}(u_{n}) − f(u)| ≤ \frac{B}{2ℓ} (\frac{1}{ℓ^{2} − Br_{n}} + \frac{1}{ℓ^{2}} ) r_{n}f(u) ■

Corollary 2.23. - If γ ∈ C^{3}([a, b]) is a simple regular curve in ℝ^{2} , then dim_{C} K(γ) = 1 .

Remark 16. - If c ∈ [a, b] and γ ∈ C([a, b]) , γ ∈ C^{3}([a, c]) , γ ∈ C^{3}([c, b]) is simple and regular on [a, c] and on [c, b] , then Proposition 2.22 and Corollary 2.23 still hold. We only give a sketch of the proof.

To prove that f^{(1)}(u) ≤ lim inf_{n} f_{r_{n}}^{(1)}(u_{n}) for every sequence (u_{n}) in C(\overline{Ω}) uniformly convergent to u one has to consider separately

f^{(1)}(u; γ|_{[a,c_{1}]}) = ∫_{a}^{c_{1}} \frac{1}{∥γ′(t)∥} (\frac{d}{dt}u(γ(t)))^{2} dt

and f^{(1)}(u; γ|_{[c_{2},b]}) defined in a similar way, where a < c_{1} < c < c_{2} < b .

By Theorem 4.7, for every ε > 0 there exist c_{1} ∈]a, c[ and c_{2} ∈]c, b[ such that

f^{(1)}(u) < f^{(1)}(u; γ|_{[a,c_{1}]}) + f^{(1)}(u; γ|_{[c_{2},b]}) + ε .

By Proposition 2.22

f^{(1)}(u) ≤ lim inf_{n}(f_{r_{n}}^{(1)}(u_{n}; γ|_{[a,c_{1}]}) + f_{r_{n}}^{(1)}(u; γ|_{[c_{2},b]})) + ε .

Since for r_{n} small enough, the r_{n}-neighborhood of K(γ|_{[a,c_{1}]}) and the one of K(γ|_{[c_{2},b]}) are disjoint and their union is contained in the r_{n}-neighborhood of K(γ) , we have

f^{(1)}(u) ≤ lim inf_{n} f_{r_{n}}^{(1)}(u_{n}) + ε .

The proof of the existence of a sequence (u_{n}) in C(\overline{Ω}) uniformly convergent to u and such that lim_{n} f_{r_{n}}^{(1)}(u_{n}) = f^{(1)}(u) is more delicate.

We can proceed in the following way. We set w_{k}(t) = u(γ(0))+∫_{0}^{t} φ_{k}(s)ds, where φ_{k} ∈ C_{c}^{∞}(]0, 1[) and ∥φ_{k} − (u ∘ γ)′∥_{L^{2}(0,1)} ≤ 1/k . Then w_{k} ∈ C^{∞}([0, 1]) , lim_{k} ∥w_{k} − u ∘ γ∥_{L^{∞}(0,1)} = 0 and lim_{k} ∥w′_{k} − (u ∘ γ)′∥_{L^{2}(0,1)} = 0 .

Now we extend w_{k} to a function v_{k} ∈ C(\overline{Ω}) in the same way we did in the proof of Proposition 2.22, except for those points near γ(c) , where the map T_{γ} is not injective. To define v_{k} in every point of the r_{k}-neighborhood of K(γ) we use Theorem 4.3.

Let q_{k}(x, y) = ∥∇v_{k}(x, y)∥ and M_{k} = ∥q_{k}∥_{L^{∞}((K(γ))_{r_{k}})} and let V_{k} = L^{2}(K(γ)_{r_{k}} \ T_{γ}(([a, c_{k}] ⋃[d_{k}, b]) × [−r_{k}, r_{k}]) where c_{k} ∈]a, c[ and d_{k} ∈]c, b[ are so that L^{2}(T_{γ}([a, c_{k}] × [−r_{k}, r_{k}]) ⋂ T_{γ}([d_{k}, b] × [−r_{k}, r_{k}])) = 0 and this intersection is not empty.

Since lim_{k} V_{k}/r_{k} = 0 there exists n_{1} ∈ ℕ^{∗} so that M_{1}V_{k}/r_{k} ≤ 1/2 for every k ∈ ℕ^{∗} k > n_{1} ; we set u_{1} = u_{2} = ... = u_{n_{1}} = v_{1} . After that, there exists n_{2} ∈ ℕ^{∗} n_{2} > n_{1} so that M_{2}V_{k}/r_{k} ≤ 1/2 for every k ∈ ℕ^{∗} k > n_{2} ; we still set u_{n_{1}+1} = u_{n_{1}+2} = ... = u_{n_{2}} = v_{1} . Let n_{3} ∈ ℕ^{∗} n_{3} > n_{2} be so that M_{3}V_{k}/r_{k} ≤ 1/3 ∀k ∈ ℕ^{∗} k > n_{3} ; we set u_{n_{2}+1} = u_{n_{2}+2} = ... = u_{n_{3}} = v_{2} and so on.

The sequence (u_{n}) so constructed has the desired properties.

2.7 Relations with the immersion of K in ℝ^{N+M}

Let M ∈ ℕ^{∗} , J : ℝ^{N} → ℝ^{N} ×ℝ^{M} , J(x) = (x, 0) and H = J(K) ; moreover Ω′ ⊆ ℝ^{N+M} is open, bounded and so that Ω = Ω′ ⋂(ℝ^{N} × {0}^{M}) .

We study relations between f_{−}^{(α)}, f_{+}^{(α)}, f^{(α)} and g_{−}^{(α)}, g_{+}^{(α)}, g^{(α)} , where g_{−}^{(α)}, g_{+}^{(α)}, g^{(α)}, g_{r}^{(α)} are defined as f_{−}^{(α)}, f_{+}^{(α)}, f^{(α)}, f_{r}^{(α)} with H and Ω′ instead of K and Ω .

Proposition 2.24. - Let α ∈]0, N] and c_{M} = L^{M}(B_{ℝ^{M}}(0, 1)) ; then g_{−}^{(α)}(v) ≤ c_{M}f_{−}^{(α)}(v ∘ J) for every v ∈ C(\overline{Ω′}) .

In particular \underline{dim_{C}}H ≤ \underline{dim_{C}}K . PROOF. - Let v ∈ C(\overline{Ω′}) ; by Proposition 2.9 1) there exists a sequence (r_{n}) converging to 0 in ]0, +∞[ and a sequence (u_{n}) in C(\overline{Ω}) such that lim_{n} ∥u_{n} − v ∘ J∥_{L^{∞}(Ω)} = 0 and f_{−}^{(α)}(v ∘ J) = lim inf_{n} f_{r_{n}}^{(α)}(u_{n}) . We can suppose r_{n} > r_{n+1} for every n ∈ ℕ^{∗} .

Let n ∈ ℕ^{∗} n > 2 , we define 3δ_{n} = min{r_{n−1} − r_{n}, r_{n−2} − r_{n−1}} , C_{n} = \{(x, y) ∈ ℝ^{N} × ℝ^{M} | (x, y) ∈ \overline{Ω′} , r_{n−1} ≤ ∥y∥ ≤ r_{n−2}\} and D_{n} = ℝ^{N} × B_{ℝ^{M}}(0, r_{n−2}) .

The set C_{n} is compact and contained into its open δ_{n}-neighborhood. Then there exist k_{n} ∈ ℕ^{∗} and (x_{1}^{(n)}, y_{1}^{(n)}), (x_{2}^{(n)}, y_{2}^{(n)}), ..., (x_{k_{n}}^{(n)}, y_{k_{n}}^{(n)}) ∈ C_{n} so that C_{n} ⊆ ⋃_{i=1}^{k_{n}} D_{ℝ^{N+M}}((x_{i}^{(n)}, y_{i}^{(n)}), δ_{n}) .

By the lemma on the partition of the unity there exist φ_{0}^{(n)}, φ_{1}^{(n)}, ..., φ_{k_{n}}^{(n)} ∈ C^{∞}(ℝ^{N+M}) so that 0 ≤ φ_{i}^{(n)}(x, y) ≤ 1 and ∑_{j=0}^{k_{n}} φ_{j}^{(n)}(x, y) = 1 for every i ∈ ℕ^{∗} i ≤ k_{n} and for every (x, y) ∈ ℝ^{N} × ℝ^{M} ; moreover, for 1 ≤ i ≤ k_{n} , the support of φ_{i}^{(n)} is contained in D_{ℝ^{N+M}}((x_{i}^{(n)}, y_{i}^{(n)}), δ_{n}) and the support of φ_{0}^{(n)} does not intersect C_{n} . We define ψ_{n}, ω_{n} : ℝ^{N} ×ℝ^{M} → ℝ ψ_{n}(x, y) = φ_{0}^{(n)}(x, y)χ_{D_{n}}(x, y) and ω_{n}(x, y) = φ_{0}^{(n)}(x, y)(1 − χ_{D_{n}}(x, y)) .

We define the function w_{n} : Ω′ → ℝ

w_{n}(x, y) = ψ_{n}(x, y)u_{n}(x) + (ω_{n}(x, y) + ∑_{i=1}^{(k_{n})} φ_{i}^{(n)}(x, y))v(x, y) .

For every n ∈ ℕ^{∗} n > 2 w_{n} ∈ C(\overline{Ω′}) because the support of φ_{0}^{(n)} does not intersect ∂D_{n} . If y ∈ B_{ℝ^{M}}(0, r_{n}) and x ∈ ℝ^{N} is so that (x, y) ∈ Ω′ then, for every i ∈ ℕ^{∗} i ≤ k_{n} , φ_{i}^{(n)}(x, y) = ω_{n}(x, y) = 0 . It follows that ψ_{n}(x, y) = 1 and w_{n}(x, y) = u_{n}(x) . Otherwise, if ∥y∥ > r_{n−2} then ψ_{n}(x, y) = 0 and w_{n}(x, y) = v(x, y) .

2.7. RELATIONS WITH THE IMMERSION OF K IN ℝ^{N+M} 79

Let ε > 0 ; there exists δ > 0 such that |v(x′, y′) − v(x′′, y′′)| < ε/2 for all (x′, y′), (x′′, y′′) ∈ Ω′ such that ∥(x′, y′) − (x′′, y′′)∥ < δ . Moreover there exists n′_{ε} ∈ ℕ^{∗} such that ∥u_{n} − v ∘ J∥_{L^{i}nfty(Ω)} < ε/2 for all n ∈ ℕ^{∗} such that n > n′_{ε} and there exists n′′_{ε} ∈ ℕ^{∗} such that r_{n−2} < δ for all n ∈ ℕ^{∗} with n > n′′_{ε} .

Then, if n > max{n′_{ε}, n′′_{ε}} and (x, y) ∈ Ω′ , ∥y∥ ≤ r_{n−2} we have

|w_{n}(x, y) − v(x, y)| = ψ_{n}(x, y)|u_{n}(x) − v(x, y)| ≤

|u_{n}(x) − v(x, 0)| + |v(x, 0) − v(x, y)| ≤ ∥u_{n} − v ∘ J∥_{L^{∞}(Ω)} + \frac{ε}{2} < ε ,

that is lim_{n} ∥w_{n} − v∥_{L^{∞}(Ω′)} = 0 .

Now, let n ∈ ℕ^{∗} , we have

c_{M}f_{r_{n}}^{(α)}(u_{n}) = c_{M}f_{r_{n}}^{(α)}(w_{n} ∘J) = \frac{1}{r_{n}^{N+M−α}} ∫_{B_{ℝ^{M}} (0,r_{n})} ∫_{K_{rn}} ∥∇(w_{n} ∘J)(x)∥^{2}dxdy

≥ \frac{1}{r_{n}^{N+M−α}} ∫_{H_{rn}} ∑_{i=1}^{N} \frac{∂w_{n}}{∂x_{i}} (x, 0)^{2}dxdy =

\frac{1}{r_{n}^{N+M−α}} ∫_{H_{rn}} ∥∇w_{n}(x, y)∥^{2}dxdy = g_{r_{n}}^{(α)}(w_{n}) .

By Proposition 2.9 1)

g_{−}^{(α)}(v) ≤ lim inf_{n} g_{r_{n}}^{(α)}(w_{n}) ≤ c_{M} lim inf_{n} f_{r_{n}}^{(α)}(u_{n}) = c_{M}f_{−}^{(α)}(v ∘ J) ■

If K^{∘}≠ ∅ and K^{∘} is of class C^{1} then we have a representation of g^{(α)}, with α = dim K .

Proposition 2.25. - If K = \overline{K^{∘}} , K^{∘} is non-empty and of class C^{1} , then

g^{(N)}(v) = c_{M}f^{(N)}(v ∘ J) for every v ∈ C(\overline{Ω′}).

PROOF. - Let v ∈ C(\overline{Ω′}) and let (r_{n}) be a sequence converging to 0 in ]0, +∞[ , so that K_{r_{n}} ⊆ Ω and r_{n} > r_{n+1} ∀n ∈ ℕ^{∗} . We prove that C_{M}f^{(N)}(v ∘ J) ≤ lim inf_{n} g_{r_{n}}^{(N)}(v_{n}) for every sequence (v_{n}) in C(\overline{Ω′}) such that lim_{n} ∥v_{n} − v∥_{L^{∞}(Ω′)} = 0 .

We first suppose that v ∘ J ∈ H^{1}(K^{∘}) and v_{n} ∈ H^{1}(H_{r_{n}}^{∘}) for every n ∈ ℕ^{∗} .

Let ε > 0 ; there exist φ_{1}, φ_{2}, ..., φ_{N} ∈ C_{c}^{∞}(K^{∘}) so that for every

i ∈ ℕ^{∗} i ≤ N ∥φ_{i} − \frac{∂(v∘J)}{∂x_{i}}∥_{L^{2}(K^{∘})} < \frac{ε}{c_{M} \sqrt{N}} .

Let n′_{ε} ∈ ℕ^{∗} be so that c_{M}∥v − v_{n}∥_{L^{∞}(Ω′)}L^{N}(K^{∘}) ∑_{i=1}^{N} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})} < ε

for every n ∈ ℕ^{∗} n > n′_{ε} .

Then for n ∈ ℕ^{∗} n > n′_{ε} we have

c_{M}f^{(N)}(v ∘ J) = c_{M} |∑_{i=1}^{N} (\frac{∂v}{∂x_{i}} , \frac{∂v}{∂x_{i}} )_{L^{2}(K^{∘})}| ≤

c_{M} |∑_{i=1}^{N} (\frac{∂v}{∂x_{i}} , \frac{∂v}{∂x_{i}} − φ_{i})_{L^{2}(K^{∘})}| + c_{M} |∑_{i=1}^{N} (\frac{∂v}{∂x_{i}} , φ_{i})_{L^{2}(K^{∘})}| ≤

c_{M} ∑_{i=1}^{N} |∫_{K^{∘}} v(x, 0)\frac{∂φ_{i}}{∂x_{i}} (x)dx| + c_{M} ∑_{i=1}^{N} ∥\frac{∂v}{∂x_{i}} ∥_{L^{2}(K^{∘})} ∥\frac{∂v}{∂x_{i}} − φ_{i}∥_{L^{2}(K^{∘})} ≤

c_{M} ∑_{i=1}^{N} |∫_{K^{∘}}(v(x, 0) − v_{n}(x, 0))\frac{∂φ_{i}}{∂x_{i}} (x)dx| + c_{M} ∑_{i=1}^{N} |∫_{K^{∘}} v_{n}(x, 0)\frac{∂φ_{i}}{∂x_{i}} (x)dx| +

c_{M} \sqrt{∑_{i=1}^{N} ∥\frac{∂v}{∂x_{i}} ∥_{L^{2}(K^{∘})}^{2}}\sqrt{∑_{i=1}^{N} ∥\frac{∂v}{∂x_{i}} − φ_{i}∥_{L^{2}(K^{∘})}^{2}} <

\frac{1}{r_{n}^{M}} ∑_{i=1}^{N} |∫_{B_{ℝ^{M}} (0,r_{n})} ∫_{K^{∘}} v_{n}(x, 0)\frac{∂φ_{i}}{∂x_{i}} (x)dxdy| + ε + ε\sqrt{f^{(N)}(v ∘ J)} .

Since the function v is uniformly continuous on Ω′ , there exists δ > 0

so that |v(x′, y′) − v(x′′, y′′)| < (3c_{M}L^{N}(K^{∘}) ∑_{i=1}^{n} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})})^{−1}ε for every

(x′, y′), (x′′, y′′) ∈ Ω′ with ∥(x′, y′) − (x′′, y′′)∥ < δ . Moreover there exists n_{δ} ∈ ℕ^{∗} so that r_{n} < δ for every n ∈ ℕ^{∗} n > n_{δ} . Let n′′_{ε} ∈ ℕ^{∗} be so that

∥v_{n}−v∥_{L^{∞}(Ω′)} < (3c_{M}L^{N}(K^{∘}) ∑_{i=1}^{n} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})})^{−1}ε for every n ∈ ℕ^{∗} n > n′′_{ε} .

For n ∈ ℕ^{∗} n > max{n_{δ}, n′′_{ε}} and for every x ∈ K , y ∈ B_{ℝ^{M}}(0, r_{n}) we have

|v_{n}(x, 0) − v_{n}(x, y)| ≤ |v_{n}(x, 0) − v(x, 0)| + |v(x, 0) − v(x, y)|+

|v(x, y) − v_{n}(x, y)| < \frac{ε}{c_{M}L^{N}(K^{∘}) ∑_{i=1}^{n} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})}} .

2.7. RELATIONS WITH THE IMMERSION OF K IN ℝ^{N+M} 81

Let n_{ε} = max{n′_{ε}, n′′_{ε}, n_{δ}} ; for n ∈ ℕ^{∗} n > n_{ε} we have

|\frac{1}{r_{n}^{M}} ∑_{i=1}^{N} |∫_{K^{∘}×B_{ℝ^{M}} (0,r_{n})} v_{n}(x, 0)\frac{∂φ_{i}}{∂x_{i}} (x)dxdy| +

−\frac{1}{r_{n}^{M}} ∑_{i=1}^{N} |∫_{K^{∘}×B_{ℝ^{M}} (0,r_{n})} v_{n}(x, y)\frac{∂φ_{i}}{∂x_{i}} (x)dxdy|| ≤

\frac{1}{r_{n}^{M}} ∑_{i=1}^{N} ∫_{K^{∘}×B_{ℝ^{M}} (0,r_{n})} |v_{n}(x, 0) − v_{n}(x, y)| |\frac{∂φ_{i}}{∂x_{i}} (x)| dxdy < ε .

Then c_{M}f^{(N)}(v ∘ J) <

\frac{1}{r_{n}^{M}} ∑_{i=1}^{N} |∫_{K^{∘}×B_{ℝ^{M}} (0,r_{n})} v_{n}(x, y)\frac{∂φ_{i}}{∂x_{i}} (x)dxdy| + (2 + \sqrt{f^{(N)}(v ∘ J)})ε ≤

\frac{1}{r_{n}^{M}} ∑_{i=1}^{N} ∥\frac{∂v_{n}}{∂x_{i}} ∥_{L^{2}(K^{∘}×B_{ℝ^{M}} (0,r_{n}))} ∥φ_{i}∥_{L^{2}(K^{∘}×B_{ℝ^{M}} (0,r_{n}))} + (2 + \sqrt{f^{(N)}(v ∘ J)})ε ≤

\sqrt{\frac{1}{r_{n}^{M}} ∑_{i=1}^{N} ∥\frac{∂v_{n}}{∂x_{i}} ∥_{L^{2}(K^{∘}×B_{ℝ^{M}} (0,r_{n}))}^{2}}\sqrt{\frac{1}{r_{n}^{M}} ∑_{i=1}^{N} ∫_{K^{∘}×B_{ℝ^{M}} (0,r_{n})} φ_{i}(x)^{2}dxdy}+

(2 + \sqrt{f^{(N)}(v ∘ J)})ε ≤

\sqrt{\frac{1}{r_{n}^{M}} (∑_{i=1}^{N} ∥\frac{∂v_{n}}{∂x_{i}} ∥_{L^{2}(H^{∘}_{rn})}^{2} + ∑_{i=1}^{M} ∥\frac{∂v_{n}}{∂y_{i}} ∥_{L^{2}(H^{∘}_{rn})}^{2})}\sqrt{c_{M} ∑_{i=1}^{N} ∥φ_{i}∥_{L^{2}(K^{∘})}^{2}}+

(2 + \sqrt{f^{(N)}(v ∘ J)})ε ≤

\sqrt{g_{r_{n}}^{(N)}(v_{n})}\sqrt{c_{M} ∑_{i=1}^{N} (∥φ_{i} − \frac{∂v}{∂x_{i}} ∥_{L^{2}(K^{∘})} + ∥\frac{∂v}{∂x_{i}} ∥_{L^{2}(K^{∘})})^{2}}+(2+\sqrt{f^{(N)}(v ∘ J)})ε

< \sqrt{g_{r_{n}}^{(N)}(v_{n})}\sqrt{c_{M} ∑_{i=1}^{N} (ε + ∥\frac{∂v}{∂x_{i}} ∥_{L^{2}(K^{∘})})^{2}} + (2 + \sqrt{f^{(N)}(v ∘ J)})ε .

If n → +∞ , then

c_{M}f^{(N)}(v ∘ J) ≤ \sqrt{lim inf_{n} g_{r_{n}}^{(N)}(v_{n})}\sqrt{c_{M} ∑_{i=1}^{N} (ε + ∥\frac{∂v}{∂x_{i}} ∥_{L^{2}(K^{∘})})^{2}}+

(2 + \sqrt{f^{(N)}(v ∘ J)}ε .

If ε → 0 we have c_{M}f^{(N)}(v ∘ J) ≤ lim inf_{n} g_{r_{n}}^{(N)}(v_{n}) .

Now we suppose v ∘ J ∉ H^{1}(K^{∘}) and v_{n} ∈ H^{1}(H_{r_{n}}^{∘}) for every n ∈ ℕ^{∗} .

Let φ_{1}, φ_{2}, ..., φ_{N} ∈ C_{c}^{∞}(K^{∘}) with ∥φ_{i}∥_{L^{2}(K^{∘})} ≤ 1 for every i ∈ ℕ^{∗} i ≤ N

and let n′_{ε}, n′′_{ε}, n_{δ} ∈ ℕ^{∗} be as before, with regard to ε = 1 . By the same arguments as before, we can show that, for every n ∈ ℕ^{∗} with n > n_{ε} = max{n′_{ε}, n′′_{ε}, n_{δ}} ,

c_{M} ∑_{i=1}^{N} |∫_{K^{∘}} v(x, 0)\frac{∂φ_{i}}{∂x_{i}} (x)dx| ≤ \sqrt{c_{M}Ng_{r_{n}}^{(N)}(v_{n})} + 2

and then, if n → +∞ ,

∑_{i=1}^{N} |∫_{K^{∘}} v(x, 0)\frac{∂φ_{i}}{∂x_{i}} (x)dx| ≤ \sqrt{\frac{N}{c_{M}} lim inf_{n} g_{r_{n}}^{(N)}(v_{n})} + 2  (2.4)

for every φ_{1}, φ_{2}, ..., φ_{N} ∈ C_{c}^{∞}(K^{∘}) with ∥φ_{i}∥_{L^{2}(K^{∘})} ≤ 1 ∀i ∈ ℕ^{∗} i ≤ N .

We define the linear functionals g_{i} : C_{c}^{∞}(K^{∘}) → ℝ

< g_{i}, φ >= ∫_{K^{∘}} v(x, 0)\frac{∂φ_{i}}{∂x_{i}} (x)dx i ∈ ℕ^{∗} i ≤ N .

Let us suppose that sup{| < g_{i}, φ > | | φ ∈ C_{c}^{∞}(K^{∘}) , ∥φ∥_{L^{2}(K^{∘})} ≤ 1} =

C_{i} < +∞ ∀i ≤ N ; then, as done at the end of the proof of Proposition 2.18, we can prove that v ∘ J ∈ H^{1}(K^{∘}) . Then there exists i ∈ ℕ^{∗} such that C_{i} = +∞ and by (2.4) lim_{n} g_{r_{n}}^{(N)}(v_{n}) = +∞ .

To conclude we have to prove that, if v ∘ J ∈ H^{1}(K^{∘}) , there exists a sequence (v_{n}) in C(\overline{Ω′}) so that lim_{n} ∥v_{n}−v∥_{L^{∞}(Ω′)} = 0 and lim_{n} g_{r_{n}}^{(N)}(v_{n}) = c_{M}f^{(N)}(v ∘ J) .

By Proposition 2.18 there exists a sequence (u_{n}) in C(\overline{Ω}) such that lim_{n} ∥u_{n} − v ∘ J∥_{L^{∞}(Ω)} = 0 and lim_{n} f_{r_{n}}^{(N)}(u_{n}) = f^{(N)}(v ∘ J) .

To extend each function (u_{n}) in Ω′ we consider the sets A_{n} = \overline{Ω} × B_{ℝ^{M}}(0, r_{n}) (for simplicity we suppose Ω′ ⋂(ℝ^{N} × B_{ℝ^{M}}(0, r_{n})) ⊆ A_{n}).

For every n ∈ ℕ^{∗} A_{n−2}\ A_{n−1}^{∘} is compact and contained into its open δ_{n}-neighborhood, with 3δ_{n} = min{r_{n−1} − r_{n}, r_{n−2} − r_{n−1}} . Then there exist

k_{n} ∈ ℕ^{∗} and (x_{1}^{(n)}, y_{1}^{(n)}), (x_{2}^{(n)}, y_{2}^{(n)}), ..., (x_{k_{n}}^{(n)}, y_{k_{n}}^{(n)}) ∈ A_{n−2}\ A_{n−1}^{∘} so that

A_{n−2}\ A_{n−1}^{∘}⊆ ⋃_{i=1}^{k_{n}} D_{ℝ^{N} ×ℝ^{M}}((x_{i}^{(n)}, y_{i}^{(n)}), δ_{n}) .

By the lemma on the partition of unity there exist φ_{0}^{(n)}, φ_{1}^{(n)}, ..., φ_{k_{n}}^{(n)} ∈ C^{∞}(ℝ^{N+M}) so that ∀(x, y) ∈ ℝ^{N} × ℝ^{M} ∀i ∈ ℕ i ≤ k_{n} we have 0 ≤ φ_{i}^{(n)}(x, y) ≤ 1 and ∑_{j=0}^{k_{n}} φ_{j}^{(n)}(x, y) = 1 ; moreover, for 1 ≤ i ≤ k_{n} , the support of φ_{i}^{(n)} is contained in D_{ℝ^{N} ×ℝ^{M}}((x_{i}^{(n)}, y_{i}^{(n)}), δ_{n}) and the support

of φ_{0}^{(n)} does not intersect A_{n−2}\ A_{n−1}^{∘} . We define ψ_{n}, ω_{n} : ℝ^{N} × ℝ^{M} → ℝ ψ_{n}(x, y) = φ_{0}^{(n)}(x, y)χ_{A_{n−2}}(x, y) , ω_{n}(x, y) = φ_{0}^{(n)}(x, y)(1 − χ_{A_{n−2}}(x, y)) .

Now the function v_{n} : Ω′ → ℝ is defined by

v_{n}(x, y) = ψ_{n}(x, y)u_{n}(x) + (ω_{n}(x, y) + ∑_{i=1}^{k_{n}} φ_{i}^{(n)}(x, y))v(x, y) .

We prove that lim_{n} ∥v_{n} −v∥_{L^{∞}(Ω′)} = 0 . Let ε > 0 ; there exists n′_{ε} ∈ ℕ^{∗} so that ∥u_{n} − v ∘ J∥_{L^{∞}(Ω)} < ε for every n ∈ ℕ^{∗} with n > n′_{ε} .

Since v is uniformly continuous on Ω′ , there exists δ > 0 so that |v(x′, y′) − v(x′′, y′′)| < ε for every (x′, y′), (x′′, y′′) ∈ Ω′ with ∥(x′, y′) − (x′′, y′′)∥ < δ . Moreover, let n′′_{ε} ∈ ℕ^{∗} be so that r_{n−2} < δ for every n ∈ ℕ^{∗} n > n′′_{ε} .

Let n ∈ ℕ^{∗} n > n_{ε} = max{n′_{ε}, n′′_{ε}} and let (x, y) ∈ Ω′ . If (x, y) ∉ A_{n−2} then v_{n}(x, y) = v(x, y) .

If (x, y) ∈ A_{n−2} , then

|v_{n}(x, y) − v(x, y)| = |ψ_{n}(x, y)u_{n}(x) − ψ_{n}(x, y)v(x, y)| ≤

|u_{n}(x) − v(x, y)| ≤ |u_{n}(x) − v(x, 0)| + |v(x, 0) − v(x, y)| < 2ε .

Moreover, if (x, y) ∈ A_{n} , then v_{n}(x, y) = u_{n}(x) .

Finally, for every n ∈ ℕ^{∗} n > 2

g_{r_{n}}^{(N)}(v_{n}) ≤ \frac{1}{r_{n}^{M}} ∫_{K_{rn}×B_{ℝ^{M}} (0,r_{n})} ∑_{i=1}^{N} \frac{∂u_{n}}{∂x_{i}} (x)dxdy = c_{M}f_{r_{n}}^{(N)}(u_{n})

and then lim_{n} g_{r_{n}}^{(N)}(v_{n}) ≤ c_{M}f^{(N)}(v ∘ J) ■

Corollary 2.26. - If K^{∘}≠ ∅ then dim_{C} H = N .

2.8 An attempt about fractal sets

In this section we study the conjecture on the cartesian product between the set of sections 1.3.4 and 1.5.1 and the closed unit interval. We call C the set constructed in those sections and so K = C × [0, 1] .

For every n ∈ ℕ^{∗} and k ∈ I_{n} × {1, 2} , k = (i, j) we set g_{k}^{(n)} : ℝ^{2} → ℝ^{2}

g_{k}^{(n)}(x, y) = (f_{i}^{(n)}(x), \frac{1}{2}(y + j − 1)) ,

where the functions f_{i}^{(n)} are as in sections 1.3.4, 1.5.1. Moreover we set, for every n ∈ ℕ^{∗} , G_{n} = {g_{k}^{(n)} | k ∈ I_{n} × {1, 2}} .

It is easy to see that for every non-empty compact set B ⊆ ℝ^{2} K = lim_{n}(G_{1} ∘ G_{2} ∘ ⋯ G_{n})(B) in the sense of the Hausdorff metric. It can also be shown that if μ_{C} is the measure on C constructed in section 1.5.1, Proposition 1.22, then μ_{K} = μ_{C} × L^{1} = H^{d} × L^{1} where μ_{K} is obtained by setting l_{k}^{(n)} = (2m_{n})^{−1} ∀n ∈ ℕ^{∗} ∀k ∈ I_{n} × {1, 2} .

Our target is to compute dim_{C} K . By Propositions 1.7 and 2.17 we already know that dim_{C} K ≤ 1 + \overline{dim_{B}}C . It follows that if the sequence (m_{n}) is bounded then dim_{C} K ≤ 1 + d .

Now we need a technical lemma.

Lemma 2.4. - Let K ⊆ ℝ^{N} K ≠ ∅ be compact and let μ be a Radon measure supported on K . We set

L^{2}(K, μ) = {u : K → ℝ^{∗} | u is μ-measurable and ∫_{K} u(x)^{2}dμ < +∞}

and

∥u∥_{L^{2}(K,μ)} = \sqrt{∫_{K} u(x)^{2}dμ} .

Let H ⊆ K be compact with μ(H) = 0 (possibly H = ∅).

Then, for every u ∈ L^{2}(K, μ) and for every ε > 0 there exists a function φ ∈ C^{∞}(ℝ^{N}) so that ∥u − φ∥_{L^{2}(K,μ)} < ε and the support of φ does not intersect H . PROOF. - Since μ is a Radon measure and K is compact we have μ(ℝ^{N}) = μ(K) < +∞ .

Let u ∈ L^{2}(K; μ) and ε > 0 ; by Theorem 4.8 there exists w ∈ C_{c}(K) = C(K) such that ∥u − w∥_{L^{2}(K,μ)} < ε/3 .

By Theorem 4.2 there exists v ∈ C(ℝ^{N}) such that v(x) = w(x) ∀x ∈ K. Let Ω ⊆ ℝ^{N} be an open bounded set so that K ⊆ Ω ; by Theorem 4.5

there exists a real polynomial q on ℝ^{N} such that ∥q − v∥_{L^{∞}(Ω)} < \frac{ε}{3\sqrt{μ(K)}} .

Since μ is a Radon measure and μ(H) = 0 there exists V ⊆ ℝ^{N} open such that H ⊆ V and μ(V) < \frac{ε^{2}}{9∥q∥_{L^{∞}(Ω)}^{2}} ; we can suppose V ⊆ Ω . Let

δ > 0 be so that H_{3δ} ⊆ V (this is possible because H is compact and

ℝ^{N} \ V is closed); since H_{4δ}\ H_{2δ}^{∘} is compact there exist n ∈ ℕ^{∗} and

x_{1}, x_{2}, ..., x_{n} ∈ H_{4δ}\ H_{2δ}^{∘} so that H_{4δ}\ H_{2δ}^{∘}⊆ ⋃_{i=1}^{n} D_{ℝ^{N}}(x_{i}, δ) .

By the lemma on the partition of the unity there exist φ_{0}, φ_{1}, ..., φ_{n} ∈ C^{∞}(ℝ^{N}) so that for every x ∈ ℝ^{N} , for every i ∈ ℕ i ≤ n we have 0 ≤ φ_{i}(x) ≤ 1 and ∑_{i=1}^{n} φ_{i}(x) = 1 ; moreover for i > 0 the support of φ_{i} is contained in D_{ℝ^{N}}(x_{i}, δ) and the support of φ_{0} is contained in H_{2δ} ⋃(ℝ^{N} \ H_{4δ}) .

Then the function φ is defined by

φ(x) = (φ_{0}(x)(1 − χ_{H_{3δ}}(x)) + ∑_{i=1}^{n} φ_{i}(x))q(x) .

Since the support of φ_{0} does not intersect ∂(H_{3δ}) , it follows that φ ∈ C^{∞}(ℝ^{N}) ; moreover, for every x ∈ H_{δ} and i ∈ ℕ^{∗} i ≤ n we have φ_{i}(x) = 0 and then φ(x) = 0 .

If x ∈ ℝ^{N} \ H_{3δ} , then φ(x) = (∑_{i=0}^{n} φ_{i}(x))q(x) = q(x) .

For every x ∈ ℝ^{N}

|φ(x)−q(x)| ≤ |φ_{0}(x)(1−χ_{H_{3δ}}(x))q(x)−φ_{0}(x)q(x)|+| ∑_{i=0}^{n} φ_{i}(x)q(x)−q(x)| =

χ_{H_{3δ}}(x)|q(x)| ≤ |q(x)| .

Finally

∥u − φ∥_{L^{2}(K,μ)} ≤ ∥u − v∥_{L^{2}(K,μ)} + ∥v − q∥_{L^{2}(K,μ)} + ∥q − φ∥_{L^{2}(K,μ)} <

\frac{ε}{3}+∥v−q∥_{L^{∞}(Ω)}\sqrt{μ(K)}+\sqrt{∫_{V} (q(x) − φ(x))^{2}dμ} < \frac{2}{3}ε+∥q∥_{L^{∞}(Ω)}\sqrt{μ(V)} < ε ■

Definition. - For every x ∈ ℝ we call J_{x} : ℝ → ℝ^{2} the inclusion J_{x}(y) = (x, y) .

Remark 17. - Let r > 0 ; by Proposition 4.4 there exists a continuous function ω : ℝ^{2} → [0, 1] such that ω(x, y) = 0 for all (x, y) ∈ ℝ^{2} \ Ω_{r} and ω(x, y) = 1 for all (x, y) ∈ \overline{Ω} .

Let u ∈ C(\overline{Ω}) ; by Theorem 4.2 there exists a function v ∈ C(ℝ^{2}) such that v(x, y) = u(x, y) for every (x, y) ∈ \overline{Ω} .

We set w(x, y) = ω(x, y)v(x, y) so that w extends u and w ∈ C_{c}(ℝ^{2}). Let the function f_{u} : ℝ → ℝ^{∗} be defined as

f_{u}(x) = \{ \begin{matrix}∫_{0}^{1} (\frac{d}{dy}w(J_{x}(y)))^{2} dy & if u ∘ J_{x} ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} ;

we prove that f_{u} is μ_{C}-measurable.

To do it we consider the functions g : ℝ → C([0, 1]) and E : C([0, 1]) → ℝ^{∗} defined by

g(x)(y) = w(x, y)

and

E(φ) = \{ \begin{matrix}∫_{0}^{1} φ′(t)^{2}dt & if φ ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} ;

on C([0, 1]) we consider the usual norm ∥ · ∥_{L^{∞}(0,1)} .

In section 3.2 we will see, independently on the present section, that E is the Γ-limit of a sequence E_{n} of functionals on C([0, 1]) ; then, by Proposition 2.13 E is lower semicontinuous.

Since w is uniformly continuous, for every ε > 0 there exists δ > 0 so that |w(x_{1}, y_{1}) − w(x_{2}, y_{2})| < ε for every (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ ℝ^{2} with ∥(x_{1}, y_{1}) − (x_{2}, y_{2})∥ < δ . Let x_{1}, x_{2} ∈ ℝ ; if |x_{1} − x_{2}| < δ , then for every y ∈ [0, 1] ∥(x_{1}, y) − (x_{2}, y)∥ < δ and |g(x_{1})(y) − g(x_{2})(y)| < ε . It follows ∥g(x_{1}) − g(x_{2})∥_{L^{∞}(0,1)} < ε .

It is easy to see that f_{u} = E ∘ g . It follows that f_{u} is lower semicontinuous and then it is μ_{C}-measurable.

By Remark 17 we can justify the following definition.

Definition. - We define a functional f : C(\overline{Ω}) → ℝ^{∗} by

f(u) = \{ \begin{matrix}∫_{C} ∫_{0}^{1} (\frac{d}{dy}u(J_{x}(y)))^{2} dydμ_{C}(x) & if u ∘ J_{x} ∈ H^{1}(0, 1) \\ & for μ_{C}-almost every x ∈ C \\ +∞ & otherwise \end{matrix} .

We will study relations between f and f_{−}^{(α)}, f_{+}^{(α)} with α = dim_{C} K . We recall (Remark 28 of the first chapter) that for every n ∈ ℕ^{∗} , t_{n} is

the lenght of each interval of the pre-Cantor set (𝔉_{1} ∘ 𝔉_{2} ∘ ⋯ ∘ 𝔉_{n})([0, 1]) and d_{n} is the minimum dinstance between two consecutive intervals.

Lemma 2.5. - Let α ∈]0, 2] . Then, for every u ∈ C(\overline{Ω}) we have

f(u) ≤ sup_{δ>0} lim inf_{r→0} \frac{r^{2−α}}{(2r + t_{n(r)}) ∏_{k=1}^{n(r)} m_{k}} inf\{f_{r}^{(α)}(v) | v ∈ C(\overline{Ω}) ,

∥u − v∥_{L^{∞}(Ω)} < δ} ,

where, for every r ∈]0, d_{1}/2[ , n(r) ∈ ℕ^{∗} is so that d_{n(r)+1} ≤ 2r < d_{n(r)} . PROOF. - Let u ∈ C(\overline{Ω}) . We set the following notation:

• for every (x, y) ∈ K w(x, y) = \frac{d}{dy}u(J_{x}(y)) if it make sense;

• H = C × {0, 1} ; H is in some sense the border of K ;

• ν is the Dirac delta on ℝ , centered on 0 : ν(A) = 0 if 0 ∉ A , ν(A) = 1 if 0 ∈ A ;

• Ω_{1}, Ω_{2} ⊆ ℝ^{2} are open sets such that K ⊆ Ω_{1} ⊆ \overline{Ω}_{1} ⊆ Ω_{2} ⊆ \overline{Ω}_{2} ⊆ Ω ;

• E = {φ ∈ C^{∞}(ℝ^{2}) | the support of φ does not intersect H} ; E plays the role of C_{c}^{∞}(K^{∘}) . By Lemma 2.4 E is dense in L^{2}(K, L^{1} × μ_{C}) ;

• ε > 0 is given.

By Proposition 4.4 there exists a continuous function ω : ℝ^{2} → [0, 1] so that ω(x, y) = 0 if and only if (x, y) ∈ ℝ^{2} \ Ω_{2} and ω(x, y) = 1 if and only if (x, y) ∈ \overline{Ω}_{1} .

We first suppose f(u) < +∞ . In this case w ∈ L^{2}(K, L^{1} × μ_{C}) ; let φ ∈ E be so that ∥w −

φ∥_{L^{2}(K,L^{1}×μ_{C})} < ε min\{1, \sqrt{1/f(u)}\} ( ∥w − φ∥_{L^{2}(K,L^{1}×μ_{C})} < ε if f(u) = 0 ).

Since u is uniformly continuous on Ω , there exists σ > 0 so that |u(x_{1}, y_{1}) − u(x_{2}, y_{2})| < ε/(3∥\frac{∂φ}{∂y}∥_{L^{∞}(Ω)}) for every (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Ω such that ∥(x_{1}, y_{1}) − (x_{2}, y_{2})∥ < σ .

With these assumptions we have

f(u) = |(w, w)_{L^{2}(K,L^{1}×μ_{C})}| ≤ |(w, w − φ)_{L^{2}(K,L^{1}×μ_{C})}| + |(w, φ)_{L^{2}(K,L^{1}×μ_{C})}| ≤

∥w∥_{L^{2}(K,L^{1}×μ_{C})}∥w − φ∥_{L^{2}(K,L^{1}×μ_{C})} + |∫_{C} ∫_{0}^{1} \frac{d}{dy} u(J_{x}(y))φ(x, y)dydμ_{C}(x)| <

|∫_{C} ∫_{0}^{1} ω(x, y)u(x, y)\frac{∂φ}{∂y}(x, y)dydμ_{C}(x)| + ε .

The function x → ∫_{0}^{1} ω(x, y)u(x, y)\frac{∂φ}{∂y}(x, y)dy belongs to C_{c}(ℝ) and then, by Proposition 1.18 there exists n′_{ε} ∈ ℕ^{∗} such that, for every n ∈ ℕ^{∗} with n > n′_{ε} , we have

|∫_{C} ∫_{0}^{1} ω(x, y)u(x, y)\frac{∂φ}{∂y}(x, y)dydμ_{C}(x) +

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)}) ∫_{C} ∫_{0}^{1} ω(x, y)u(x, y)\frac{∂φ}{∂y}(x, y)dydf_{i_{1}i_{2}...i_{n}♯}ν(x)| < ε.

By the same reason there exists n′′_{ε} ∈ ℕ^{∗} such that, for every n ∈ ℕ^{∗} with n > n′′_{ε} , we have

|∫_{C} ∫_{0}^{1} ω(x, y)φ(x, y)^{2}dydμ_{C}(x) +

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}}(∏_{k=1}^{n} l_{i_{k}}^{(k)}) ∫_{C} ∫_{0}^{1} ω(x, y)φ(x, y)^{2}dydf_{i_{1}i_{2}...i_{n}♯}ν(x)| < ε . (2.5)

Recalling that l_{j}^{(k)} = 1/m_{k} ∀k ∈ ℕ^{∗} ∀j ∈ I_{n}, we have, for n > n′_{ε} , f(u) <

|\frac{1}{∏_{k=1}^{n} m_{k}} ∑_{i_{1}=1}^{m_{1}} ... ∑_{i_{n}=1}^{m_{n}} ∫_{C} ∫_{0}^{1} u(f_{i_{1}i_{2}...i_{n}}(x), y)\frac{∂φ}{∂y}(f_{i_{1}i_{2}...i_{n}}(x), y)dydν(x)| + 2ε ≤

\frac{1}{∏_{k=1}^{n} m_{k}} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} |∫_{0}^{1} u(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)\frac{∂φ}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)dy|+2ε.

Now, let δ > 0 with δ < ε/(3∥\frac{∂φ}{∂y}∥_{L^{∞}(Ω)}) and let r > 0 , n = n(r) ∈ ℕ^{∗} be so that n > n′_{ε} , n > n′′_{ε} , t_{n} + r < σ and d_{n+1} ≤ 2r < d_{n} . Suppose

that v ∈ C(\overline{Ω}) is so that v ∈ H^{1}(K_{r}^{∘}) and ∥u − v∥_{L^{∞}(Ω)} < δ . Then, for every (i_{1}, i_{2}, ..., i_{n}) ∈ ∏_{k=1}^{n} I_{k} ,

|∫_{0}^{1} u(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)\frac{∂φ}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)dy +

− ∫_{0}^{1} v(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)\frac{∂φ}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)dy| ≤

∥u − v∥_{L^{∞}(Ω)} ∥\frac{∂φ}{∂y}∥_{L^{∞}(Ω)} < δ ∥\frac{∂φ}{∂y}∥_{L^{∞}(Ω)} < \frac{ε}{3} .

It follows f(u) <

\frac{1}{∏_{k=1}^{n} m_{k}} ∑_{i_{1}=1}^{m_{1}} ... ∑_{i_{n}=1}^{m_{n}} |∫_{0}^{1} v(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)\frac{∂φ}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)dy| + \frac{7}{3}ε .

Let (i_{1}, i_{2}, ..., i_{n}) ∈ ∏_{k=1}^{n} I_{k} ; if x ∈ [p_{n}(i_{1}, i_{2}, ..., i_{n}) − r, p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}+r] and y ∈ [0, 1] , then ∥(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)−(x, y)∥ = |p_{n}(i_{1}, i_{2}, ..., i_{n})− x| ≤ t_{n} +r < σ and |v(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)−v(x, y)| ≤ |v(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)− u(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)| + |u(p_{n}(i_{1}, i_{2}, ..., i_{n}), y) − u(x, y)| + |u(x, y) − v(x, y)| < 2δ + ε/(3∥\frac{∂φ}{∂y}∥_{L^{∞}(Ω)}) < ε/∥\frac{∂φ}{∂y}∥_{L^{∞}(Ω)} .

Then

|∫_{0}^{1} v(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)\frac{∂φ}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)dy +

−\frac{1}{t_{n} + 2r} ∫_{p_{n}(i_{1},i_{2},...,i_{n})−r}^{p_{n}(i_{1},i_{2},...,i_{n})+t_{n}+r} ∫_{0}^{1} v(x, y)\frac{∂φ}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)dydx| ≤

\frac{1}{t_{n} + 2r} ∫_{p_{n}(i_{1},i_{2},...,i_{n})−r}^{p_{n}(i_{1},i_{2},...,i_{n})+t_{n}+r} ∫_{0}^{1} |v(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)+

−v(x, y)| |\frac{∂φ}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)| dydx < ε .

It follows

f(u) < \frac{10}{3}ε + \frac{1}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ...

∑_{i_{n}=1}^{m_{n}} |∫_{p_{n}(i_{1},i_{2},...,i_{n})−r}^{p_{n}(i_{1},i_{2},...,i_{n})+t_{n}+r} ∫_{0}^{1} v(x, y)\frac{∂φ}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)dydx| =

\frac{10}{3}ε + \frac{1}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ...

∑_{i_{n}=1}^{m_{n}} |∫_{p_{n}(i_{1},i_{2},...,i_{n})−r}^{p_{n}(i_{1},i_{2},...,i_{n})+t_{n}+r} ∫_{0}^{1} \frac{∂v}{∂y} (x, y)φ(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)dydx| .

The sets of the type [p_{n}(i_{1}, i_{2}, ..., i_{n}) − r, p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n} + r] × [0, 1] are disjoint because 2r < d_{n} . Let

K_{n,r} = ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ... ⋃_{i_{n}=1}^{m_{n}}([p_{n}(i_{1}, i_{2}, ..., i_{n}) − r, p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n} + r] × [0, 1])

90 CHAPTER 2. A CONJECTURE OF E. DE GIORGI and ψ(x, y) = φ(p_{n}(i_{1}, i_{2}, ..., i_{n}), y) if x ∈ [p_{n}(i_{1}, i_{2}, ..., i_{n})−r, p_{n}(i_{1}, i_{2}, ..., i_{n})+ t_{n} + r] , then

f(u) < \frac{1}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} ∫_{K_{n,r}} |\frac{∂v}{∂y} (x, y)ψ(x, y)| dydx + \frac{10}{3}ε ≤

\frac{1}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} ∥\frac{∂v}{∂y} ∥_{L^{2}(K_{n,r})} ∥ψ∥_{L^{2}(K_{n,r})} + \frac{10}{3}ε =

\sqrt{\frac{1}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} ∫_{K_{n,r}} \frac{∂v}{∂y} (x, y)^{2}dxdy}

\sqrt{\frac{1}{∏_{k=1}^{n} m_{k}} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} ∫_{0}^{1} φ(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)^{2}dy} + \frac{10}{3}ε ≤

\sqrt{\frac{1}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} ∫_{K_{r}} ∥∇v(x, y)∥^{2}dxdy}

\sqrt{\frac{1}{∏_{k=1}^{n} m_{k}} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} ∫_{C} ∫_{0}^{1} ω(x, y)φ(x, y)^{2}dydf_{i_{1}i_{2}...i_{n}♯}ν(x)} + \frac{10}{3}ε <

\sqrt{\frac{r^{2−α}}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} f_{r}^{(α)}(v)}\sqrt{∫_{C} ∫_{0}^{1} φ(x, y)^{2}dydμ_{C}(x) + ε} + \frac{10}{3}ε ,

where the last inequality is due to (2.5).

Now,

∫_{C} ∫_{0}^{1} φ(x, y)^{2}dydμ_{C}(x) = ∥φ∥_{L^{2}(K,L^{1}×μ_{C})}^{2} ≤
(∥φ − w∥_{L^{2}(K,L^{1}×μ_{C})} + ∥w∥_{L^{2}(K,L^{1}×μ_{C})})^{2} < (ε + \sqrt{f(u)})^{2} ;

then

f(u) < \frac{10}{3}ε + \sqrt{ε + (ε + \sqrt{f(u)})^{2}}\sqrt{\frac{r^{2−α}}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} f_{r}^{(α)}(v)} .

If v ∉ H^{1}(K_{r}^{∘}) , then f_{r}^{(α)}(v) = +∞ ; then

f(u) < \frac{10}{3}ε + \sqrt{ε + (ε + \sqrt{f(u)})^{2}}

\sqrt{\frac{r^{2−α}}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} inf\{f_{r}^{(α)}(v) | v ∈ C(\overline{Ω}) , ∥u − v∥_{L^{∞}(Ω)} < δ\}} .

Let g(α, r, δ, u) be the term under the last square root; if r → 0 and δ → 0 , we have

f(u) < \frac{10}{3}ε + \sqrt{ε + (ε + \sqrt{f(u)})^{2}}\sqrt{sup_{δ>0} lim inf_{r→0} g(α, r, δ, u)} .

Finally, by letting ε → 0 , we obtain the desired result.

Now we suppose f(u) = +∞ .

Let φ ∈ E be so that ∥φ∥_{L^{2}(K,L^{1}×μ_{C})} ≤ 1 and let σ, δ, n′_{ε}, n′′_{ε}, r, n, v be as in the previous case, with respect to the new φ instead of the old one. By using the same arguments as before we can show that

|∫_{C} ∫_{0}^{1} u(x, y)\frac{∂φ}{∂y}(x, y)dydμ_{C}(x)| <

\frac{7}{3}ε + \sqrt{ε + ∥φ∥_{L^{2}(K,L^{1}×μ_{C})}}\sqrt{\frac{r^{2−α}}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} f_{r}^{(α)}(v)} ≤

\frac{7}{3}ε + \sqrt{ε + 1}\sqrt{\frac{r^{2−α}}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} f_{r}^{(α)}(v)}

for every v ∈ C(\overline{Ω}) such that ∥u − v∥_{L^{∞}(Ω)} < δ .

Then, if r → 0 and δ → 0 ,

|∫_{C} ∫_{0}^{1} u(x, y)\frac{∂φ}{∂y}(x, y)dydμ_{C}(x)| ≤ \frac{7}{3}ε + \sqrt{1 + ε}\sqrt{sup_{δ>0} lim inf_{r→0} g(α, r, δ, u)}

for every φ ∈ E with ∥φ∥_{L^{2}(K,L^{1}×μ_{C})} ≤ 1 .

To conclude we have to prove that M =

sup \{|∫_{C} ∫_{0}^{1} u(x, y)\frac{∂φ}{∂y}(x, y)dydμ_{C}(x)| | φ ∈ E, ∥φ∥_{L^{2}(K,L^{1}×μ_{C})} ≤ 1\} = ∞.

Let D = {x ∈ C | u ∘ J_{x} ∉ H^{1}(0, 1)} ; we consider two different cases. \underline{1. μ_{C}(D) = 0 .} Suppose that M < +∞ . For every φ ∈ E with ∥φ∥_{L^{2}(K,L^{1}×μ_{C})} ≤ 1 ,

|∫_{C} ∫_{0}^{1} w(x, y)φ(x, y)dydμ_{C}(x)| = |∫_{C} ∫_{0}^{1} u(x, y)\frac{∂φ}{∂y}(x, y)dydμ_{C}(x)| ≤ M .

Since E is dense in L^{2}(K, L^{1} × μ_{C}) , it follows \sqrt{f(u)} =

sup {|∫_{C} ∫_{0}^{1} w(x, y)φ(x, y)dydμ_{C}(x)| | φ ∈ E , ∥φ∥_{L^{2}(K,L^{1}×μ_{C})} ≤ 1} ≤ M.

\underline{2. μ_{C}(D) > 0 .} Let x ∈ D ; for every ψ ∈ C_{c}^{∞}(]0, 1[) with ∥ψ∥_{L^{2}(0,1)} ≤ 1 there exists | ∫_{0}^{1} u(x, y)ψ′(y)dy| < +∞ . We define a linear functional g : C_{c}^{∞}(]0, 1[) → ℝ , < g, ψ >= ∫_{0}^{1} u(x, y)ψ′(y)dy . Let us suppose that

sup {|∫_{0}^{1} u(x, y)ψ′(y)dy| | ψ ∈ C_{c}^{∞}(]0, 1[) , ∥ψ∥_{L^{2}(0,1)} ≤ 1} = M_{x} < +∞ ;

then g is continuous on C_{c}^{∞}(]0, 1[) with respect to the L^{2}-norm. Since g is linear it is uniformly continuous and by Theorem 4.1, there exists h : L^{2}(0, 1) → ℝ uniformly continuous such that h(ψ) =< g, ψ > for every ψ ∈ C_{c}^{∞}(]0, 1[) . Since h is continuous on L^{2}(]0, 1[) , C_{C}^{∞}(]0, 1[) is dense in L^{2}(]0, 1[) and g is linear it is easy to see that h is linear. Then there exists η ∈ L^{2}(0, 1) such that < h, q >= ∫_{0}^{1} η(y)q(y)dy ∀q ∈ L^{2}(0, 1) and then

∫_{0}^{1} u(x, y)ψ′(y)dy =< g, ψ >=< h, ψ >= ∫_{0}^{1} η(y)ψ(y)dy ∀ψ ∈ C_{c}^{∞}(]0, 1[) ;

that is u ∘ J_{x} ∈ H^{1}(0, 1) and \frac{d(u∘J_{x})}{dy} = −η . Therefore M_{x} = +∞ .

Let R ∈]0, +∞[ ; there exists ψ_{x} ∈ C_{c}^{∞}(]0, 1[) so that ∥ψ_{x}∥_{L^{2}(0,1)} ≤ 1 and ∫_{0}^{1} u(x, y)ψ′(y)dy > 8R/μ_{C}(D) . The function F (t) = ∫_{0}^{1} u(t, y)ψ′_{x}(y)dy is continuous and then there exists δ_{x} ∈]0, 1] such that F (t) > 4R/μ_{C}(D) for every t ∈ [x − δ_{x}, x + δ_{x}] .

Let 𝔄 = {[x − r, x + r] | x ∈ D , 0 < r ≤ δ_{x}} ; by Theorem 4.11 (with A = D and U = ℝ ) there exist a sequence (x_{n}) in D and a sequence (r_{n}) in ]0, 1] so that μ_{C}(D \⋃_{n=1}^{+∞}[x_{n} −r_{n}, x_{n} +r_{n}]) = 0 and then μ_{C}(D) = μ_{C}(D ⋂ ⋃_{n=1}^{+∞}[x_{n} − r_{n}, x_{n} + r_{n}]) + μ_{C}(D \ ⋃_{n=1}^{+∞}[x_{n} − r_{n}, x_{n} + r_{n}]) = ∑_{n=1}^{+∞} μ_{C}(D ⋂[x_{n} − r_{n}, x_{n} + r_{n}]) because these intervals are disjoint.

Then there exists n ∈ ℕ^{∗} such that ∑_{k=1}^{n} μ_{C}(D ⋂[x_{k} − r_{k}, x_{k} + r_{k}]) > μ_{C}(D)/2 .

We can suppose that x_{1} < x_{2} < ... < x_{n} ; we set 2δ′ = min{x_{k+1} − r_{k+1} − x_{k} − r_{k} | k ∈ ℕ^{∗} , k < n} .

Let k ∈ ℕ^{∗} k < n ; there exists s′′_{k} ∈]x_{k} + r_{k}, x_{k} + r_{k} + δ′] so that μ_{C}([x_{k} + r_{k}, s′′_{k}]) < R/(2n∥u∥_{L^{∞}(Ω)}∥ψ′_{x_{k}}∥_{L^{∞}(0,1)}) .

Indeed, if x_{k} +r_{k} ∉ C , then we can choose s′′_{k} so that [x_{k} +r_{k}, s′′_{k}] ⋂ C = ∅ and then its measure is 0 . Else there exists i ∈ I such that {x_{k} + r_{k}} = ⋂_{h=1}^{+∞}[p_{h}(π_{1,2,...,h}(i)), p_{h}(π_{1,2,...,h}(i)) + t_{h}] . Let h_{0} ∈ ℕ^{∗} be so that t_{h}^{d} < R/(2n∥u∥_{L^{∞}(Ω)}∥ψ′_{x_{k}}∥_{L^{∞}(0,1)}) for all h ≥ h_{0} . If π_{h}(i) = m_{h} ∀h > h_{0} then x_{k} + r_{k} = p_{h_{0}}(π_{1,2,...,h_{0}}(i)) + t_{h_{0}} and we can choose s′′_{k} = x_{k} + r_{k} + d_{h_{0}}/2 ; still μ_{C}([x_{k} + r_{k}, s′′_{k}]) = 0 . Otherwise, by setting s′′_{k} = min{p_{h_{0}}(π_{1,2,...,h_{0}}(i)) + t_{h_{0}}, x_{k} + r_{k} + δ′} , we have μ_{C}([x_{k} + r_{k}, s′′_{k}]) ≤ μ_{C}([p_{h_{0}}(π_{1,2,...,h_{0}}(i)), p_{h_{0}}(π_{1,2,...,h_{0}}(i)) + t_{h_{0}}]) = t_{h_{0}}^{d} by Proposition 1.22 and Lemma 1.6.

By the same arguments there exists s′_{k} ∈ [x_{k} − r_{k} − δ′, x_{k} − r_{k}[ so that μ_{C}([s′_{k}, x_{k} − r_{k}]) < R/(2n∥u∥_{L^{∞}(Ω)}∥ψ′_{x_{k}}∥_{L^{∞}(0,1)}) with k ∈ ℕ^{∗} 1 < k ≤ n . Moreover we set s′_{1} = −1 and s′′_{n} = 2 .

Let k ∈ ℕ^{∗} k ≤ n ; by using the lemma on the partition of the unity it is possible to construct a function η_{k} ∈ C_{c}^{∞}(ℝ) so that η_{k}(x) ∈ [0, 1] ∀x ∈ ℝ , η_{k}(x) = 1 ∀x ∈ [x_{k} − r_{k}, x_{k} + r_{k}] , η_{k}(x) = 0 ∀x ∈ ℝ\]s′_{k}, s′′_{k}[ . The technique needed in the construction of η_{k} has been used many times in this chapter and so we omit the details.

We set φ_{k}(x, y) = η_{k}(x)ψ_{x_{k}}(y) ; it is clear that φ_{k} ∈ E . Let φ = ∑_{k=1}^{n} φ_{k} ; φ ∈ E and since the supports of the functions φ_{k} are disjoint, it follows

∫_{C} ∫_{0}^{1} φ(x, y)^{2}dydμ_{C}(x) = ∑_{k=1}^{n} ∫_{C} ∫_{0}^{1} φ_{k}(x, y)^{2}dydμ_{C}(x) =

∑_{k=1}^{n} ∫_{s′_{k}}^{s′′_{k}} η_{k}(x)^{2}dμ_{C} ∫_{0}^{1} ψ_{x_{k}}(y)^{2}dy ≤ ∑_{k=1}^{n} μ_{C}([s′_{k}, s′′_{k}]) = μ_{C}(⋃_{k=1}^{n}[s′_{k}, s′′_{k}]) ≤ 1 .

Finally

∫_{C} ∫_{0}^{1} u(x, y)\frac{∂φ}{∂y}(x, y)dydμ_{c}(x) = ∑_{k=1}^{n} ∫_{C} ∫_{0}^{1} u(x, y)\frac{∂φ_{k}}{∂y} (x, y)dydμ_{c}(x) =

∑_{k=1}^{n} ∫_{s′_{k}}^{s′′_{k}} η_{k}(x) ∫_{0}^{1} u(x, y)ψ′_{x_{k}}(y)dydμ_{C}(x) =

∑_{k=1}^{n} (∫_{x_{k}−r_{k}}^{x_{k}+r_{k}} ∫_{0}^{1} u(x, y)ψ′_{x_{k}}(y)dydμ_{C}(x)+

∫_{[s′_{k},x_{k}−r_{k}] ⋃[x_{k}+r_{k},s′′_{k}]} η_{k}(x) ∫_{0}^{1} u(x, y)ψ′_{x_{k}}(y)dydμ_{C}(x)) ≥

∑_{k=1}^{n} (\frac{4R}{μ_{C}(D)}μ_{C}([x_{k} − r_{k}, x_{k} + r_{k}])+

− |∫_{s′_{k}}^{x_{k}−r_{k}} η_{k}(x) ∫_{0}^{1} u(x, y)ψ′_{x_{k}}(y)dydμ_{C}(x)| +

− |∫_{x_{k}+r_{k}}^{s′′_{k}} η_{k}(x) ∫_{0}^{1} u(x, y)ψ′_{x_{k}}(y)dydμ_{C}(x)|) ≥

\frac{4R}{μ_{C}(D)} ∑_{k=1}^{n} μ_{C}([x_{k} − r_{k}, x_{k} + r_{k}])+

− ∑_{k=1}^{n} ∥u∥_{L^{∞}(Ω)}∥ψ′_{x_{k}}∥_{L^{∞}(0,1)}(μ_{C}([s′_{k}, x_{k} − r_{k}]) + μ_{C}([x_{k} + r_{k}, s′′_{k}])) >

2R − 2n\frac{R}{2n} = R ■

Lemma 2.6. - Let u ∈ C(\overline{Ω}) with f(u) < +∞ . Then there exists a sequence (v_{n}) in C^{∞}(\overline{Ω}) such that lim_{n} ∥v_{n} − u∥_{L^{∞}(Ω)} = 0 and lim_{n} f(v_{n}) = f(u) . PROOF. - Let A = {x ∈ C | u ∘ J_{x} ∉ H^{1}(0, 1)} ; by the definition of f , μ_{C}(A) = 0 .

We define a function g : ℝ → ℝ ,

g(x) = \{ \begin{matrix}∫_{0}^{1} (\frac{d}{dy}u(J_{x}(y)))^{2} dy & x ∈ C \ A \\ 0 & otherwise \end{matrix} .

For every Borel set E ⊆ ℝ and for every F ⊆ E with μ_{C}(E \ F) = 0 we set

ν(F) = ∫_{E} g(x)dμ_{C} .

By Theorem 4.12 there exists a set B ⊆ C so that μ_{C}(B) = 0 and

lim_{r→0} \frac{ν([x − r, x + r])}{μ_{C}([x − r, x + r])} = g(x) ∀x ∈ C \ B .  (2.6)

We set D = A ⋃ B so that μ_{C}(D) = 0 .

We remark that by Theorem 4.6 ν is a Borel measure. Since μ_{C} is Borel regular, it is easy to see that ν is Borel regular. Moreover ν(ℝ) = f(u) < +∞ and then, by Remark 24 of the first chapter, ν is a Radon measure.

Now, let ε > 0 ; since u is uniformly continuous on Ω , there exists δ_{1} > 0 such that |u(x_{1}, y_{1}) − u(x_{2}, y_{2})| < ε for every (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Ω with ∥(x_{1}, y_{1}) − (x_{2}, y_{2})∥ < δ_{1} .

By (2.6), for every x ∈ C \ D there exists r_{x} ∈]0, δ_{1}/4[ such that for every r ∈]0, r_{x}[ ,

|g(x) − \frac{ν([x − r, x + r])}{μ_{C}([x − r, x + r])}| < ε .

Let 𝔄 = {[x − r, x + r] | x ∈ C \ D , 0 < 3r < r_{x}} ; by Theorem 4.11 there exists a sequence (x_{n}) in C \ D and a sequence (r_{n}) in ℝ so that:

1. 0 < 3r_{n} < r_{x_{n}} ∀n ∈ ℕ^{∗} ;

2. [x_{n} − r_{n}, x_{n} + r_{n}] ⋂[x_{k} − r_{k}, x_{k} + r_{k}] = ∅ ∀n, k ∈ ℕ^{∗} n ≠ k ;

3. μ_{C}((C \ D) \ (⋃_{n=1}^{+∞}[x_{n} − r_{n}, x_{n} + r_{n}])) = 0 .

Then ∑_{n=1}^{+∞} μ_{C}([x_{n} − r_{n}, x_{n} + r_{n}]) = 1 and ∑_{n=1}^{+∞} ν([x_{n} − r_{n}, x_{n} + r_{n}]) = f(u) < +∞ ; therefore there exist n′_{ε}, n′′_{ε} ∈ ℕ^{∗} so that ∑_{k=m+1}^{+∞} μ_{C}([x_{k} − r_{k}, x_{k} + r_{k}]) < ε and ∑_{k=n+1}^{+∞} ν([x_{k} − r_{k}, x_{k} + r_{k}]) < ε for every m, n ∈ ℕ^{∗} with m > n′_{ε} , n > n′′_{ε} . We fix m = 1 + max{n′_{ε}, n′′_{ε}} .

It is not restrictive to suppose x_{n} < x_{n+1} ∀n ∈ ℕ^{∗} n < m . Let 3δ_{2} = min{x_{n+1} − r_{n+1} − x_{n} − r_{n} | n ∈ ℕ^{∗} n < m} .

Let n ∈ ℕ^{∗} n ≤ m . Since μ_{C} is a Radon measure there exist s′_{n}, s′′_{n} ∈]0, r_{x_{n}}/3[⋂]0, δ_{2}[ so that

μ_{C}(]x_{n} − r_{n} − s′_{n}, x_{n} + r_{n} + s′′_{n}[) − μ_{C}([x_{n} − r_{n}, x_{n} + r_{n}]) < \frac{ε}{mM_{n}} ,

where M_{n} = max{1, g(x_{n})} .

If x_{n} + r_{n} ∉ C then we set r′′_{n} = r_{n} . Else, let i^{(n)} ∈ I be so that

{x_{n} + r_{n}} = ⋂_{k=1}^{+∞}[p_{k}(i_{1}^{(n)}, i_{2}^{(n)}, ..., i_{k}^{(n)}), p_{k}(i_{1}^{(n)}, i_{2}^{(n)}, ..., i_{k}^{(n)}) + t_{k}] .

Let k′′_{n} ∈ ℕ^{∗} be so that t_{k} < s′′_{n} for every k ∈ ℕ^{∗} k > k′′_{n} ; then we set r′′_{n} = p_{k′′_{n}+1}(i_{1}^{(n)}, i_{2}^{(n)}, ..., i_{k′′_{n}+1}^{(n)}) + t_{k′′_{n}+1} − x_{n} . It follows x_{n} + r_{n} ≤ x_{n} + r′′_{n} < x_{n} + r_{n} + s′′_{n} .

We proceed in the same way to handle x_{n} −r_{n} and to define r′_{n} . If x_{n} − r_{n} ∉ C then we set r′_{n} = r_{n} ; otherwise r′_{n} = x_{n} − p_{k′_{n}+1}(j_{1}^{(n)}, j_{2}^{(n)}, ..., j_{k′_{n}+1}^{(n)}) where j^{(n)} ∈ I is so that

{x_{n} − r_{n}} = ⋂_{k=1}^{+∞}[p_{k}(j_{1}^{(n)}, j_{2}^{(n)}, ..., j_{k}^{(n)}), p_{k}(j_{1}^{(n)}, j_{2}^{(n)}, ..., j_{k}^{(n)}) + t_{k}]

and k′_{n} ∈ ℕ^{∗} is so that t_{k} < s′_{n} for every k ∈ ℕ^{∗} k > k′_{n} . It follows x_{n} − r_{n} − s′_{n} < x_{n} − r′_{n} ≤ x_{n} − r_{n} .

Let k = 1 + max{max{k′_{n}, k′′_{n}} | n ∈ ℕ^{∗} n ≤ m} ; there exist h ∈ ℕ^{∗} h < ∏_{j=1}^{k} m_{j} and i^{(1)}, i^{(2)}, ..., i^{(h)} ∈ I such that

C\⋃_{n=1}^{m}[x_{n}−r′_{n}, x_{n}+r′′_{n}] = C ⋂ ⋃_{n=1}^{h}[p_{k}(i_{1}^{(n)}, i_{2}^{(n)}, ..., i_{k}^{(n)}), p_{k}(i_{1}^{(n)}, i_{2}^{(n)}, ..., i_{k}^{(n)})+t_{k}].

To simplify the notation we set q_{n} = p_{k}(i_{1}^{(n)}, i_{2}^{(n)}, ..., i_{k}^{(n)}) for every n ∈ ℕ^{∗} n ≤ h . We remark that

μ_{C}(⋃_{n=1}^{h}[q_{n}, q_{n} + t_{k}]) ≤ μ_{C}( ⋃_{n=m+1}^{+∞}[x_{n} − r_{n}, x_{n} + r_{n}]) < ε

and the same inequality holds with ν instead of μ_{C} .

Let for every n ∈ ℕ^{∗} n ≤ h , c_{n} = sup{c ∈ ℝ | c ≤ g(x) for μ_{C}-almost every x ∈ [q_{n}, q_{n} + t_{k}]} ; then, for every n ∈ ℕ^{∗} n ≤ h there exists z_{n} ∈ [q_{n}, q_{n} + t_{k}] ⋂ C so that u ∘ J_{z_{n}} ∈ H^{1}(0, 1) and c_{n} ≤ g(z_{n}) < c_{n} + ε .

Now we construct a function v ∈ C^{∞}(\overline{Ω}) such that ∥v − u∥_{L^{∞}(Ω)} and |f(v) − f(u)| are small.

Let 2δ_{3} be the minimum of the mutual dinstances between the m + h intervals of the form [x_{n} − r′_{n}, x_{n} + r′′_{n}] and [q_{n}, q_{n} + t_{k}] . If 4δ_{3} > δ_{1} we set δ_{3} = δ_{1}/4 .

By using the lemma on the partition of the unity it is possible to construct, for every n, l ∈ ℕ^{∗} n ≤ m , l ≤ h , two functions ψ_{n}, η_{l} ∈ C_{c}^{∞}(ℝ^{2}) so that the following conditions are satisfied:

1. ψ_{n}(x, y), η_{l}(x, y) ∈ [0, 1] ∀(x, y) ∈ ℝ^{2} ;

2. ψ_{n}(x, y) = 1 ∀x ∈ [x_{n} − r′_{n}, x_{n} + r′′_{n}] ∀y ∈ [0, 1] ;

3. ψ_{n}(x, y) = 0 ∀(x, y) ∈ ℝ^{2} \ ([x_{n} − r′_{n}, x_{n} + r′′_{n}] × [0, 1])_{δ_{3}} ;

4. η_{l}(x, y) = 1 ∀x ∈ [q_{l}, q_{l} + t_{k}] ∀y ∈ [0, 1] ;

5. η_{l}(x, y) = 0 ∀(x, y) ∈ ℝ^{2} \ ([q_{l}, q_{l} + t_{k}] × [0, 1])_{δ_{3}} .

Moreover ω(x, y) = 1 − ∑_{n=1}^{m} ψ_{n}(x, y) − ∑_{n=1}^{h} η_{n}(x, y) .

For every x ∈ C \ D there exists φ_{x} ∈ C_{c}^{∞}(]0, 1[) such that

∥φ_{x} − \frac{d(u ∘ J_{x})}{dy}∥_{L^{2}(0,1)} < ε .

By the Stone Weierstrass theorem there exists a polinomial p on ℝ^{2} such that ∥u − p∥_{L^{∞}(Ω)} < ε .

The function v is defined as

v(x, y) = p(x, y)ω(x, y) + ∑_{n=1}^{m} ψ_{n}(x, y)(u(x_{n}, 0) + ∫_{0}^{y} φ_{x_{n}}(t)dt)+

∑_{n=1}^{h} η_{n}(x, y)(u(z_{n}, 0) + ∫_{0}^{y} φ_{z_{n}}(t)dt) .

It is clear that v ∈ C^{∞}(\overline{Ω}) .

Let us estimate ∥u − v∥_{L^{∞}(Ω)} . If y ∈ [0, 1] , x ∈ [x_{n} − r′_{n}, x_{n} + r′′_{n}] n ∈ ℕ^{∗} n ≤ m then

|u(x, y)−v(x, y)| ≤ |u(x, y)−u(x_{n}, y)|+|u(x_{n}, y)−u(x_{n}, 0)−∫_{0}^{y} φ_{x_{n}}(t)dt| <

ε + ∫_{0}^{1} |\frac{d}{dy} u(J_{x_{n}}(t)) − φ_{x_{n}}(t)| dt ≤ ε + ∥\frac{d(u ∘ J_{x_{n}})}{dy} − φ_{x_{n}} ∥_{L^{2}(0,1)} < 2ε .

We used the fact that |x − x_{n}| < r_{x_{n}} < δ_{1}/4 .

If y ∈ [0, 1] , x ∈ [q_{n}, q_{n} + t_{k}] n ∈ ℕ^{∗} n ≤ h then |x − z_{n}| ≤ t_{k} < δ_{1}/12 and as before we can prove that |u(x, y) − v(x, y)| < 2ε .

If (x, y) ∈ Ω \ ((⋃_{n=1}^{m}([x_{n} − r′_{n}, x_{n} + r′′_{n}] × [0, 1])_{δ_{3}}) ⋃(⋃_{n=1}^{h}([q_{n}, q_{n} + t_{k}] × [0, 1])_{δ_{3}})) then |u(x, y) − v(x, y)| = |u(x, y) − p(x, y)| < ε .

If for some n ∈ ℕ^{∗} n ≤ m , (x, y) ∈ ([x_{n} − r′_{n}, x_{n} + r′′_{n}] × [0, 1])_{δ_{3}} \ ([x_{n} − r′_{n}, x_{n} + r′′_{n}] × [0, 1]) , then we consider two cases: if 0 ≤ y ≤ 1 then

|u(x, y) − v(x, y)| =

|u(x, y) − p(x, y)(1 − ψ_{n}(x, y)) − ψ_{n}(x, y)(u(x_{n}, 0) + ∫_{0}^{y} φ_{x_{n}}(t)dt)| <

ε + ψ_{n}(x, y)(|p(x, y) − u(x, y)| + |u(x, y) − u(x_{n}, 0) − ∫_{0}^{y} φ_{x_{n}}(t)dt|) <

2ε + |u(x, y) − u(x_{n}, y)| + |u(x_{n}, y) − u(x_{n}, 0) − ∫_{0}^{y} φ_{x_{n}}(t)dt| < 4ε because |x − x_{n}| ≤ r_{x_{n}} + δ_{3} < δ_{1}/2 . Else, let us suppose, for example, −δ_{3} ≤ y < 0 , then

|u(x, y) − v(x, y)| = |u(x, y) − p(x, y) + ψ_{n}(x, y)p(x, y) − ψ_{n}(x, y)u(x_{n}, 0)| ≤

|u(x, y) − p(x, y)| + ψ_{n}(x, y)(|p(x, y) − u(x, y)| + |u(x, y) − u(x, 0)|+

|u(x, 0) − u(x_{n}, 0)|) < 4ε .

The remaining cases are treated in the same way and the result is ∥u − v∥_{L^{∞}(Ω)} < 4ε .

To conclude the proof we have to show that |f(v) − f(u)| < cε for some constant c ∈ ℝ . Indeed we have |f(v) − f(u)| =

|∑_{n=1}^{m} ∫_{x_{n}−r′_{n}}^{x_{n}+r′′_{n}} ∫_{0}^{1} φ_{x_{n}}(y)^{2}dydμ_{C} + ∑_{n=1}^{h} ∫_{q_{n}}^{q_{n}+t_{k}} ∫_{0}^{1} φ_{z_{n}}(y)^{2}dydμ_{C} − ν(C)| =

|∑_{n=1}^{m} μ_{C}([x_{n} − r′_{n}, x_{n} + r′′_{n}])∥φ_{x_{n}}∥_{L^{2}(0,1)}^{2}+

∑_{n=1}^{h} μ_{C}([q_{n}, q_{n} + t_{k}])∥φ_{z_{n}}∥_{L^{2}(0,1)}^{2} − ∑_{n=1}^{+∞} ν([x_{n} − r_{n}, x_{n} + r_{n}])| .

For every n ∈ ℕ^{∗} n ≤ m we have

|∥φ_{x_{n}}∥_{L^{2}(0,1)}^{2} − g(x_{n})| ≤

∥φ_{x_{n}} − \frac{d(u ∘ J_{x_{n}})}{dy}∥_{L^{2}(0,1)} (∥φ_{x_{n}}∥_{L^{2}(0,1)} + ∥\frac{d(u ∘ J_{x_{n}})}{dy}∥_{L^{2}(0,1)}) <

ε (∥φ_{x_{n}} − \frac{d(u ∘ J_{x_{n}})}{dy}∥_{L^{2}(0,1)} + 2\sqrt{g(x_{n})}) < ε^{2} + 2ε\sqrt{g(x_{n})} .

The same argument holds for z_{n} , n ∈ ℕ^{∗} n ≤ h .

It follows

|f(v) − f(u)| < ∑_{n=1}^{m} μ_{C}([x_{n} − r′_{n}, x_{n} + r′′_{n}])(ε^{2} + 2ε\sqrt{g(x_{n})})+

|∑_{n=1}^{m} μ_{C}([x_{n} − r′_{n}, x_{n} + r′′_{n}])g(x_{n}) − ∑_{n=1}^{m} ν([x_{n} − r_{n}, x_{n} + r_{n}])| +

∑_{n=1}^{h} μ_{C}([q_{n}, q_{n} + t_{k}])(ε^{2} + 2ε\sqrt{g(z_{n})}) + ∑_{n=1}^{h} μ_{C}([q_{n}, q_{n} + t_{k}])g(z_{n}) + ε <

ε^{2} ∑_{n=1}^{m} μ_{C}([x_{n} − r′_{n}, x_{n} + r′′_{n}]) + 2ε ∑_{n=1}^{m} μ_{C}([x_{n} − r′_{n}, x_{n} + r′′_{n}])\sqrt{g(x_{n})}+

∑_{n=1}^{m}(μ_{C}([x_{n} − r′_{n}, x_{n} + r′′_{n}]) − μ_{C}([x_{n} − r_{n}, x_{n} + r_{n}]))g(x_{n})+

|∑_{n=1}^{m} μ_{C}([x_{n} − r_{n}, x_{n} + r_{n}])g(x_{n}) − ∑_{n=1}^{m} ν([x_{n} − r_{n}, x_{n} + r_{n}])| +

ε^{2} ∑_{n=1}^{h} μ_{C}([q_{n}, q_{n} + t_{k}]) + 2ε ∑_{n=1}^{h} μ_{C}([q_{n}, q_{n} + t_{k}])\sqrt{g(z_{n})}+

∑_{n=1}^{h} μ_{C}([q_{n}, q_{n} + t_{k}])(c_{n} + ε) + ε .

Let a ∈ ℕ^{∗} a ≤ m and n′_{1}, n′_{2}, ..., n′_{a} ∈ ℕ^{∗} be the indeces such that g(x_{n′_{i}}) ≤ 1 and let n′′_{1}, n′′_{2}, ..., n′′_{m−a} ∈ ℕ^{∗} be the others. In the same way we call h′_{1}, h′_{2}, ..., h′_{b} ∈ ℕ^{∗} b ∈ ℕ^{∗} b ≤ h the indeces such that g(z_{h′_{i}}) ≤ 1 and let h′′_{1}, h′′_{2}, ..., h′′_{h−b} ∈ ℕ^{∗} the others.

Then

|f(v) − f(u)| < ε^{2}μ_{C}(⋃_{n=1}^{m}[x_{n} − r′_{n}, x_{n} + r′′_{n}]) + 2ε ∑_{i=1}^{a} μ_{C}([x_{n′_{i}} − r′_{n′_{i}}, x_{n′_{i}} + r′′_{n′_{i}}])+

2ε ∑_{i=1}^{m−a} μ_{C}([x_{n′′_{i}} − r′_{n′′_{i}}, x_{n′′_{i}} + r′′_{n′′_{i}}])g(x_{n′′_{i}}) + ∑_{n=1}^{m} \frac{εg(x_{n})}{mM_{n}}+

∑_{n=1}^{m} μ_{C}([x_{n} − r_{n}, x_{n} + r_{n}]) |g(x_{n}) − \frac{ν([x_{n} − r_{n}, x_{n} + r_{n}])}{μ_{C}([x_{n} − r_{n}, x_{n} + r_{n}])}| + ε^{2}μ_{C}(C)+

2ε ∑_{i=1}^{b} μ_{C}([q_{h′_{i}}, q_{h′_{i}} + t_{k}]) + 2ε ∑_{i=1}^{h−b} μ_{C}([q_{h′′_{i}}, q_{h′′_{i}} + t_{k}])g(z_{h′′_{i}})+

∑_{n=1}^{h} μ_{C}([q_{n}, q_{n} + t_{k}])c_{n} + ε ∑_{n=1}^{h} μ_{C}([q_{n}, q_{n} + t_{k}]) + ε <

2ε ∑_{i=1}^{m−a}(μ_{C}([x_{n′′_{i}} − r′_{n′′_{i}}, x_{n′′_{i}} + r′′_{n′′_{i}}]) − μ_{C}([x_{n′′_{i}} − r_{n′′_{i}}, x_{n′′_{i}} + r_{n′′_{i}}]))g(x_{n′′_{i}})+

100 CHAPTER 2. A CONJECTURE OF E. DE GIORGI

2ε ∑_{i=1}^{m−a} μ_{C}([x_{n′′_{i}}−r_{n′′_{i}}, x_{n′′_{i}}+r_{n′′_{i}}])g(x_{n′′_{i}})+ε ∑_{n=1}^{m} μ_{C}([x_{n}−r_{n}, x_{n}+r_{n}])+3εμ_{C}(C)+

2ε ∑_{i=1}^{h−b} μ_{C}([q_{h′′_{i}}, q_{h′′_{i}} + t_{k}])(c_{h′′_{i}} + ε) + ∑_{n=1}^{h} ν([q_{n}, q_{n} + t_{k}]) + 4ε + 2ε^{2} <

2ε ∑_{i=1}^{m−a} μ_{C}([x_{n′′_{i}} − r_{n′′_{i}}, x_{n′′_{i}} + r_{n′′_{i}}]) |g(x_{n′′_{i}}) − \frac{ν([x_{n′′_{i}} − r_{n′′_{i}}, x_{n′′_{i}} + r_{n′′_{i}}])}{μ_{C}([x_{n′′_{i}} − r_{n′′_{i}}, x_{n′′_{i}} + r_{n′′_{i}}])}| +

2ε ∑_{i=1}^{m−a} \frac{εg(x_{n′′_{i}})}{mM_{n′′_{i}}} +2ε ∑_{i=1}^{m−a} ν([x_{n′′_{i}} −r_{n′′_{i}}, x_{n′′_{i}} +r_{n′′_{i}}])+2ε ∑_{i=1}^{h−b} μ_{C}([q_{h′′_{i}}, q_{h′′_{i}} +t_{k}])c_{h′′_{i}}+

2ε^{2} ∑_{i=1}^{h−b} μ_{C}([q_{h′′_{i}}, q_{h′′_{i}} + t_{k}]) + ν(⋃_{n=1}^{h}[q_{n}, q_{n} + t_{k}]) + 8ε + 2ε^{2} ≤

2ε^{2} ∑_{i=1}^{m−a} μ_{C}([x_{n′′_{i}} − r_{n′′_{i}}, x_{n′′_{i}} + r_{n′′_{i}}]) + 2εν(C) + 2ε ∑_{i=1}^{h−b} ν([q_{h′′_{i}}, q_{h′′_{i}} + t_{k}])+

2ε^{2}μ_{C}(C) + 9ε + 4ε^{2} ≤

2εν(⋃_{i=1}^{h−b}[q_{h′′_{i}}, q_{h′′_{i}} + t_{k}]) + 2εf(u) + 9ε + 8ε^{2} < (2f(u) + 9 + 10ε)ε ■

Remark 18. - Let, for every r ∈]0, d_{1}/2[ , n = n(r) be so that d_{n+1} ≤ 2r < d_{n} . It is easy to see that

(t_{n} + 2r) ∏_{k=1}^{n} m_{k} < L^{2}(K_{r}) < (2r + 1)(t_{n} + 2r) ∏_{k=1}^{n} m_{k} ,

from which

lim_{r→0} \frac{L^{2}(K_{r})}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} = 1 .  (2.7)

Let α ∈ [0, 2] : by (2.7) we have

\underline{M^{(α)}}(K) = 2^{α−2} lim inf_{r→0} \frac{t_{n} + 2r}{r^{2−α}} ∏_{k=1}^{n} m_{k}  (2.8)

and

\overline{M^{(α)}}(K) = 2^{α−2} lim sup_{r→0} \frac{t_{n} + 2r}{r^{2−α}} ∏_{k=1}^{n} m_{k} .  (2.9)

2.8. AN ATTEMPT ABOUT FRACTAL SETS 101 Proposition 2.27. - Let α ∈ [0, 2] and u ∈ C(\overline{Ω}) .

1. If \underline{M^{(α)}}(K) = +∞ , then f_{−}^{(α)}(u) = +∞ whenever f(u) > 0 ;

2. if 0 < \underline{M^{(α)}}(K) < +∞ , then f_{−}^{(α)}(u) = 2^{2−α}\underline{M^{(α)}}(K)f(u) ;

3. if \underline{M^{(α)}}(K) = 0 , then f_{−}^{(α)}(u) = 0 whenever f(u) < +∞ ;

4. if \overline{M^{(α)}}(K) = +∞ , then f_{+}^{(α)}(u) = +∞ whenever f(u) > 0 ;

5. if 0 < \overline{M^{(α)}}(K) < +∞ , then f_{+}^{(α)}(u) = 2^{2−α}\overline{M^{(α)}}(K)f(u) ;

6. if \overline{M^{(α)}}(K) = 0 , then f_{+}^{(α)}(u) = 0 whenever f(u) < +∞ . PROOF. - For every r ∈]0, d_{1}/2[ we set n = n(r) so that d_{n+1} ≤ 2r < d_{n} . \underline{Proof of 1 and 4.} Let M ∈]0, f(u)[ ; by Lemma 2.5 there exists δ > 0 so that

\frac{1}{2}M < lim inf_{r→0} \frac{r^{2−α}}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} inf\{f_{r}^{(α)}(w) | w ∈ C(\overline{Ω}) , ∥u−w∥_{L^{∞}(Ω)} < δ\} .

Then there exists r_{0} > 0 so that

\frac{1}{4}M < \frac{r^{2−α}}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} inf\{f_{r}^{(α)}(w) | w ∈ C(\overline{Ω}) , ∥u − w∥_{L^{∞}(Ω)} < δ\}

for every r ∈]0, r_{0}] and therefore

M \frac{t_{n} + 2r}{4r^{2−α}} ∏_{k=1}^{n} m_{k} < inf\{f_{r}^{(α)}(w) | w ∈ C(\overline{Ω}) , ∥u−w∥_{L^{∞}(Ω)} < δ\} ∀r ∈]0, r_{0}] .

If r → 0 , by (2.8) and (2.9), we have 1 and 4.

\underline{Proof of 2 and 5.} Let ε > 0 ; by Lemma 2.5 there exists δ > 0 such that

f(u) < ε + lim inf_{r→0} \frac{r^{2−α}}{(t_{n} + 2r) ∏_{h=1}^{n} m_{h}} inf\{f_{r}^{(α)}(w) |

w ∈ C(\overline{Ω}) , ∥u − w∥_{L^{∞}(Ω)} < δ\} ;

then

f(u)\frac{t_{n} + 2r}{r^{2−α}} ∏_{h=1}^{n} m_{h} < 2ε\frac{t_{n} + 2r}{r^{2−α}} ∏_{h=1}^{n} m_{h}+

inf\{f_{r}^{(α)}(w) | w ∈ C(\overline{Ω}) , ∥u − w∥_{L^{∞}(Ω)} < δ\} .  (2.10)

By (2.8)

2^{2−α}\underline{M^{(α)}}(K)f(u) ≤ 2^{3−α}ε\underline{M^{(α)}}(K) + Γ- lim inf_{r→0} f_{r}^{(α)}(u) .

If ε → 0 then

2^{2−α}\underline{M^{(α)}}(K)f(u) ≤ Γ- lim inf_{r→0} f_{r}^{(α)}(u) .

Let (r_{k}) be a sequence converging to 0 in ]0, d_{1}/2[ , such that

lim_{k} \frac{t_{n} + 2r_{k}}{r_{k}^{2−α}} ∏_{h=1}^{n} m_{h} = 2^{2−α}\overline{M^{(α)}}(K) ;

by (2.10) and (2.9), we have

2^{2−α}\overline{M^{(α)}}(K)f(u) ≤ 2^{3−α}ε\overline{M^{(α)}}(K)+

lim sup_{k} inf\{f_{r_{k}}^{(α)}(w) | w ∈ C(\overline{Ω}) , ∥u − w∥_{L^{∞}(Ω)} < δ\} .

Let (u_{k}) be a sequence in C(\overline{Ω}) such that lim_{k} ∥u_{k} − u∥_{L^{∞}(Ω)} = 0 ; then

2^{2−α}\overline{M^{(α)}}(K)f(u) ≤ 2^{3−α}ε\overline{M^{(α)}}(K) + lim sup_{k} f_{r_{k}}^{(α)}(u_{k})

and by letting ε → 0 , the condition 2. (a) of Proposition 2.9 is satisfied.

To prove the opposite inequality we first suppose u ∈ C^{1}(\overline{Ω}) . We first define a function w : Ω×]0, d_{1}/2[→ ℝ in the following way: given r ∈ ]0, d_{1}/2[ , the ∏_{h=1}^{n} m_{h} sets of the form [p_{n}(i_{1}, i_{2}, ..., i_{n})−r, p_{n}(i_{1}, i_{2}, ..., i_{n})+ t_{n} + r] × [−t_{n}/2 − r, 1 + t_{n}/2 + r] are pairwise disjoint; let 3δ_{r} > 0 be the minimum of their mutual distance. Let

K(r) = ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ... ⋃_{i_{n}=1}^{m_{n}}([p_{n}(i_{1}, i_{2}, ..., i_{n})−r, p_{n}(i_{1}, i_{2}, ..., i_{n})+t_{n}+r]×[−r, 1+r])

and let K(r, δ_{r}) = K(r)_{δ_{r}} be its closed δ_{r}-neighborhood.

By using the lemma on the partition of the unity it is possible to construct a function w_{r} : Ω → ℝ so that:

1. w_{r} ∈ C^{1}(\overline{Ω}) ; 2.8. AN ATTEMPT ABOUT FRACTAL SETS 103

2. for every (i_{1}, i_{2}, ..., i_{n}) ∈ ∏_{h=1}^{n} I_{h} , w_{r}(x, y) = u(p_{n}(i_{1}, i_{2}, ..., i_{n}), y) ∀x ∈ [p_{n}(i_{1}, i_{2}, ..., i_{n}) − r, p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n} + r] ∀y ∈ [−r, 1 + r];

3. w_{r}(x, y) = u(x, y) ∀(x, y) ∈ Ω \ K(r, δ_{r}) ;

4. lim_{r→0} ∥u − w_{r}∥_{L^{∞}(Ω)} = 0 . We set w(x, y, r) = w_{r}(x, y) ; we remark that w is not continuous on r. Now we set the following notation:

• (r_{k}) is a sequence converging to 0 in ]0, d_{1}/2[ and such that

lim_{k} \frac{t_{n} + 2r_{k}}{r_{k}^{2−α}} ∏_{h=1}^{n} m_{h} = 2^{2−α}\underline{M^{(α)}}(K) ,

where n = n(r_{k}) ;

• (R_{k}) is a sequence converging to 0 in ]0, d_{1}/2[ ;

• for every k ∈ ℕ^{∗} , v_{k}, V_{k} : Ω → ℝ are defined by v_{k}(x, y) = w(x, y, r_{k}) , V_{K}(x, y) = w(x, y, R_{k}) ;

• M = ∥\frac{∂u}{∂y}∥_{L^{∞}(Ω)}^{2} ;

• for every r ∈]0, d_{1}/2[ ,

A_{n} = ⋃_{i_{1}=1}^{m_{1}} ⋃_{i_{2}=1}^{m_{2}} ... ⋃_{i_{n}=1}^{m_{n}}([p_{n}(i_{1}, i_{2}, ..., i_{n})−r, p_{n}(i_{1}, i_{2}, ..., i_{n})+t_{n}+r]×[0, 1]);

• for every k ∈ ℕ^{∗}

c_{k} = \frac{1}{r_{k}^{2−α}} ∫_{K_{r_{k}} \A_{n}} ∥∇v_{k}(x, y)∥^{2}dxdy

with n = n(r_{k}) ; moreover we define C_{k} in the same way, with R_{k} and V_{k} instead of r_{k} and v_{k} (of course n = n(R_{k}));

• Ω_{1}, Ω_{2} ⊆ ℝ^{2} are open sets such that K ⊆ Ω_{1} ⊆ \overline{Ω_{1}} ⊆ Ω_{2} ⊆ \overline{Ω_{2}} ⊆ Ω ;

• ν is the Dirac delta on ℝ , centered on 0 ;

• ε > 0 is given.

The sequences (c_{k}) and (C_{k}) tend to 0 ; indeed, for every k ∈ ℕ^{∗}

c_{k} ≤ \frac{1}{r_{k}^{2−α}} ∑_{i=1}^{m_{1}} ... ∑_{i=n}^{m_{n}} ∫_{p_{n}(i_{1},i_{2},...,i_{n})−r_{k}}^{p_{n}(i_{1},i_{2},...,i_{n})+t_{n}+r_{k}} (∫_{−r_{k}}^{0} \frac{∂u}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)^{2}dy+

∫_{1}^{1+r_{k}} \frac{∂u}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)^{2}dy) dx ≤ 2Mr_{k} \frac{t_{n} + 2r_{k}}{r_{k}^{2−α}} ∏_{h=1}^{n} m_{h} .

By (2.8), lim_{k} c_{k} = 0 . By the same arguments, using (2.9) instead of (2.8), one proves that lim_{k} C_{k} = 0 .

By Proposition 4.4 there exists a continuous function ω : ℝ^{2} → [0, 1] so that ω(x, y) = 0 if and only if (x, y) ∈ ℝ^{2} \ Ω_{2} and ω(x, y) = 1 if and only if (x, y) ∈ \overline{Ω}_{1} .

The function x → ∫_{0}^{1} ω(x, y)\frac{∂u}{∂y}(x, y)^{2}dy belongs to C_{c}(ℝ) and then, by Proposition 1.18, there exists k_{ε} ∈ ℕ^{∗} so that, for every k ∈ ℕ^{∗} with k > k_{ε} we have

|∫_{C} ∫_{0}^{1} ω(x, y)\frac{∂u}{∂y}(x, y)^{2}dydμ_{C}(x)+

− ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{k}=1}^{m_{k}}(∏_{h=1}^{k} l_{i_{h}}^{(h)}) ∫_{C} ∫_{0}^{1} ω(x, y)\frac{∂u}{∂y}(x, y)^{2}df_{i_{1}i_{2}...i_{k}♯}ν(x)| < ε . (2.11)

To prove statement 2 for u ∈ C^{1}(\overline{Ω}) we prove the condition 1. (b) of Proposition 2.9.

Let k ∈ ℕ^{∗} be so that n = n(r_{k}) > k_{ε} ; recalling that l_{i}^{(h)} = 1/m_{h} ∀h ∈ ℕ^{∗} ∀i ∈ I_{h} , we have

|f_{r_{k}}^{(α)}(v_{k}) − 2^{2−α}\underline{M^{(α)}}(K)f(u)| <

|\frac{1}{r_{k}^{2−α}} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} ∫_{p_{n}(i_{1},i_{2},...,i_{n})−r_{k}}^{p_{n}(i_{1},i_{2},...,i_{n})+t_{n}+r_{k}} ∫_{0}^{1} ∥∇v_{k}(x, y)∥^{2}dxdy+

−\frac{2^{2−α}}{∏_{h=1}^{n} m_{h}} \underline{M^{(α)}}(K) ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} ∫_{C} ∫_{0}^{1} \frac{∂u}{∂y}(f_{i_{1}i_{2}...i_{n}}(x), y)^{2}dydν(x)| +

c_{k} + ε2^{2−α}\underline{M^{(α)}}(K) ≤ c_{k} + ε2^{2−α}\underline{M^{(α)}}(K)+

|\frac{t_{n} + 2r_{k}}{r_{k}^{2−α}} − \frac{2^{2−α}}{∏_{h=1}^{n} m_{h}} \underline{M^{(α)}}(K)| ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} ∫_{0}^{1} \frac{∂u}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)^{2}dy ≤

M |\frac{t_{n} + 2r_{k}}{r_{k}^{2−α}} ∏_{h=1}^{n} m_{h} − 2^{2−α}\underline{M^{(α)}}(K)| + c_{k} + ε2^{2−α}\underline{M^{(α)}}(K) .

If k → +∞ , by (2.8), we have lim_{k} |f_{r_{k}}^{(α)}(v_{k}) − 2^{2−α}\underline{M^{(α)}}(K)f(u)| ≤ ε2^{2−α}\underline{M^{(α)}}(K) for every ε > 0 . Then statement 2. follows by statement 1. of Proposition 2.9.

To prove statement 5. for u ∈ C^{1}(\overline{Ω}) we prove statement 2. (b) of Proposition 2.9.

Let k ∈ ℕ^{∗} ; we have f_{R_{k}}^{(α)}(V_{k}) =

\frac{1}{R_{k}^{2−α}} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} ∫_{p_{n}(i_{1},i_{2},...,i_{n})−R_{k}}^{p_{n}(i_{1},i_{2},...,i_{n})+t_{n}+R_{k}} ∫_{0}^{1} \frac{∂u}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)^{2}dydx+c_{k} =

\frac{t_{n} + 2R_{k}}{R_{k}^{2−α}} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} ∫_{0}^{1} \frac{∂u}{∂y}(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)^{2}dy + c_{k} =

\frac{t_{n} + 2R_{k}}{R_{k}^{2−α}} (∏_{h=1}^{n} m_{h}) ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ...

∑_{i_{n}=1}^{m_{n}}(∏_{h=1}^{n} l_{i_{h}}^{(h)}) ∫_{C} ∫_{0}^{1} ω(x, y)\frac{∂u}{∂y}(f_{i_{1}i_{2}...i_{n}}(x), y)^{2}dydν + c_{k} .

Now, if k is so that n = n(R_{k}) > k_{ε} , by (2.11), we have

f_{R_{k}}^{(α)}(V_{k}) < \frac{t_{n} + 2R_{k}}{R_{k}^{2−α}} (∏_{h=1}^{n} m_{h}) ∫_{C} ∫_{0}^{1} ω(x, y)\frac{∂u}{∂y}(x, y)^{2}dydμ_{C} + ε + c_{k} =

\frac{t_{n} + 2R_{k}}{R_{k}^{2−α}} (∏_{h=1}^{n} m_{h})f(u) + ε + c_{k} .

By (2.9) lim sup_{k} f_{R_{k}}^{(α)}(V_{k}) ≤ 2^{2−α}\overline{M^{(α)}}(K)f(u) + ε . If ε → 0 , by Proposition 2.9 we have 5.

Now we consider u ∈ C(\overline{Ω}) with f(u) < +∞ . Let us prove 2. By Lemma 2.6 there exists a sequence (v_{n}) in C^{∞}(\overline{Ω})

so that lim_{n} ∥v_{n} − u∥_{L^{∞}(Ω)} = 0 and lim_{n} f(v_{n}) = f(u) .

Let n ∈ ℕ^{∗} ; we have proved that there exist a sequence (r_{k}^{(n)}) in ]0, d_{1}/2[ and a sequence (w_{k}^{(n)}) in C^{1}(\overline{Ω}) so that lim_{k} r_{k}^{(n)} = 0 , lim_{k} ∥w_{k}^{(n)}− v_{n}∥_{L^{∞}(Ω)} = 0 and lim_{k} f_{r_{k}^{(n)}}^{(α)}(w_{k}^{(n)}) = 2^{2−α}\underline{M^{(α)}}(K)f(v_{n}) .

Then there exist k′_{n}, k′′_{n}, k′′′_{n} ∈ ℕ^{∗} so that the following implications hold:

• k > k′_{n} ⟹ r_{k}^{(n)} < \frac{1}{n} ;

• k > k′′_{n} ⟹ ∥w_{k}^{(n)} − v_{n}∥_{L^{∞}(Ω)} < \frac{1}{n} ;

• k > k′′′_{n} ⟹ |f_{r_{k}^{(n)}}^{(α)}(w_{k}^{(n)}) − 2^{2−α}\underline{M^{(α)}}(K)f(v_{n})| < \frac{1}{n} .

Let k_{n} = 1 + max{k′_{n}, k′′_{n}, k′′′_{n}} ; we have lim_{n} r_{k_{n}}^{(n)} = 0 and for every n ∈ ℕ^{∗} , ∥u − w_{k_{n}}^{(n)}∥_{L^{∞}(Ω)} < ∥u − v_{n}∥_{L^{∞}(Ω)} + n^{−1} , from which lim_{n} ∥u − w_{k_{n}}^{(n)}∥_{L^{∞}(Ω)} = 0 . Moreover

|f_{r_{kn}^{(n)}}^{(α)}(w_{k_{n}}^{(n)}) − 2^{2−α}\underline{M^{(α)}}(K)f(u)| < \frac{1}{n} + 2^{2−α}\underline{M^{(α)}}(K)|f(v_{n}) − f(u)|

and then it tends to 0 as n → +∞ . Again by Proposition 2.9, statement 2 follows.

Let us prove 5. By Lemma 2.6 there exists a sequence (v_{n}) in C^{∞}(\overline{Ω}) such that ∥u − v_{n}∥_{L^{∞}(Ω)} < n^{−1} and 2^{2−α}\overline{M^{(α)}}(K)|f(u) − f(v_{n})| < n^{−1} for every n ∈ ℕ^{∗} .

Let (R_{k}) be a sequence converging to 0 in ]0, d_{1}/2[ . For every n ∈ ℕ^{∗} there exists a sequence (V_{k}^{(n)}) in C^{1}(\overline{Ω}) so that lim_{k} ∥V_{k}^{(n)} − v_{n}∥_{L^{∞}(Ω)} = 0 and

lim sup_{k} f_{R_{k}}^{(α)}(V_{k}^{(n)}) ≤ 2^{2−α}\overline{M^{(α)}}(K)f(v_{n}) < 2^{2−α}\overline{M^{(α)}}(K)f(u) + \frac{1}{n} .

Then there exist k′_{n}, k′′_{n} ∈ ℕ^{∗} so that ∥V_{k}^{(n)} − v_{n}∥_{L^{∞}(Ω)} < n^{−1} for every k ∈ ℕ^{∗} k > k′_{n} and

f_{R_{k}}^{(α)}(V_{k}^{(n)}) < 2^{2−α}\overline{M^{(α)}}(K)f(u) + \frac{2}{n}

for every k ∈ ℕ^{∗} k > k′′_{n} . We set k_{n} = 1 + max{k′_{n}, k′′_{n}} and w_{k} = V_{k}^{(n)} for every k ∈ ℕ^{∗} with k_{n} ≤ k < k_{n+1} .

For every k ∈ ℕ^{∗} there exists n_{k} ∈ ℕ^{∗} so that

∥w_{k}−u∥_{L^{∞}(Ω)} = ∥V_{k}^{(n_{k})}−u∥_{L^{∞}(Ω)} ≤ ∥V_{k}^{(n_{k})}−v_{n_{k}}∥_{L^{∞}(Ω)}+∥v_{n_{k}} −u∥_{L^{∞}(Ω)} < \frac{2}{n_{k}}

and

f_{R_{k}}^{(α)}(w_{k}) < 2^{2−α}\overline{M^{(α)}}(K)f(u) + \frac{2}{n_{k}} .

Moreover lim_{k} n_{k} = +∞ and then, by Proposition 2.9, 5 follows.

\underline{Proof of 3. and 6.} Let u ∈ C(\overline{Ω}) with f(u) < +∞ . If u ∈ C^{1}(\overline{Ω}) , then we can show that f_{−}^{(α)}(u) = 0 in the same way we proved statement 2.

So we suppose u ∉ C^{1}(\overline{Ω}) ; let (v_{n}) be a sequence in C^{1}(\overline{Ω}) such that lim_{n} ∥v_{n} − u∥_{L^{∞}(Ω)} = 0 ; for any n ∈ ℕ^{∗} f_{−}^{(α)}(v_{n}) = 0 and then there exists a sequence (r_{k}^{(n)}) converging to 0 in ]0, +∞[ and a sequence (u_{k}^{(n)}) in C(\overline{Ω}) such that lim_{k} ∥u_{k}^{(n)} − v_{n}∥_{L^{∞}(Ω)} = 0 and 0 = f_{−}^{(α)}(v_{n}) = lim_{k} f_{r_{k}^{(n)}}^{(α)}(u_{k}^{(n)}). By a standard diagonal argument it is possible to construct

a strictly increasing sequence (k_{n}) in ℕ^{∗} so that lim_{n} ∥u_{k_{n}}^{(n)} − u∥_{L^{∞}(Ω)} = 0 and lim_{n} f_{r_{kn}^{(n)}}^{(α)}(u_{k_{n}}^{(n)}) = 0 . By Proposition 2.9 f_{−}^{(α)}(u) = 0 .

Statement 6 can be proved in the same way ■

Corollary 2.28. - We have \underline{dim_{B}}K = \underline{dim_{C}}K ≤ \overline{dim_{C}}K = \overline{dim_{B}}K . PROOF. - Let α ∈]0, \underline{dim_{B}}K[ (β ∈]0, \overline{dim_{B}}K[); then \underline{M^{(α)}}(K) = +∞ (\overline{M^{(β)}}(K) = +∞) and by Propositon 2.27 there exists u ∈ C(\overline{Ω}) such that f_{−}^{(α)}(u) = +∞ > 0 (f_{+}^{(β)}(u) = +∞ > 0). Then \underline{dim_{C}}K ≥ α (\overline{dim_{C}}K ≥ β). It follows \underline{dim_{C}}K ≥ \underline{dim_{B}}K (\underline{dim_{C}}K ≥ \underline{dim_{B}}K).

If α ∈]\underline{dim_{B}}K, 2[ , then \underline{M^{(α)}}(K) = 0 and then f_{−}^{(α)}(u) = 0 ∀u ∈ C^{1}(\overline{Ω}). By Remark 8 \underline{dim_{C}}K ≤ α . Then \underline{dim_{C}}K ≤ \underline{dim_{B}}K . The last equality follows by Proposition 2.17 ■

Remark 19. - If the sequence (m_{n}) is bounded then dim_{C} K = dim_{B} K = dim K = d + 1 .

If the sequence (m_{n}) is not bounded then it may happen that \overline{dim_{C}}K = \overline{dim_{B}}K > dim K (see Example 5 in the first chapter).

Example 1. - Let q ∈ ℕ^{∗} q ≥ 3 , d = log_{q} 2 and m_{n} = 2 for every n ∈ ℕ^{∗} . Then C is the q-adic Cantor set. In this case t_{n} = q^{−n} for every n ∈ ℕ^{∗} .

By applying (2.8) and (2.9), an easy calculation shows that

\underline{M^{(d+1)}}(K) = 2^{d−1} lim_{n} \frac{t_{n} + 2r′_{n}}{r′_{n}^{1−d}} 2^{n} = \frac{1}{d(1 − d)^{1−d}} , with r′_{n} = \frac{1 − d}{2dq^{n}}

and

\overline{M^{(d+1)}}(K) = 2^{d−1} lim_{n} \frac{t_{n} + 2r′′_{n}}{r′′_{n}^{1−d}} 2^{n} = \frac{q − 1}{(q − 2)^{1−d}} , with r′′_{n} = \frac{q − 2}{2q^{n}} .

By Proposition 2.27 we have

f_{−}^{(d+1)}(u) = \frac{1}{d} (\frac{2}{1 − d})^{1−d} f(u)

and

f_{+}^{(d+1)}(u) = (q − 1) (\frac{2}{q − 2})^{1−d} f(u)

for every u ∈ C(\overline{Ω}) .

In particular dim_{C} K = d + 1 .

Example 2. - Let K be the set of the previous example, with q = 3 ; then dim_{C} K = 1 + log_{3} 2 .

Let S ⊆ ℝ^{2} be the non-empty compact set that is invariant with respect to the eight similitudes h_{i}(x, y) = (x/3, y/3) + (a_{i}, b_{i}) , where (a_{i}, b_{i}) i ∈ {1, 2, ..., 8} are the points (0, 0), (1/3, 0), (2/3, 0), (0, 1/3), (2/3, 1/3), (0, 2/3), (1/3, 2/3), (2/3, 2/3) ; S is called Sierpinski Carpet. By Theorem 1.19 dim S = log_{3} 8 = dim_{B} S .

It is clear that K ⊆ S ; then log_{3} 6 ≤ dim_{C} S ≤ log_{3} 8 .

Remark 20. - Let K be the set of Example 1. In this case C is invariant under the family of similitudes 𝔉 = {f_{1}, f_{2}} , f_{i}(x) = x/q + (i − 1)(1 − 1/q). By Remark 25 in the first chapter, also μ_{C} is invariant under 𝔉 , in the sense that μ_{C} = (f_{1♯}μ_{C} + f_{2♯}μ_{C})/2 .

Let G = {g_{1}, g_{2}, g_{3}, g_{4}} be the family of similitudes with g_{1}(x, y) = (f_{1}(x), y/2) , g_{2}(x, y) = (f_{2}(x), y/2) , g_{3}(x, y) = (f_{2}(x), (y +1)2) , g_{4}(x, y) = (f_{1}(x), (y + 1)2) ; then K is invariant under G . We prove that so is the functional f , in the sense that f(u) = ∑_{i=1}^{4} f(u ∘ g_{i}) for every u ∈ C(\overline{Ω}) .

Suppose that f(u) < +∞ : we have

∑_{i=1}^{4} f(u ∘ g_{i}) = ∑_{i=1}^{4} ∫_{C} ∫_{0}^{1} (\frac{∂(u ∘ g_{1})}{∂y}(x, y))^{2} dydμ_{C}(x) =

\frac{1}{4} ∑_{i=1}^{4} ∫_{C} ∫_{0}^{1} (\frac{∂u}{∂y}(g_{i}(x, y)))^{2} dydμ_{C}(x) .

Let, for example i = 1 ; then

∫_{C} ∫_{0}^{1} (\frac{∂u}{∂y}(g_{1}(x, y)))^{2} dydμ_{C}(x) = ∫_{C} ∫_{0}^{1} (\frac{∂u}{∂y} (f_{1}(x), \frac{1}{2}y))^{2} dydμ_{C}(x) =

2 ∫_{f_{1}(C)} ∫0^{\frac{1}{2}} (\frac{∂u}{∂y}(x, y))^{2} dydf_{1♯}μ_{C}(x) .

Let for every A ⊆ ℝ and r, s ∈ ℝ , rA + s = {rx + s | x ∈ A} .

Then, for every A ⊆ C , μ_{C}(f_{1}(A)) = (μ_{C}(A) + μ_{C}(f_{2}^{−1}(f_{1}(A))))/2 = (μ_{C}(A) + μ_{C}(A + 1 − q))/2 = μ_{C}(A)/2 because A + 1 − q ⊆ [1 − q, 2 − q] and then C ⋂(A + 1 − q) = ∅ .

Now, if A ⊆ f_{1}(C) , f_{1♯}μ_{C}(A) = μ_{C}(f_{1}^{−1}(A)) = 2μ_{C}(f_{1}(f_{1}^{−1}(A))) = 2μ_{C}(A) and then

∫_{C} ∫_{0}^{1} (\frac{∂u}{∂y}(g_{1}(x, y)))^{2} dydμ_{C}(x) = 4 ∫_{f_{1}(C)} ∫0^{\frac{1}{2}} (\frac{∂u}{∂y}(x, y))^{2} dydμ_{C}(x) =

4 ∫_{g_{1}(K)} (\frac{∂u}{∂y}(x, y))^{2} d(L^{1} × μ_{C}) .

In the same way we can show that, for 1 < i ≤ 4 ,

∫_{C} ∫_{0}^{1} (\frac{∂u}{∂y}(g_{i}(x, y)))^{2} dydμ_{C}(x) = 4 ∫_{g_{i}(K)} (\frac{∂u}{∂y}(x, y))^{2} d(L^{1} × μ_{C}) .

Since

⋃_{i=1}^{4} g_{i}(K) = K  (2.12)

and (L^{1} × μ_{C})(g_{i}(K) ⋂ g_{j}(K)) = 0 for 1 ≤ i < j ≤ 4 , we have

∑_{i=1}^{4} f(u ∘ g_{i}) = ∑_{i=1}^{4} ∫_{g_{i}(K)} (\frac{∂u}{∂y}(x, y))^{2} d(L^{1} × μ_{C}) = f(u) .

If f(u) = +∞ , by (2.12), there exists i ∈ {1, 2, 3, 4} such that F (u ∘ g_{i}) = +∞ .

110 CHAPTER 2. A CONJECTURE OF E. DE GIORGI

Invariant energy for the Koch curve

3.1 Introduction

We prove the Γ-convergence of two sequences of functionals on the Koch curve.

The first one is constructed by means of finite differences, while the second one is a sequence of integral functionals.

First we need some considerations about the analogous functional on the closed unit interval [0, 1] . Indeed there is an homeomorphism between the Koch curve and [0, 1] .

3.2 Γ-convergence on the closed unit interval

Given M ∈ ℕ^{∗} M ≥ 2 , the closed unit interval [0, 1] is invariant with respect to the family G = {g_{1}, g_{2}, ..., g_{M}} of contractive similitudes given by g_{i}(t) = (t + i − 1)/M . Moreover G satisfies the open set condition (for example, O =]0, 1[).

Now we define a sequence of functionals on ]0, 1[ . We consider the space C([0, 1]) equipped with the norm ∥u∥_{L^{∞}(0,1)} .

Definition. - The sequence E_{n} : C([0, 1]) → ℝ , n ∈ ℕ is defined by

\{ \begin{matrix}E_{o}(u) = (u(1) − u(0))^{2} \\ E_{n}(u) = M ∑_{j=1}^{M} E_{n−1}(u ∘ g_{j}) \end{matrix} .

Remark 1. - For every n ∈ ℕ and every u ∈ C([0, 1])

E_{n}(u) = M^{n} ∑_{j=1}^{M^{n}} (u (\frac{j}{M^{n}} ) − u (\frac{j − 1}{M^{n}}))^{2} .  (3.1)

The statement is obvious for n = 0 ; let us suppose that it is true for n ∈ ℕ , then

E_{n+1}(u) = M ∑_{j=1}^{M} M^{n} ∑_{i=1}^{M^{n}} (u (g_{j} (\frac{i}{M^{n}} )) − u (g_{j} (\frac{i − 1}{M^{n}})))^{2} =

M^{n+1} ∑_{j=1}^{M} ∑_{i=1}^{M^{n}} (u (\frac{i}{M^{n+1}} + \frac{j − 1}{M}) − u (\frac{i − 1}{M^{n+1}} + \frac{j − 1}{M^{n}}))^{2} =

M^{n+1} ∑_{k=1}^{M^{n+1}} (u (\frac{k}{M^{n+1}} ) − u (\frac{k − 1}{M^{n+1}} ))^{2} .

Definition. - We define a functional F : C([0, 1]) → ℝ

F (u) = \{ \begin{matrix}∫_{0}^{1} u′(t)^{2}dt & u ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} .

Proposition 3.1. - For every u ∈ C([0, 1]) , F (u) = Γ- lim_{n} E_{n}(u) . PROOF. - Let u ∈ C([0, 1]) ; we prove that F (u) ≤ lim inf_{n} E_{n}(u_{n}) for every sequence (u_{n}) in C([0, 1]) uniformly convergent to u .

First we suppose u ∈ H^{1}(0, 1) . Let (u_{n}) be a sequence of continuous functions uniformly convergent to u , we set, for every n ∈ ℕ^{∗} , v_{n} : [0, 1] → ℝ

v_{n}(t) = u_{n} (\frac{j − 1}{M^{n}}) + (u_{n} (\frac{j}{M^{n}} ) − u_{n} (\frac{j − 1}{M^{n}})) (M^{n}t − j + 1) (3.2)

if t ∈ [(j − 1)M^{−n}, jM^{−n}] .

It is easy to see that each function v_{n} is continuous, piecewise linear and v_{n}(jM^{−n}) = u_{n}(jM^{−n}) for every n, j ∈ ℕ^{∗} j ≤ M^{n} . It follows that E_{n}(v_{n}) = E_{n}(u_{n}) ∀n ∈ ℕ^{∗} .

We prove that lim_{n} ∥u − v_{n}∥_{L^{∞}(0,1)} = 0 . Let ε > 0 and n′_{ε} ∈ ℕ^{∗} be so that ∥u_{n} − u∥_{L^{∞}(0,1)} < ε ∀n > n_{ε} . Since u is uniformly continuous there exists δ > 0 so that |u(s) − u(t)| < ε for every s, t ∈ [0, 1] with |s − t| < δ . Let n′′_{ε} ∈ ℕ^{∗} be so that M^{−n′′_{ε}} < δ and n_{ε} = max{n′_{ε}, n′′_{ε}} ; then, for every n ∈ ℕ^{∗} n > n_{ε} and t ∈ [0, 1] we choose j ∈ ℕ^{∗} so that (j − 1)M^{−n} ≤ t ≤ jM^{−n} and we obtain, by (3.2),

|v_{n}(t) − u(t)| ≤ |u_{n} (\frac{j − 1}{M^{n}}) − u(t)| +

M^{n} |u_{n} (\frac{j}{M^{n}} ) − u_{n} (\frac{j − 1}{M^{n}})| |t − \frac{j − 1}{M^{n}} | ≤ |u_{n} (\frac{j − 1}{M^{n}}) − u (\frac{j − 1}{M^{n}})| +

|u (\frac{j − 1}{M^{n}}) − u(t)| + |u_{n} (\frac{j}{M^{n}} ) − u (\frac{j}{M^{n}} )| + |u (\frac{j}{M^{n}} ) − u (\frac{j − 1}{M^{n}})| +

|u (\frac{j − 1}{M^{n}}) − u_{n} (\frac{j − 1}{M^{n}})| < 5ε .

Now, let ε > 0 ; since u′ ∈ L^{2}(0, 1) , there exists φ ∈ C_{c}^{∞}(]0, 1[) so that ∥u′ − φ∥_{L^{2}(0,1)} < ε and \sqrt{F (u)}∥u′ − φ∥_{L^{2}(0,1)} < ε . Moreover there exists n_{ε} ∈ ℕ^{∗} such that ∥u − v_{n}∥_{L^{∞}(0,1)} < ε/∥φ′∥_{L^{∞}(0,1)} for every n > n_{ε}. Let n > n_{ε} , we have

F (u) = |(u′, u′)_{L^{2}(0,1)}| ≤ |(u′, u′ − φ)_{L^{2}(0,1)}| + |(u′, φ)_{L^{2}(0,1)}| ≤
|∫_{0}^{1} u′(t)φ(t)dt| + ∥u′∥_{L^{2}(0,1)}∥u′ − φ∥_{L^{2}(0,1)} < |∫_{0}^{1} u(t)φ′(t)dt| + ε ≤

|∫_{0}^{1}(u(t) − v_{n}(t))φ′(t)dt| + |∫_{0}^{1} v_{n}(t)φ′(t)dt| + ε ≤

|∫_{0}^{1} v′_{n}(t)φ(t)dt|+∥u−v_{n}∥_{L^{∞}(0,1)}∥φ′∥_{L^{∞}(0,1)} +ε < ∥v′_{n}∥_{L^{2}(0,1)}∥φ∥_{L^{2}(0,1)} +2ε ≤

(∥φ − u′∥_{L^{2}(0,1)} + ∥u′∥_{L^{2}(0,1)})\sqrt{∑_{j=1}^{M^{n}} ∫_{(j−1)M^{−n}}^{jM^{−n}} v′_{n}(t)^{2}dt} + 2ε <

(ε + \sqrt{F (u)})\sqrt{M^{2n} ∑_{j=1}^{M^{n}} M^{−n} (u_{n} (\frac{j}{M^{n}} ) − u_{n} (\frac{j − 1}{M^{n}}))^{2}} + 2ε =

(ε + \sqrt{F (u)})\sqrt{E_{n}(u_{n})} + 2ε .

By letting n → +∞ we have

F (u) ≤ (ε + \sqrt{F (u)})\sqrt{lim inf_{n} E_{n}(u_{n})} + 2ε .

Now, if ε → 0 we have the desired inequality.

Now we suppose u ∉ H^{1}(0, 1) . Let (u_{n}) be a sequence of continuous functions uniformly converging to u and let φ ∈ C_{c}^{∞}(]0, 1[) with ∥φ∥_{L^{2}(0,1)} ≤ 1 . Let n_{0} ∈ ℕ^{∗} be so that ∥u−v_{n}∥_{L^{∞}(0,1)} < 1/∥φ′∥_{L^{∞}(0,1)} ∀n > n_{0} ; we have, for n > n_{0} ,

|∫_{0}^{1} u(t)φ′(t)dt| ≤ |∫_{0}^{1}(u(t) − v_{n}(t))φ′(t)dt| + |∫_{0}^{1} v_{n}(t)φ′(t)dt| ≤

|∫_{0}^{1} v′_{n}(t)φ(t)dt| + ∥u − v_{n}∥_{L^{∞}(0,1)}∥φ′∥_{L^{∞}(0,1)} ≤
∥v′_{n}∥_{L^{2}(0,1)}∥φ∥_{L^{2}(0,1)} + 1 ≤ \sqrt{E_{n}(u_{n})} + 1 ;

where v_{n} is as in (3.2). It follows

|∫_{0}^{1} u(t)φ′(t)dt| ≤ 1 + \sqrt{lim inf_{n} E_{n}(u_{n})}

for every φ ∈ C_{c}^{∞}(]0, 1[) with ∥φ∥_{L^{2}(0,1)} ≤ 1 .

We define a linear functional g : C_{c}^{∞}(]0, 1[) → ℝ < g, φ >= ∫_{0}^{1} u(t)φ′(t)dt .

Let us suppose that sup{| ∫_{0}^{1} u(t)φ′(t)dt| | φ ∈ C_{c}^{∞}(]0, 1[) ∥φ∥_{L^{2}(0,1)} ≤ 1} = C < +∞ ; then g is continuous on C_{c}^{∞}(]0, 1[) with respect to the L^{2}- norm. Since g is linear it is uniformly continuous and by Theorem 4.1, there exists h : L^{2}(0, 1) → ℝ uniformly continuous such that h(φ) =< g, φ > for every φ ∈ C_{c}^{∞}(]0, 1[) . Since h is continuous on L^{2}(]0, 1[) , C_{C}^{∞}(]0, 1[) is dense in L^{2}(]0, 1[) and g is linear it is easy to see that h is linear. Then there exists w ∈ L^{2}(0, 1) such that < h, v >= ∫_{0}^{1} w(t)v(t)dt ∀v ∈ L^{2}(0, 1) and then

∫_{0}^{1} u(t)φ′(t)dt =< g, φ >=< h, φ >= ∫_{0}^{1} w(t)φ(t)dt ∀φ ∈ C_{c}^{∞}(]0, 1[) ;

that is u ∈ H^{1}(0, 1) and u′ = −w . Then C = +∞ and lim_{n} E_{n}(u_{n}) = +∞.

Now we prove that, if u ∈ H^{1}(0, 1) , there exists a sequence (u_{n}) of continuous functions uniformly convergent to u and such that lim_{n} E_{n}(u_{n}) = F (u) .

Let (φ_{n}) be a sequence in C_{c}^{∞}(]0, 1) such that lim_{n} ∥u′ −φ_{n}∥_{L^{2}(0,1)} = 0; for every n ∈ ℕ^{∗} we define w_{n} : [0, 1] → ℝ w_{n}(t) = u(0) + ∫_{0}^{t} φ_{n}(s)ds .

Of course w′_{n} = φ_{n} ∀n ∈ ℕ^{∗} ; moreover, for every n ∈ ℕ^{∗} and for every t ∈ [0, 1] we have

|u(t) − w_{n}(t)| = |u(0) + ∫_{0}^{t} u′(s)ds − u(0) − ∫_{0}^{t} φ_{n}(s)ds| ≤

∫_{0}^{1} |u′(s) − φ_{n}(s)|ds ≤ ∥u′ − φ_{n}∥_{L^{2}(0,1)} .

Then lim_{n} ∥u − w_{n}∥_{L^{∞}(0,1)} = 0 .

Let us consider w_{1} : w′_{1}^{2} is uniformly continuous and then there exists δ_{1} > 0 such that |w′_{1}(t)^{2}−w′_{1}(s)^{2}| < 1 for every s, t ∈ [0, 1] with |s−t| < δ_{1}. Moreover there exists h_{1} ∈ ℕ^{∗} such that M^{−h_{1}} < δ_{1} . We set n_{1} = h_{1} and u_{1} = u_{2} = ⋯ = u_{n_{1}} = w_{1} .

By recurrence we define a sequence (n_{k}) of natural numbers and a sequence (u_{n}) in C^{1}([0, 1]) . Suppose to have defined n_{1}, n_{2}, ..., n_{k} and u_{1}, u_{2}, ..., u_{n_{k}} for a given k ∈ ℕ^{∗} ; then there exists δ_{k+1} > 0 such that |w′_{k+1}(t)^{2} − w′_{k+1}(s)^{2}| < 1/(k + 1) for every s, t ∈ [0, 1] with |s − t| < δ_{k+1} and there exists h_{k+1} ∈ ℕ^{∗} such that M^{−h_{k+1}} < δ_{k+1} . We set n_{k+1} = max{1 + n_{k}, h_{k+1}} and u_{n_{k}+1} = u_{n_{k}+2} = ⋯ = u_{n_{k+1}} = w_{k+1} .

We consider the new sequence (u_{n}) so constructed; we still have lim_{n} ∥u_{n}− u∥_{L^{∞}(0,1)} = 0 and lim_{n} ∥u′_{n} − u′∥_{L^{2}(0,1)} = 0 . We have to prove that lim_{n} E_{n}(u_{n}) = F (u) .

For every n ∈ ℕ^{∗}

|F (u)−E_{n}(u_{n})| ≤ |F (u)−F (u_{n})|+|F (u_{n})−E_{n}(u_{n})| = |∥u′∥_{L^{2}(0,1)}^{2}−∥u′_{n}∥_{L^{2}(0,1)}^{2}|

+ |∑_{j=1}^{M^{n}} ∫_{(j−1)M^{−n}}^{jM^{−n}} u′_{n}(t)^{2}dt − M^{n} ∑_{j=1}^{M^{n}} (u_{n} (\frac{j}{M^{n}} ) − u_{n} (\frac{j − 1}{M^{n}}))^{2}| .

Let n ∈ ℕ^{∗} , since u_{n} ∈ C^{1}([0, 1]) , for every j ∈ ℕ^{∗} j ≤ M^{n} there exist s_{j}^{(n)}, t_{j}^{(n)} ∈ [(j − 1)M^{−n}, jM^{−n}] so that ∫_{(j−1)M^{−n}}^{jM^{−n}} u′_{n}(t)^{2}dt = M^{−n}u′_{n}(s_{j}^{(n)})^{2}

and u_{n}(jM^{−n}) − u_{n}((j − 1)M^{−n}) = M^{−n}u′_{n}(t_{j}^{(n)}) .

Then

|F (u) − E_{n}(u_{n})| ≤ |∥u′∥_{L^{2}(0,1)} − ∥u′_{n}∥_{L^{2}(0,1)}|(2∥u′∥_{L^{2}(0,1)} + ∥u′_{n} − u′∥_{L^{2}(0,1)})+

| ∑_{j=1}^{M^{n}} M^{−n}(u′_{n}(s_{j}^{(n)})^{2} − u′_{n}(t_{j}^{(n)})^{2})| .

Let ε > 0 : there exists k_{ε} ∈ ℕ^{∗} so that ∥u′ − w′_{k}∥_{L^{2}(0,1)} < ε and k^{−1} < ε for every k > k_{ε} . For every n ∈ ℕ^{∗} n > n_{k_{ε}+1} there exists k ∈ ℕ^{∗} k > k_{ε} such that u_{n} = w_{k} and then ∥u′ − u′_{n}∥_{L^{2}(0,1)} < ε ; moreover |s_{j}^{(n)}−t_{j}^{(n)}| ≤ M^{−n} ≤ M^{−n_{k}} ≤ M^{−h_{k}} < δ_{k} , from which |u′_{n}(s_{j}^{(n)})−u′_{n}(t_{j}^{(n)})| = |w′_{k}(s_{j}^{(n)}) − w′_{k}(t_{j}^{(n)})| < k^{−1} < ε .

Then, for n > n_{k_{ε}+1}

|F (u) − E_{n}(u)| < ε(ε + 2\sqrt{F (u)}) + ε ■

Remark 2. - The functional F is proportional to G(F) , in the sense that for every u ∈ C([0, 1]) F (u) = M ∑_{j=1}^{M} F (u ∘ g_{j}) .

Indeed, for every u ∈ H^{1}(0, 1)

M ∑_{j=1}^{M} F (u ∘ g_{j}) = M ∑_{j=1}^{M} ∫_{0}^{1} (\frac{d}{dt}u (\frac{t + j − 1}{M}))^{2} dt =

\frac{1}{M} ∑_{j=1}^{M} ∫_{0}^{1} u′ (\frac{t + j − 1}{M})^{2} dt = ∑_{j=1}^{M} ∫_{(j−1)/M}^{j/M} u′(s)^{2}ds = F (u) .

3.3 Γ-convergence on the Koch curve

Let 𝔉 = {f_{1}, f_{2}, f_{3}, f_{4}} where f_{1}, f_{2}, f_{3}, f_{4} : ℝ^{2} → ℝ^{2} are the four similitudes f_{1}(x, y) = \frac{1}{3}(x, y) , f_{2}(x, y) = \frac{1}{6}(x − \sqrt{3}y + 2, \sqrt{3}x + y) , f_{3}(x, y) = \frac{1}{6}(x + \sqrt{3}y + 3, −\sqrt{3}x + y + \sqrt{3}) , f_{4}(x, y) = \frac{1}{3}(x + 2, y) . The Koch curve is the unique non-empty compact set which is invariant with respect to 𝔉 .

The functions f_{1}, f_{2}, f_{3}, f_{4} are contractive similitudes with Lipschitz constant equal to 1/3 ; moreover 𝔉 satisfies the open set condition (for example one can take the open set O to be the open triangle with vertices (0, 0), (1, 0), (\frac{1}{2}, \frac{\sqrt{3}}{6})). By Theorem 1.19 K is a d-set with d = log_{3} 4 .

As said in the title K is a curve, in the sense that there is a continuous one to one function γ : [0, 1] → K . Moreover γ^{−1} is continuous.

Before defining the sequences of functionals on K we prove our last claim.

Let I_{n} = {1, 2, 3, 4} ∀n ∈ ℕ^{∗} and I = ∏_{n=1}^{+∞} I_{n} = {1, 2, 3, 4}^{ℕ^{∗}} ; as done in the first chapter, we define p : I → ℝ^{2} p(i) = lim_{n} f_{i_{1}i_{2}...i_{n}}(0, 0) , where i_{j} = π_{j}(i) ∀j ∈ ℕ^{∗} . Then K = p(I) .

In the same way [0, 1] = q(I) , where q(i) = g_{i_{1}i_{2}...i_{n}}(0) (choose M = 4 in the previous section).

Our target is to define γ as p ∘ q^{−1} but p and q are not injective; therefore we have to prove that, for every k, h ∈ I , p(k) = p(h) if and only if q(k) = q(h) .

For every n ∈ ℕ^{∗} the functions p_{n} : ∏_{j=1}^{n} I_{j} → ℝ^{2} , q_{n} : ∏_{j=1}^{n} I_{j} → ℝ are defined as in the first chapter, that is p_{n}(i_{1}, i_{2}, ..., i_{n}) = f_{i_{1}i_{2}...i_{n}}(0, 0) and q_{n}(i_{1}, i_{2}, ..., i_{n}) = g_{i_{1}i_{2}...i_{n}}(0) .

We set P ⊆ ℝ^{2} to be the closed triangle with vertices (0, 0), (1, 0), (\frac{1}{2}, \frac{\sqrt{3}}{6}). It is easy to see that 𝔉(P) ⊆ P and for every i, j ∈ {1, 2, 3, 4} with i < j , f_{i}(P) ⋂ f_{j}(P) = {f_{i}(1, 0)} = {f_{j}(0, 0)} if j = i+1 and f_{i}(P) ⋂ f_{j}(P) = ∅ if j > i + 1 .

In the same way we set Q = [0, 1] . Then G(Q) = Q and for every i, j ∈ {1, 2, 3, 4} with i < j , g_{i}(Q) ⋂ g_{j}(Q) = {g_{i}(1)} = {g_{j}(0)} if j = i+1 and g_{i}(q) ⋂ g_{j}(Q) = ∅ if j > i + 1 .

Lemma 3.1. - For every i ∈ I we have the following:

1. p(i) = (0, 0) if and only if π_{j}(i) = 1 ∀j ∈ ℕ^{∗} ,

2. p(i) = (1, 0) if and only if π_{j}(i) = 4 ∀j ∈ ℕ^{∗} ,

3. q(i) = 0 if and only if π_{j}(i) = 1 ∀j ∈ ℕ^{∗} ,

4. q(i) = 1 if and only if π_{j}(i) = 4 ∀j ∈ ℕ^{∗} .

PROOF. - We prove 1; the proofs of 2,3,4 are similar. Let i ∈ I and i_{j} = π_{j}(i) ∀j ∈ ℕ^{∗} . It is clear that if i_{j} = 1 ∀j ∈ ℕ^{∗} , p(i) = (0, 0) (since (0, 0) is the fixed point of f_{1}).

Conversely, let n_{0} = min{j ∈ ℕ^{∗} | i_{j} > 1} ; then (0, 0) ∉ f_{i_{n_{0}}}(P) ; indeed (0, 0) ∈ f_{1}(P) and f_{1}(P) ⋂ f_{2}(P) = \{(\frac{1}{3}, 0)\} , f_{1}(P) ⋂ f_{i_{n_{0}}}(P) = ∅ if n_{0} > 2 .

Since f_{i_{n_{0}}}(P) is closed, there exists δ > 0 such that ∥f_{i_{n_{0}}}(x, y)∥ ≥ δ for every (x, y) ∈ P and then ∥f_{i_{1}i_{2}...i_{n_{0}}}(x, y)∥ ≥ δ3^{1−n_{0}} for every (x, y) ∈ P . Since f_{i_{n_{0}+1}i_{n_{0}+2}...i_{n}}(0, 0) ∈ f_{i_{n_{0}+1}i_{n_{0}+2}...i_{n}}(P) ⊆ P for every n > n_{0} , it follows ∥f_{i_{1}i_{2}...i_{n}}(0, 0)∥ ≥ δ3^{1−n_{0}} ∀n ∈ ℕ^{∗} n > n_{0} and then ∥p(i)∥ ≥ δ3^{1−n_{0}} > 0 ■

Lemma 3.2. - For every k, h ∈ I p(k) = p(h) if and only if q(k) = q(h).

Moreover ♯p^{−1}(x, y) ≤ 2 and ♯q^{−1}(t) ≤ 2 for every (x, y) ∈ K and for every t ∈ [0, 1] . PROOF. -Let k, h ∈ I be so that p(k) = p(h) ; we suppose h ≠ k , otherwise the statement is obvious.

Let, for every j ∈ ℕ^{∗} k_{j} = π_{j}(k) and h_{j} = π_{j}(h) and let n_{0} = min{j ∈ ℕ^{∗} | k_{j} ≠ h_{j}} ; we suppose k_{n_{0}} < h_{n_{0}} .

If h_{n_{0}}−k_{n_{0}} > 1 then there exists δ > 0 such that ∥(x_{1}, y_{1})−(x_{2}, y_{2})∥ ≥ δ for every (x_{1}, y_{1}) ∈ f_{k_{n_{0}}}(P) and every (x_{2}, y_{2}) ∈ f_{h_{n_{0}}}(P) . Then, for every n ∈ ℕ^{∗} n > n_{0} ,

∥f_{k_{1}k_{2}...k_{n}}(0, 0) − f_{h_{1}h_{2}...h_{n}}(0, 0)∥ =

\frac{1}{3^{n_{0}−1}} ∥f_{k_{n_{0}}}(f_{k_{n_{0}+1}k_{n_{0}+2}...k_{n}}(0, 0)) − f_{h_{n_{0}}}(f_{h_{n_{0}+1}h_{n_{0}+2}...h_{n}}(0, 0))∥ ≥ \frac{δ}{3^{n_{0}−1}}

and then ∥p(k) − p(h)∥ ≥ δ3^{1−n_{0}} > 0 . Then h_{n_{0}} = k_{n_{0}} + 1 ; we prove that k_{n} = 4 and h_{n} = 1 for every n ∈ ℕ^{∗} n > n_{0} .

The function f_{k_{1}k_{2}...k_{n_{0}−1}} is injective and p(k) ∈ f_{k_{1}k_{2}...k_{n_{0}}}(P) , p(h) ∈ f_{h_{1}h_{2}...h_{n_{0}}}(P) ; by our assumption we have

p(h) = p(k) ∈ f_{k_{1}k_{2}...k_{n_{0}−1}}(f_{k_{n_{0}}}(P) ⋂ f_{h_{n_{0}}}(P)) =
{f_{k_{1}k_{2}...k_{n_{0}−1}}(f_{k_{n_{0}}}(1, 0))} = {f_{k_{1}k_{2}...k_{n_{0}−1}}(f_{h_{n_{0}}}(0, 0))} .

Then lim_{n} p_{n−n_{0}}(k_{n_{0}+1}, k_{n_{0}+2}, ..., k_{n}) = (1, 0) and lim_{n} p_{n−n_{0}}(h_{n_{0}+1}, h_{n_{0}+2}, ..., h_{n}) = (0, 0) ; by Lemma 3.1 k_{n} = 4 and h_{n} = 1 ∀n ∈ ℕ^{∗} n > n_{0} .

By the same lemma lim_{n} q_{n−n_{0}}(k_{n_{0}+1}, k_{n_{0}+2}, ..., k_{n}) = 1 and lim_{n} q_{n−n_{0}} (h_{n_{0}+1}, h_{n_{0}+2}, ..., h_{n}) = 0 . But g_{k_{n_{0}}}(1) = g_{h_{n_{0}}}(0) because h_{n_{0}} = k_{n_{0}} + 1 ; then q(k) = q(h) .

Now, let i ∈ I be so that p(i) = p(k) = p(h) and let m_{0} = min{j ∈ ℕ^{∗} | i_{j} ≠ k_{j}} . If m_{0} > n_{0} then i_{m_{0}} = 3 because k_{m_{0}} = 4 ; but this fact implies k_{n} = 1 ∀n ∈ ℕ^{∗} n > m_{0} and we know that k_{n} = 4 . Then it must be m_{0} < n_{0} ; but in this case k_{n} is constant for n > m_{0} and then k_{n} = 4 ∀n > m_{0} ; that is k_{n_{0}} = 4 and h_{n_{0}} = 5 . Then m_{0} = n_{0} and i = h (if i_{n_{0}} = k_{n_{0}} − 1 then k_{n_{0}+1} = 1 and not 4).

To prove the opposite implication one can exchange 𝔉, p, P with G, q, Q and repeat the same considerations ■

Definition. - We set γ : [0, 1] → K γ = p ∘ q^{−1} .

Remark 3. - By Lemma 3.2 γ is well defined and it is a one to one correspondence between [0, 1] and K .

Let C ⊆ K be closed: by Proposition 1.14 p^{−1}(C) is closed in I and then it is compact (since I is compact); then γ^{−1}(C) = q(p^{−1}(C)) is compact (and then closed) because q is continuous.

In the same way we can prove that γ^{−1} is continuous. Then γ is an homeomorphism between [0, 1] and K .

Remark 4. - Let t ∈ [0, 1] , i ∈ q^{−1}(t) , i_{n} = π_{n}(i) ∀n ∈ ℕ^{∗} and j ∈ ℕ^{∗} j ≤ 4 . Moreover, let k ∈ I be so that π_{1}(k) = j and π_{n}(k) = π_{n−1}(i) ∀n ≥ 2 . Then

q(k) = lim_{n} g_{ji_{1}i_{2}...i_{n}}(0) = g_{j}(lim_{n} g_{i_{1}i_{2}...i_{n}}(0)) = g_{j}(q(i)) = g_{j}(t) ,

that is k ∈ q^{−1}(g_{j}(t)) and then, p(k) = γ(g_{j}(t)) ; but

p(k) = lim_{n} f_{ji_{1}i_{2}...i_{n}}(0, 0) = f_{j}(lim_{n} f_{i_{1}i_{2}...i_{n}}(0, 0)) = f_{j}(p(i)) = f_{j}(γ(t)) .

Then

f_{j} ∘ γ = γ ∘ g_{j} ∀j ∈ {1, 2, 3, 4}  (3.3)

Remark 5. - Given n ∈ ℕ^{∗} , the 4^{n} intervals of the form [q_{n}(i_{1}, i_{2}, ..., i_{n}), q_{n}(i_{1}, i_{2}, ..., i_{n}) + 4^{−n}[ are pairwise disjoint.

For all (i_{1}, i_{2}, ..., i_{n+1}) ∈ {1, 2, 3, 4}^{n+1} the interval [q_{n}(i_{1}, i_{2}, ..., i_{n}), q_{n}(i_{1}, i_{2}, ..., i_{n}) + 4^{−n}[ contains [q_{n}(i_{1}, i_{2}, ..., i_{n+1}), q_{n}(i_{1}, i_{2}, ..., i_{n+1}) + 4^{−n−1}[ and

[0, 1[= ⋃_{i_{1}=1}^{4} ⋃_{i_{2}=1}^{4} ... ⋃_{i_{n}=1}^{4}[q_{n}(i_{1}, i_{2}, ..., i_{n}), q_{n}(i_{1}, i_{2}, ..., i_{n}) + 4^{−n}[ .

Now we define a sequence (γ_{n}) of curves that converge uniformly to γ .

Definition. - For every n ∈ ℕ^{∗} we define γ_{n} : [0, 1] → ℝ^{2} γ_{n}(1) = (1, 0) and γ_{n}(t) = f_{i_{1}i_{2}...i_{n}}(4^{n}(t − q_{n}(i_{1}, i_{2}, ..., i_{n})), 0) if t ∈ [q_{n}(i_{1}, i_{2}, ..., i_{n}), q_{n}(i_{1}, i_{2}, ..., i_{n}) + 4^{−n}[ . Moreover γ_{0}(t) = (t, 0) .

Remark 6. - The functions q_{n} are given by

q_{n}(i_{1}, i_{2}, ..., i_{n}) = ∑_{j=1}^{n} \frac{i_{j} − 1}{4^{j}} .  (3.4)

We prove this by induction: let n ∈ ℕ^{∗} , we have g_{i_{n}}(0) = (i_{n} − 1)/4 . Let us suppose that for k ∈ ℕ^{∗} 2 ≤ k ≤ n

g_{i_{k}i_{k+1}...i_{n}}(0) = ∑_{j=k}^{n} \frac{i_{j} − 1}{4^{j−k+1}} ,  (3.5)

then

g_{i_{k−1}i_{k}...i_{n}}(0) = g_{i_{k−1}}(g_{i_{k}i_{k+1}...i_{n}}(0)) = ∑_{j=k−1}^{n} \frac{i_{j} − 1}{4^{j−k}} .

By induction (3.5) holds for any k ≤ n and in particular, for k = 1 we have (3.4).

It follows

q(i) = ∑_{j=1}^{+∞} \frac{i_{j} − 1}{4^{j}} ∀i ∈ I , with i_{j} = π_{j}(i) .  (3.6)

Let j ∈ ℕ^{∗} j ≤ 4^{n} − 1 ; by (3.4) there is a unique (i_{1}, i_{2}, ..., i_{n}) ∈ {1, 2, 3, 4}^{n} so that j4^{−n} = q_{n}(i_{1}, i_{2}, ..., i_{n}) ; then, if i ∈ I is so that π_{k}(i) = 1 ∀k > n ,

γ_{n} (\frac{j}{4^{n}} ) = γ_{n}(q_{n}(i_{1}, i_{2}, ..., i_{n})) = f_{i_{1}i_{2}...i_{n}}(0, 0) = lim_{k} f_{i_{1}i_{2}...i_{k}}(0, 0) = p(i) =

γ(q(i)) = γ(lim_{k} g_{i_{1}i_{2}...i_{k}}(0)) = γ(g_{i_{1}i_{2}...i_{n}}(0)) = γ(q_{n}(i_{1}, i_{2}, ..., i_{n})) = γ (\frac{j}{4^{n}} )

because (0, 0) is fixed for f_{1} and 0 is fixed for g_{1} .

Remark 7. - Let n ∈ ℕ , j ∈ {1, 2, 3, 4} and t ∈ [0, 1] . Then

f_{j}(γ_{n}(t)) = γ_{n+1}(g_{j}(t)) .  (3.7)

Suppose n > 0 , t < 1 , j ∈ {1, 2, 3, 4} ; let i_{1}, i_{2}, ..., i_{n} ∈ {1, 2, 3, 4} be so that t ∈ [q_{n}(i_{1}, i_{2}, ..., i_{n}), q_{n}(i_{1}, i_{2}, ..., i_{n}) + 4^{−n}[ ; by (3.4) g_{j}(t) ∈ [q_{n+1}(j, i_{1}, i_{2}, ..., i_{n}), q_{n+1}(j, i_{1}, i_{2}, ..., i_{n}) + 4^{−n−1}[ ; then

γ_{n+1}(g_{j}(t)) = f_{ji_{1}i_{2}...i_{n}}(4^{n+1}(g_{j}(t) − q_{n+1}(j, i_{1}, i_{2}, ..., i_{n})), 0) =
f_{j}(f_{i_{1}i_{2}...i_{n}}(4^{n}(t − 4q_{n+1}(j, i_{1}, i_{2}, ..., i_{n}) + j − 1), 0)) =
f_{j}(f_{i_{1}i_{2}...i_{n}}(4^{n}(t − q_{n}(i_{1}, i_{2}, ..., i_{n})), 0)) = f_{j}(γ_{n}(t)) .

Remark 8. - For every n ∈ ℕ^{∗} γ_{n} is continuous and piecewise linear.

For every t ∈]0, 1[\q_{n}(π_{1}, 2, ..., n(I)) γ_{n} is differentiable in t and ∥γ′_{n}(t)∥ = (\frac{4}{3})^{n} . The length of K(γ_{n}) is (\frac{4}{3})^{n} .

Indeed for every i_{1}, i_{2}, ..., i_{n} ∈ {1, 2, 3, 4} γ_{n}([q_{n}(i_{1}, i_{2}, ..., i_{n}), q_{n}(i_{1}, i_{2}, ..., i_{n}) + 4^{−n}[) = f_{i_{1}i_{2}...i_{n}}([0, 1[) is a segment whose length is 3^{−n} .

Proposition 3.2. - The sequence (γ_{n}) converges to γ uniformly. PROOF. -Let t ∈ [0, 1[ and i_{1}, i_{2}, ...i_{n} ∈ {1, 2, 3, 4} be so that t ∈ [q_{n}(i_{1}, i_{2}, ..., i_{n}), q_{n}(i_{1}, i_{2}, ..., i_{n}) + 4^{−n}[ ; then there exists i ∈ q^{−1}(t) so that i_{j} = π_{j}(i) ∀j ∈ ℕ^{∗} j ≤ n ; let i_{j} = π_{j}(i) even for j > n . Since [0, 1[×{0} ⊆ P and for every k ∈ ℕ^{∗} k > n f_{i_{n+1}i_{n+2}...i_{k}}(0, 0) ∈ P we have

∥γ_{n}(t) − γ(t)∥ = ∥f_{i_{1}i_{2}...i_{n}}(4^{n}(t − q_{n}(i_{1}, i_{2}, ..., i_{n})), 0) − lim_{k} f_{i_{1}i_{2}...i_{k}}(0, 0)∥ =

lim_{k} \frac{1}{3^{n}} ∥(4^{n}(t − q_{n}(i_{1}, i_{2}, ..., i_{n})), 0) − f_{i_{n+1}i_{n+2}...i_{k}}(0, 0)∥ ≤ \frac{1}{3^{n}} |P | = \frac{1}{3^{n}} .

If t = 1 then γ_{n}(t) = γ(t) = (1, 0) ∀n ∈ ℕ^{∗} ■

Now we define a functional on K . As done in the second chapter we consider an open bounded set Ω ⊆ ℝ^{2} such that K ⊆ Ω . We also suppose that γ_{n}([0, 1]) ⊆ Ω for every n ∈ ℕ^{∗} .

The space we consider is still C(\overline{Ω}) equipped with the norm ∥u∥_{L^{∞}(Ω)} .

Definition. - We define a functional E : C(\overline{Ω}) → ℝ^{∗}

E(u) = \{ \begin{matrix}∫_{0}^{1} (\frac{d}{dt}u(γ(t)))^{2} dt & u ∘ γ ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} .

Remark 9. - It is well known that the two components of γ are not Lipschitz functions. Moreover those functions have not bounded variation since K is not L^{1}-rectifiable.

Thus one can ask if there exists a non-constant function u ∈ C(\overline{Ω}) so that E(u) < +∞ .

To prove it, let v ∈ H^{1}(0, 1) ; then v ∘ γ^{−1} is continuous on K . By Theorem 4.2 there exists u ∈ C(ℝ^{2}) so that u(x, y) = v(γ^{−1}(x, y)) for all (x, y) ∈ K . It follows u ∘ γ = v ∈ H^{1}(0, 1) .

We prove that the functional E is finite on a dense subset of C(\overline{Ω}) .

Proposition 3.3. - For every u ∈ C(\overline{Ω}) there exists a sequence (u_{n}) in C(\overline{Ω}) such that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 and E(u_{n}) < +∞ for all n ∈ ℕ^{∗}. PROOF. - For every n ∈ ℕ^{∗} we consider the function v_{n} : [0, 1] → ℝ

v_{n}(t) = u (γ (\frac{j − 1}{4^{n}})) + (u (γ (\frac{j}{4^{n}} )) − u (γ (\frac{j − 1}{4^{n}}))) (4^{n}t − j + 1)

if t ∈ [(j − 1)4^{−n}, j4^{−n}] . For every n ∈ ℕ^{∗} , v_{n} is continuous and piecewise linear; then it belongs to H^{1}(0, 1) . The function v_{n} ∘ γ^{−1} : K → ℝ is continuous and by Theorem 4.2 there exists w_{n} ∈ C(ℝ^{2}) such that w_{n}(x, y) = v_{n}(γ^{−1}(x, y)) for all (x, y) ∈ K .

Let n ∈ ℕ^{∗} ; the function w_{n} is uniformly continuous on Ω and then there exists δ_{n} > 0 so that |w_{n}(x′, y′) − w_{n}(x′′, y′′)| < 1/n for every (x′, y′), (x′′, y′′) ∈ Ω with ∥(x′, y′) − (x′′, y′′)∥ < δ_{n} .

Let u ∈ C(\overline{Ω}) ; since u is uniformly continuous there exists σ_{n} > 0 so that |u(x′, y′) − u(x′′, y′′)| < 1/n for every (x′, y′), (x′′, y′′) ∈ Ω with ∥(x′, y′) − (x′′, y′′)∥ < σ_{n}.

Let r_{n} = min{δ_{n}, σ_{n}} ; K is compact and contained into its open r_{n}- neighborhood; then there exist k_{n} ∈ ℕ^{∗} and (x_{1}^{(n)}, y_{1}^{(n)}), (x_{2}^{(n)}, y_{2}^{(n)}), ..., (x_{k_{n}}^{(n)}, y_{k_{n}}^{(n)}) ∈ K such that K ⊆ ⋃_{i=1}^{k_{n}} D_{ℝ^{2}}((x_{i}^{(n)}, y_{i}^{(n)}), r_{n}) . By the lemma on the partition of the unity there exist φ_{0}^{(n)}, φ_{1}^{(n)}, ..., φ_{k_{n}}^{(n)} ∈ C^{∞}(ℝ^{2}) so that ∀(x, y) ∈ ℝ^{2} ∀i ∈ ℕ^{∗} i ≤ k_{n} we have 0 ≤ φ_{i}^{(n)}(x, y) ≤ 1 and ∑_{i=0}^{k_{n}} φ_{i}^{(n)}(x, y) = 1 ; moreover, for 1 ≤ i ≤ k_{n} the support of φ_{i}^{(n)} is contained in D_{ℝ^{2}}((x_{i}^{(n)}, y_{i}^{(n)}), r_{n}) and the support of φ_{0}^{(n)} does not intersect K .

We define u_{n} : Ω → ℝ by

u_{n}(x, y) = φ_{0}^{(n)}(x, y)u(x, y) + w_{n}(x, y) ∑_{i=1}^{k_{n}} φ_{i}^{(n)}(x, y) .

It is clear that u_{n} ∈ C(\overline{Ω}) for every n ∈ ℕ^{∗} ; moreover, if (x, y) ∈ K , then φ_{0}^{(n)}(x, y) = 0 and u_{n}(x, y) = w_{n}(x, y) ; then u_{n} ∘ γ = v_{n} ∈ H^{1}(0, 1) ∀n ∈ ℕ^{∗}.

Now we prove that lim_{n} ∥u_{n} −u∥_{L^{∞}(Ω)} = 0 . Let (x, y) ∈ Ω and n ∈ ℕ^{∗}.

If (x, y) ∉K_{r_{n}}^{∘} then u_{n}(x, y) = u(x, y) .

If (x, y) ∈K_{r_{n}}^{∘} then

|u_{n}(x, y)−u(x, y)| = |φ_{0}^{(n)}(x, y)u(x, y)+w_{n}(x, y)−φ_{0}^{(n)}(x, y)w_{n}(x, y)−u(x, y)|

= (1 − φ_{0}^{(n)}(x, y))|w_{n}(x, y) − u(x, y)| ≤ |w_{n}(x, y) − u(x, y)| .

Let ε > 0 and let δ > 0 be such that |u(γ(t)) − u(γ(s))| < ε for every t, s ∈ [0, 1] with |t − s| < δ . Let n ∈ ℕ^{∗} with 4^{−n} < δ and 2/n < ε ; if

(x, y) ∈K_{r_{n}}^{∘} , then there exists t_{n} ∈ [0, 1] such that ∥(x, y) − γ(t_{n})∥ < r_{n} and there exists j_{n} ∈ ℕ^{∗} j_{n} ≤ 4^{n} so that (j_{n} − 1)4^{−n} ≤ t_{n} ≤ j4^{−n} . Then

|u_{n}(x, y) − u(x, y)| ≤

|w_{n}(x, y) − w_{n}(γ(t_{n}))| + |w_{n}(γ(t_{n})) − u(γ(t_{n}))| + |u(γ(t_{n})) − u(x, y)| <

|u (γ (\frac{j_{n} − 1}{4^{n}})) + (u (γ (\frac{j_{n}}{4^{n}} )) − u (γ (\frac{j_{n} − 1}{4^{n}}))) (4^{n}t_{n} − j_{n} + 1)+

−u(γ(t_{n}))| + \frac{2}{n} < |u (γ (\frac{j_{n} − 1}{4^{n}})) − u(γ(t_{n}))| +

|u (γ (\frac{j_{n}}{4^{n}} )) − u (γ (\frac{j_{n} − 1}{4^{n}}))| + ε < 3ε ■

Remark 10. - The functional E is proportional to 𝔉(E) , in the sense that for every u ∈ C(\overline{Ω}) E(u) = 4 ∑_{j=1}^{4} E(u ∘ f_{j}) .

Indeed, by (3.3),

∑_{j=1}^{4} E(u ∘ f_{j}) = ∑_{j=1}^{4} ∫_{0}^{1} (\frac{d}{dt}u(f_{j}(γ(t))))^{2} dt =

∑_{j=1}^{4} ∫_{0}^{1} (\frac{d}{dt}u(γ(g_{j}(t))))^{2} dt = 4 ∑_{j=1}^{4} ∫_{(j−1)/4}^{j/4} (\frac{1}{4} \frac{d}{ds}u(γ(s)))^{2} ds = \frac{1}{4}E(u) .

Lemma 3.3. - Let u ∈ C(\overline{Ω}) and let (u_{n}) be a sequence in C(\overline{Ω}) such that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 . Then lim_{n} ∥u_{n} ∘ γ_{n} − u ∘ γ∥_{L^{∞}(0,1)} = 0 . PROOF. - Let ε > 0 . The function u is uniformly continuous Ω and then there exists δ > 0 so that |u(x_{1}, y_{1}) − u(x_{2}, y_{2})| < ε for every (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Ω such that ∥(x_{1}, y_{1}) − (x_{2}, y_{2})∥ < δ .

By Proposition 3.2 there exists n′_{ε} ∈ ℕ^{∗} so that ∥γ_{n}(t) − γ(t)∥ < δ for every t ∈ [0, 1] and for every n ∈ ℕ^{∗} n > n′_{ε} .

Let n′′_{ε} ∈ ℕ^{∗} be so that ∥u_{n} − u∥_{L^{∞}(Ω)} < ε for every n ∈ ℕ^{∗} n > n′′_{ε} and let n_{ε} = max{n′_{ε}, n′′_{ε}} ; then, for every n ∈ ℕ^{∗} n > n_{ε} and for every t ∈ [0, 1]

|u(γ(t)) − u_{n}(γ_{n}(t))| < |u(γ(t)) − u(γ_{n}(t))| + |u(γ_{n}(t)) − u_{n}(γ_{n}(t))| < 2ε ■

3.3.1 Difference functionals

Definition. - The sequence E′_{n} : C(\overline{Ω}) → ℝ , n ∈ ℕ is defined by

\{ \begin{matrix}E′_{o}(u) = (u(1, 0) − u(0, 0))^{2} \\ E′_{n}(u) = 4 ∑_{j=1}^{4} E′_{n−1}(u ∘ f_{j}) \end{matrix} .

Remark 11. - Let u ∈ C(\overline{Ω}) ; it is clear that E′_{0}(u) = E_{0}(u ∘ γ) . Suppose that for n ∈ ℕ E′_{n}(u) = E_{n}(u ∘ γ) ; then, by (3.3)

E′_{n+1}(u) = 4 ∑_{j=1}^{4} E′_{n}(u ∘ f_{j}) = 4 ∑_{j=1}^{4} E_{n}(u ∘ f_{j} ∘ γ) =

4 ∑_{j=1}^{4} E_{n}(u ∘ γ ∘ g_{j}) = E_{n+1}(u ∘ γ) .

Then E′_{n}(u) = E_{n}(u ∘ γ) for every u ∈ C(\overline{Ω}) and for every n ∈ ℕ . By Remarks 1 and 6 we also have E′_{n}(u) = E_{n}(u∘γ_{n}) for every u ∈ C(\overline{Ω})

and for every n ∈ ℕ .

Proposition 3.4. - For every u ∈ C(\overline{Ω}) Γ- lim_{n} E′_{n}(u) = E(u) . PROOF. - Apply Proposition 3.1 and the previous remark and note that E(u) = F (u ∘ γ) ■

3.3.2 Integral functionals

Definition. - The sequence E′′_{n} : C(\overline{Ω}) → ℝ^{∗} , n ∈ ℕ is defined by

\{ \begin{matrix}E′′_{0}(u) = \{ \begin{matrix}∫_{0}^{1} (\frac{d}{dt}u(γ_{0}(t)))^{2} dt & u ∘ γ_{0} ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} \\ E′′_{n}(u) = 4 ∑_{j=1}^{4} E′′_{n−1}(u ∘ f_{j}) \end{matrix} .

Remark 12. - For every u ∈ C(\overline{Ω}) we have

E′′_{n}(u) = \{ \begin{matrix}∫_{0}^{1} (\frac{d}{dt}u(γ_{n}(t)))^{2} dt & u ∘ γ_{n} ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} .  (3.8)

We prove it by induction. For n = 0 it is true by definition. Let us suppose that (3.8) holds for a given n ∈ ℕ . Then

E′′_{n+1}(u) = 4 ∑_{j=1}^{4} E′′_{n}(u ∘ f_{j}) = 4 ∑_{j=1}^{4} ∫_{0}^{1} (\frac{d}{dt}u(f_{j}(γ_{n}(t))))^{2} dt =

4 ∑_{j=1}^{4} ∫_{0}^{1} (\frac{d}{dt}u(γ_{n+1}(g_{j}(t))))^{2} dt = 16 ∑_{j=1}^{4} ∫_{(j−1)/4}^{j/4} (\frac{1}{4} \frac{d}{ds}u(γ_{n+1}(s)))^{2} ds =

∫_{0}^{1} (\frac{d}{ds}u(γ_{n+1}(s)))^{2} ds .

Proposition 3.5. - For every u ∈ C(\overline{Ω}) , E(u) = Γ- lim_{n} E′′_{n}(u) . PROOF. - Let u ∈ C(\overline{Ω}) ; first we prove that E(u) ≤ lim inf_{n} E′′_{n}(u_{n}) for every sequence (u_{n}) in C(\overline{Ω}) , uniformly convergent to u . We give two different proofs of this fact; the first one is a direct proof, while the second one uses Proposition 3.1.

\underline{First proof.} Let us suppose that u ∘ γ ∈ H^{1}(0, 1) and let (u_{n}) be a sequence in C(\overline{Ω}) so that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 and u_{n} ∘ γ_{n} ∈ H^{1}(0, 1) ∀n ∈ ℕ^{∗} .

Given ε > 0 let φ ∈ C_{c}^{∞}(]0, 1[) be so that ∥(u∘γ)′−φ∥_{L^{2}(0,1)}\sqrt{E(u)} < ε and ∥(u ∘ γ)′ − φ∥_{L^{2}(0,1)} < ε . By Lemma 3.3 there exists n_{ε} ∈ ℕ^{∗} such that ∥u_{n} ∘ γ_{n} − u ∘ γ∥_{L^{∞}(0,1)} < ε/∥φ′∥_{L^{∞}(0,1)} for every n ∈ ℕ^{∗} n > n_{ε} .

Let n ∈ ℕ^{∗} n > n_{ε} , then

E(u) = |(\frac{d(u ∘ γ)}{dt}, \frac{d(u ∘ γ)}{dt})_{L^{2}(0,1)}| ≤

|(\frac{d(u ∘ γ)}{dt}, \frac{d(u ∘ γ)}{dt} − φ)_{L^{2}(0,1)}| + |(\frac{d(u ∘ γ)}{dt}, φ)_{L^{2}(0,1)}| ≤

|∫_{0}^{1} u(γ(t))φ′(t)dt| + ∥\frac{d(u ∘ γ)}{dt}∥_{L^{2}(0,1)} ∥\frac{d(u ∘ γ)}{dt} − φ∥_{L^{2}(0,1)} <

|∫_{0}^{1}(u(γ(t)) − u_{n}(γ_{n}(t)))φ′(t)dt| + |∫_{0}^{1} u_{n}(γ_{n}(t))φ′(t)dt| + ε ≤

|∫_{0}^{1} \frac{d}{dt}u_{n}(γ_{n}(t))φ(t)dt| + ∥u ∘ γ − u_{n} ∘ γ_{n}∥_{L^{∞}(0,1)}∥φ′∥_{L^{∞}(0,1)} + ε <

∥\frac{d(u_{n} ∘ γ_{n})}{dt}∥_{L^{2}(0,1)} ∥φ∥_{L^{2}(0,1)} + 2ε ≤

(∥φ − \frac{d(u ∘ γ)}{dt}∥_{L^{2}(0,1)} + ∥\frac{d(u ∘ γ)}{dt}∥_{L^{2}(0,1)}) \sqrt{E′′_{n}(u_{n})} + 2ε <

(ε + \sqrt{E(u)})\sqrt{E′′_{n}(u_{n})} + 2ε .

By letting n → +∞ we have

E(u) ≤ (ε + \sqrt{E(u)})\sqrt{lim inf_{n} E′′_{n}(u_{n})} + 2ε .

If ε → 0 , then we obtain the desired result.

Now we suppose u ∘ γ ∉ H^{1}(0, 1) . Let (u_{n}) be a sequence in C(\overline{Ω}) such that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 .

Let ε > 0 and φ ∈ C_{c}^{∞}(]0, 1[) be so that ∥φ∥_{L^{2}(0,1)} ≤ 1 ; moreover we fix n_{ε} ∈ ℕ^{∗} so that ∥u_{n} ∘ γ_{n} − u ∘ γ∥_{L^{∞}(0,1)} < ε/∥φ′∥_{L^{∞}(0,1)} for every n ∈ ℕ^{∗} with n > n_{ε} .

For every n ∈ ℕ^{∗} n > n_{ε} we have

|∫_{0}^{1} u(γ(t))φ′(t)dt| ≤

|∫_{0}^{1}(u(γ(t)) − u_{n}(γ_{n}(t)))φ′(t)dt| + |∫_{0}^{1} u_{n}(γ_{n}(t))φ′(t)dt| ≤

|∫_{0}^{1} \frac{d}{dt}u_{n}(γ_{n}(t))φ(t)dt| + ∥u ∘ γ − u_{n} ∘ γ_{n}∥_{L^{∞}(0,1)}∥φ′∥_{L^{∞}(0,1)} <

∥\frac{d(u_{n} ∘ γ_{n})}{dt}∥_{L^{2}(0,1)} ∥φ∥_{L^{2}(0,1)} + ε < \sqrt{E′′_{n}(u_{n})} + ε .

By letting n → +∞ we have

|∫_{0}^{1} u(γ(t))φ′(t)dt| ≤ \sqrt{lim inf_{n} E′′_{n}(u_{n})} + ε

for every φ ∈ C_{c}^{∞}(]0, 1[) with ∥φ∥_{L^{2}(0,1)} ≤ 1 .

Let us suppose that sup{| ∫_{0}^{1} u(γ(t))φ′(t)dt| | φ ∈ C_{c}^{∞}(]0, 1[) ∥φ∥_{L^{2}(0,1)} ≤ 1} = C < +∞ ; as done in the proof of Propositions 3.1 and 2.22 we can show that u ∘ γ ∈ H^{1}(0, 1) . Then lim_{n} E′′_{n}(u_{n}) = +∞ .

\underline{Second proof.} Let u ∈ C(\overline{Ω}) : we compare E′′_{n}(u) with E_{n}(u ∘ γ) . Let n ∈ ℕ^{∗} , if u ∘ γ_{n} ∉ H^{1}(0, 1) then E′′_{n}(u) = +∞ > E_{n}(u ∘ γ) . If u ∘ γ_{n} ∈ H^{1}(0, 1) then, by Remark 6

E_{n}(u ∘ γ) = E_{n}(u ∘ γ_{n}) = 4^{n} ∑_{j=1}^{4^{n}} (∫_{(j−1)4^{−n}}^{j4^{−n}} \frac{d}{dt}u(γ_{n}(t))dt)^{2} ≤

∑_{j=1}^{4^{n}} ∫_{(j−1)4^{−n}}^{j4^{−n}} (\frac{d}{dt}u(γ_{n}(t)))^{2} dt = E′′_{n}(u) .

By Propositions 3.1 and 2.12 ,

E(u) = F (u ∘ γ) = Γ- lim_{n} E_{n}(u ∘ γ) ≤ Γ- lim inf_{n} E′′_{n}(u)

and then E(u) ≤ lim inf_{n} E′′_{n}(u_{n}) for every sequence (u_{n}) in C(\overline{Ω}) such that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 .

Let u ∈ C(\overline{Ω}) ; to complete the proof we have to prove that there exists a sequence (u_{n}) in C(\overline{Ω}) so that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 and lim_{n} E′′_{n}(u_{n}) = E(u) .

Let n ∈ ℕ^{∗} ; the function u∘γ∘γ_{n}^{−1} is continuous on K_{n} = γ_{n}([0, 1]) . By Theorem 4.2 there exists v_{n} ∈ C(ℝ^{2}) such that v_{n}(x, y) = u(γ(γ_{n}^{−1}(x, y))) for every (x, y) ∈ K_{n} .

The function v_{n} is uniformly continuous on Ω and then there exists δ′_{n} > 0 so that |v_{n}(x_{1}, y_{1}) − v_{n}(x_{2}, y_{2})| < 1/n for every (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Ω with ∥(x_{1}, y_{1}) − (x_{2}, y_{2})∥ < δ′_{n} .

Moreover there exists δ′′_{n} > 0 so that |u(x_{1}, y_{1}) − u(x_{2}, y_{2})| < 1/n for every (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Ω with ∥(x_{1}, y_{1}) − (x_{2}, y_{2})∥ < δ′′_{n} .

We set δ_{n} = min{δ′_{n}, δ′′_{n}} , σ_{n} = δ_{n}/4 , A_{n} = (K_{n})_{3σ_{n}} , B_{n} = (K_{n})_{2σ_{n}} .

The set A_{n}\ B_{n}^{∘} is compact and then there exist k_{n} ∈ ℕ^{∗} and (x_{1}^{(n)}, y_{1}^{(n)}),

..., (x_{k_{n}}^{(n)}, y_{k_{n}}^{(n)}) ∈ A_{n}\ B_{n}^{∘} so that A_{n}\ B_{n}^{∘}⊆ ⋃_{i=1}^{k_{n}} D_{ℝ^{2}}((x_{i}^{(n)}, y_{i}^{(n)}), σ_{n}) . By

the lemma on the partition of the unity there exist φ_{0}^{(n)}, φ_{1}^{(n)}, ..., φ_{k_{n}}^{(n)} ∈ C^{∞}(ℝ^{2}) so that 0 ≤ φ_{i}^{(n)}(x, y) ≤ 1 ∀(x, y) ∈ ℝ^{2} ∀i ∈ ℕ i ≤ k_{n} and ∑_{i=0}^{k_{n}} φ_{i}^{(n)}(x, y) = 1 ∀(x, y) ∈ ℝ^{2} ; moreover, for 1 ≤ i ≤ k_{n} the support of φ_{i}^{(n)} is contained in D_{ℝ^{2}}((x_{i}^{(n)}, y_{i}^{(n)}), σ_{n}) and the support of φ_{0}^{(n)} does

not intersect A_{n}\ B_{n}^{∘} . We set ψ_{n}, ω_{n} : ℝ^{2} → ℝ ψ_{n}(x, y) = φ_{0}^{(n)}(x, y)(1 −

χ_{B_{n}}(x, y)) , ω_{n}(x, y) = φ_{0}^{(n)}(x, y)χ_{B_{n}}(x, y) .

Since A_{n}\ B_{n}^{∘} is compact, it follows ψ_{n}, ω_{n} ∈ C(ℝ^{2}) .

We set

u_{n}(x, y) = ψ_{n}(x, y)u(x, y) + (ω_{n}(x, y) + ∑_{i=1}^{k_{n}} φ_{i}^{(n)}(x, y))v_{n}(x, y) .

Of course u_{n} ∈ C(\overline{Ω}) ; moreover, for every (x, y) ∈ K_{n} ⊆ B_{n} ψ_{n}(x, y) = 0 and then u_{n}(x, y) = v_{n}(x, y) . It follows

E′′_{n}(u_{n}) = ∫_{0}^{1} (\frac{d}{dt}u_{n}(γ_{n}(t)))^{2} dt = ∫_{0}^{1} (\frac{d}{dt}v_{n}(γ_{n}(t)))^{2} dt = E(u) .

To conclude we prove that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 .

If (x, y) ∈ Ω\ A_{n}^{∘} then ω_{n}(x, y) = φ_{i}^{(n)}(x, y) = 0 ∀i ∈ ℕ^{∗} i ≤ k_{n} and

then u_{n}(x, y) = u(x, y) .

Let ε > 0 ; there exists δ > 0 so that |u(x_{1}, y_{1}) − u(x_{2}, y_{2})| < ε for every (x_{1}, y_{1}), (x_{2}, y_{2}) ∈ Ω with ∥(x_{1}, y_{1}) − (x_{2}, y_{2})∥ < δ ; moreover there exists n_{ε} ∈ ℕ^{∗} so that ∥γ_{n}(t) − γ(t)∥ < δ for every n ∈ ℕ^{∗} with n > n_{ε} .

If (x, y) ∈A_{n}^{∘} , there exists t_{n} ∈ [0, 1] such that ∥(x, y) − γ_{n}(t_{n})∥ < δ_{n} ; then, for every n ∈ ℕ^{∗} with n > max{n_{ε}, 2/ε} we have

|u_{n}(x, y) − u(x, y)| = (ψ_{n}(x, y) + ∑_{i=1}^{k_{n}} φ_{i}^{(n)}(x, y))|v_{n}(x, y) − u(x, y)| ≤

|v_{n}(x, y) − v_{n}(γ_{n}(t_{n}))| + |v_{n}(γ_{n}(t_{n})) − u(γ_{n}(t_{n}))| + |u(γ_{n}(t_{n})) − u(x, y)| <

|u(γ(t_{n})) − u(γ_{n}(t_{n}))| + \frac{2}{n} < 2ε ■

Remark 13. - The use of Proposition 3.1 simplifies the proof of the previous result but Proposition 3.1 uses the fact that g_{i}([0, 1]) ⋂ g_{i+1}([0, 1]) = {g_{i}(1)} = {g_{i+1}(0)} for every i ∈ {1, 2, ..., M − 1} ; for this reason we conjecture that the first proof (of the first part) of Proposition 3.5 better adapts to possible generalizations in higher dimensions.

Appendix

In this chapter we recall some standard facts that have been used in the previous chapters.

4.1 Topology

Theorem 4.1. - Let X be a metric space and let X′ be a dense subspace of X .

If Y is a complete metric space and f : X′ → Y is uniformly continuous, then there exists a unique continuous function g : X → Y such that g(x) = f(x) for every x ∈ X′ . Moreover g is uniformly continuous. PROOF. - See [7], chapter 6, Theorem 5.7.

Theorem 4.2. - Suppose that K ⊆ ℝ^{N} is compact and f : K → ℝ^{M} is continuous.

Then there exists a continuous function g : ℝ^{N} → ℝ^{M} such that g(x) = f(x) ∀x ∈ K . PROOF. - See [11], chapter 1, section 1.2, Theorem 1.

Theorem 4.3. - Assume A ⊆ ℝ^{N} and let f : A → ℝ^{M} be a Lipschitz function with Lipschitz constant L . Then there exists a Lipschitz function g : ℝ^{N} → ℝ^{M} so that g(x) = f(x) for every x ∈ A and ∥g(x) − g(y)∥ ≤ L\sqrt{M}∥x − y∥ for every x, y ∈ ℝ^{N} . PROOF. - See [11], chapter 3, section 1.2, Theorem 1.

Proposition 4.4. - Let X be a T_{3} topological space satisfying the second axiom of countability. Then for every closed and disjoint sets C, D ⊆ X there exists a continuous function f : X → [0, 1] such that f(x) = 0 if and only if x ∈ C and f(x) = 1 if and only if x ∈ D . PROOF. - See [7], chapter 7, Corollary 1.5.

Theorem 4.5(Stone Weierstrass). - Let K ⊆ ℝ^{N} be compact. Then for every u ∈ C(K) there exists a sequence (p_{n}) of real polynomials such that lim_{n} ∥p_{n} − u∥_{L^{∞}(K)} = 0 . PROOF. - See [7], chapter 6, Theorem 8.9.

Lemma 4.1(Partition of unity). - Let A be a subset of ℝ^{N} and let {A_{i} ⊆ ℝ^{N} | i ∈ I} be a family of open subsets of ℝ^{N} such that A ⊆ ⋃_{i∈I} A_{i}.

Then there exist a family of functions {φ_{j} ∈ C_{c}^{∞}(ℝ^{N}) | j ∈ J} with the following properties:

• for every j ∈ J and every x ∈ ℝ^{N} , 0 ≤ φ(x) ≤ 1 ;

• if K ⊆ A , \overline{K} is compact and \overline{K} ⊆ A , then there exist n ∈ ℕ^{∗} and j_{1}, j_{2}, ..., j_{n} ∈ J such that φ_{j}(x) = 0 ∀x ∈ K and ∀j ∈ J\{j_{1}, j_{2}, ..., j_{n}} ;

• for every j ∈ J there exists i ∈ I such that the support of φ_{j} is contained in A_{i} ;

• for every x ∈ A , sup{∑_{j∈J′} φ_{j}(x) | J′ ⊆ J , ♯J′ < +∞} = 1 .

PROOF. - See [1], chapter 3, Theorem 3.14.

4.2 Measure theory

Theorem 4.6. - Let (Ω, 𝔐, μ) be a measure space and f : Ω → [0, +∞] a measurable function on Ω . For every A ∈ 𝔐 we set

ν(A) = ∫_{A} f(x)dμ .

Then ν is a measure. PROOF. - See [2], appendix, page 378, Theorem 10.

Theorem 4.7(Lebesgue). - Let (Ω, 𝔐, μ) be a measure space and D ∈ 𝔐. Let (u_{n}) be a sequence of functions from Ω to ℝ^{∗} and for every x ∈ D , u(x) = lim_{n} u_{n}(x) .

If there exists a function v ∈ L^{1}(D) such that

|u_{n}(x)| ≤ v(x) ∀x ∈ D ∀n ∈ ℕ^{∗} ,

then

lim_{n} ∫_{D} u_{n}(x)dμ = ∫_{D} u(x)dμ .

PROOF. - See [2], appendix, page 385, Theorem 13.

Theorem 4.8. - Let X be a locally compact Hausdorff space and μ a Radon measure on X .

Let p ∈ [1, +∞[ and L^{p}(X, μ) = {u : X → ℝ^{∗} | u is μ-measurable and ∫_{X} u(x)^{p}dμ < +∞} .

Then C_{c}(X) is dense in L^{p}(X, μ) . PROOF. - See [22], chapter 3, Theorem 3.14.

Theorem 4.9(Tonelli). - Let (Ω_{1}, 𝔐_{1}, μ_{1}) and (Ω_{2}, 𝔐_{2}, μ_{2}) be σ-finite measure spaces and let u : Ω_{1} × Ω_{2} → [0, +∞[ be a measurable function. Then the following facts hold:

• u is measurable with respect to each argument;

• the function v(x) = ∫_{Ω_{2}} u(x, y)dμ_{2} is measurable;

• the function w(y) = ∫_{Ω_{1}} u(x, y)dμ_{1} is measurable;

• ∫_{Ω_{1}×Ω_{2}} u(x, y)d(μ_{1}×μ_{2}) = ∫_{Ω_{1}} ∫_{Ω_{2}} u(x, y)dμ_{2}dμ_{1} = ∫_{Ω_{2}} ∫_{Ω_{1}} u(x, y)dμ_{1}dμ_{2}.

PROOF. - See [2], appendix, page. 400.

Theorem 4.10(Fubini). - Let (Ω_{1}, 𝔐_{1}, μ_{1}) and (Ω_{2}, 𝔐_{2}, μ_{2}) be σ-finite measure spaces and let u ∈ L^{1}(Ω_{1} × Ω_{2}) . Then the following facts hold:

• for almost every y ∈ Ω_{2} the functions v_{1}(x) = u(x, y) and w_{1}(x) = ∫_{Ω_{2}} u(x, y)dμ_{2} belong to L^{1}(Ω_{1}) ;

• for almost every x ∈ Ω_{1} the functions v_{2}(y) = u(x, y) and w_{2}(y) = ∫_{Ω_{1}} u(x, y)dμ_{1} belong to L^{1}(Ω_{2}) ;

• ∫_{Ω_{1}×Ω_{2}} u(x, y)d(μ_{1}×μ_{2}) = ∫_{Ω_{1}} ∫_{Ω_{2}} u(x, y)dμ_{2}dμ_{1} = ∫_{Ω_{2}} ∫_{Ω_{1}} u(x, y)dμ_{1}dμ_{2}.

PROOF. - See [2], appendix, page. 400.

The following is a corollary of the Besicovitch’s covering theorem.

Theorem 4.11. - Let μ be a Borel measure on ℝ^{N} and 𝔄 any collection of nondegenerate closed balls. Let A denote the set of centers of the balls in 𝔄 . Assume that μ(A) < +∞ and inf{r | B_{ℝ^{N}}(x, r) ∈ 𝔄} = 0 for each x ∈ A .

Then, for every open set U ⊆ ℝ^{N} there exists a countable collection 𝔅 of disjoint balls of 𝔄 such that ⋃_{B∈𝔅} B ⊆ U and μ((A ⋂ U)\⋃_{B∈𝔅} B) = 0. PROOF. - See [11], section 1.5.2, Corollary 1.

Theorem 4.12(Lebesgue Besicovitch). - Let μ be a Radon measure on ℝ^{N} and f ∈ L_{loc}^{1}(ℝ^{N}, μ) (that is ∫_{K} |f(x)|dμ < +∞ for every compact set K ⊆ ℝ^{N}). Then

lim_{r→0} \frac{1}{μ(B_{ℝ^{N}}(x, r))} ∫_{B_{ℝ^{N}} (x,r)} f(y)dμ(y) = f(x)

for μ-almost every x ∈ ℝ^{N} . PROOF. - See [11], section 1.7.1, Theorem 1.

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