Construction of energies on compact subsets of ℝ^{N}

Giorgio Follo (follo@science.unitn.it)

Department of Mathematics ”E. Brioschi”, Politecnico di Milano, Via Bonardi, 9, 20133 Milano, Italy

Abstract. We study the possibility to construct quadratic forms on compact subsets of ℝ^{N}. The method we use is the following: if K ⊆ ℝ^{N} is compact, r > 0 , α ≥ 0 and u is a continuous function, let f_{r}^{(α)}(u) be the Dirichlet integral on the r-neighborhood of K , normalized by a factor r^{α−N} ; we take the Γ-limit, with respect to the uniform convergence, of f_{r}^{(α)}(u) , as r → 0 .

We also define a dimension of the compact set K to be the infimum of the numbers α so that f_{r}^{(α)} Γ-converges to 0 . We call it capacity dimension and prove that it is less than or equal to the box dimension.

An integral representation for the Γ-limit of f_{r}^{(α)} is given for some K and for α equal to the capacity dimension.

Keywords: Calculus of variations, Dirichlet forms, fractals, Γ-convergence, I. F. S., potential theory.

Mathematics Subject Classification (2000): Primary: 49J45, 31C15; Secondary: 49J53, 28A75, 28A78.

1. Introduction

We describe a method proposed by E. De Giorgi to construct energies on compact subsets of ℝ^{N} and test it on some cases.

Let Ω ⊆ ℝ^{N} be open and bounded and let K ⊆ Ω be nonempty and compact. We consider the space C(\overline{Ω}) of the uniformly continuous functions on Ω and a family of functionals f_{r}^{(α)} : C(\overline{Ω}) → ℝ^{∗} depending on two parameters α ≥ 0 and r > 0 . These functionals are defined as f_{r}^{(α)}(u) = r^{α−N} ∫_{K_{r}} ∥∇u(x)∥^{2}dx if u ∈ H^{1}(K_{r}) ⋂ C(\overline{Ω})

and f_{r}^{(α)}(u) = +∞ otherwise (K_{r} is the r-neighborhood of K). Then we call f_{−}^{(α)} and f_{+}^{(α)} the lower and upper Γ-limit of f_{r}^{(α)} as

r → 0 and f^{(α)} the Γ-limit if it exists. We can think to f_{−}^{(α)}, f_{+}^{(α)}, f^{(α)} as functionals defined on C(K) ; indeed f_{−}^{(α)}(u) = f_{−}^{(α)}(v) (f_{+}^{(α)}(u) = f_{+}^{(α)}(v)) if u = v on K .

Moreover the domain of finiteness of f^{(α)} is a Banach space with respect to the norm ∥ · ∥_{C(K)} + \sqrt{f^{(α)}} (and the same for f_{−}^{(α)}, f_{+}^{(α)}).

We also define a notion of dimension related to the functionals f_{−}^{(α)} and f_{+}^{(α)} . We call lower (upper) capacity dimension of K the infimum of the numbers α so that f_{−}^{(α)} (f_{+}^{(α)}) is identically 0 . We proved that it is less than or equal to the lower (upper) box dimension of K . If the lower and the upper capacity dimension of K coincide, we call capacity dimension of K the common value.

If K is the closure of an open bounded regular set, then its capacity dimension is N ; if K is a simple regular curve then it is 1 and it is 0 if K is totally disconnected. In the above cases an integral representation for f^{(α)} (with α to be the capacity dimension of K) is computed in Theorems 4.1 and 4.5 and Proposition 4.6.

We also studied how the capacity dimensions of K change if K is embedded in ℝ^{N+M} ; we expect that it does not change; but we proved it only for regular sets (Theorem 4.3). However, we proved that in general, it does not increase.

Finally we studied an example in which the capacity dimensions are not integers. For this set we found an integral representation for f_{−}^{(α)}, f_{+}^{(α)} (Theorem 4.7) and the capacity dimensions are equal to the box ones that can be grater than the Hausdorff dimension.

Let us suppose that C is the classical Cantor set. It is well known that it is self similar; that is there exist 2 contractive similitudes f_{1}, f_{2} : ℝ → ℝ and an open set O ⊆ ℝ so that C = f_{1}(C) ⋃ f_{2}(C) , f_{1}(O), f_{2}(O) ⊆ O and f_{1}(O) ⋂ f_{2}(O) = ∅ (see [13]). The set K = C × [0, 1] is self similar with respect to a new system g_{1}, g_{2}, ..., g_{6} of contractive similitudes. Then f_{−}^{(α)}, f_{+}^{(α)} are invariant with respect to g_{1}, g_{2}, ..., g_{6} , in the sense that f_{−}^{(α)}(u) = ⋃_{i=1}^{6} f_{−}^{(α)}(u ∘ g_{i}) for every u ∈ C(\overline{Ω}) and the same fact holds for f_{+}^{(α)} .

It would be important to know if there exist M positive constants λ_{1}, λ_{2}, ..., λ_{M} ∈]0, +∞[ so that

f_{−}^{(α)}(u) = ∑_{i=1}^{M} λ_{i}f_{−}^{(α)}(u ∘ f_{i}) for every u ∈ C(\overline{Ω})  (1)

(and the same formula for f_{+}^{(α)}) any time K is self similar with respect to the same family f_{1}, f_{2}, ..., f_{M} . At the moment we do not know the answer to this question.

Quadratic forms satisfying (1) have been used, for example, in [16].

2. Notation and preliminary results

In this note ℕ = {0, 1, 2, ...} is the set of all non negative integer numbers, ℕ^{∗} = ℕ\{0} and ℝ^{∗} = [−∞, +∞] is the extended real line.

By X we indicate a topological space and for every x ∈ X , U(x) denote the set of all open neighborhoods of x .

The closed and open balls in ℝ^{N} will be indicated by the symbols

B_{ℝ^{N}} (x_{0}, r) = {x ∈ ℝ^{N} | ∥x − x_{0}∥ ≤ r} ,
D_{ℝ^{N}} (x_{0}, r) = {x ∈ ℝ^{N} | ∥x − x_{0}∥ < r} .

The diameter of a subset A of ℝ^{N} is indicated by |A| ; that is |A| = sup{∥x − y∥ | x, y ∈ A} 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_{ℝ^{N}} (x, r).

The Lebesgue measure of A is L^{N}(A) , dim A is its Hausdorff dimension and χ_{A} : ℝ^{N} → ℝ is the charateristic function of A ; that is χ_{A}(x) = 1 if x ∈ A and χ_{A}(x) = 0 otherwise.

If A ⊆ ℝ^{N} is open and T : A → ℝ^{N} is differentiable, DT : A → ℝ^{N×N} is the Jacobian of T .

Let K ⊆ ℝ^{N} and u : K → ℝ be a bounded function. For every δ ∈ [0, +∞[ we set

ω(u, δ) = sup{|u(x) − u(y)| | x, y ∈ K , ∥x − y∥ ≤ δ} .

The function ω(u, ·) is called the modulus of continuity of u . It is well known that if K is compact and u ∈ C(K) then the

function ω(u, ·) : [0, +∞[→ ℝ is increasing and continuous from the right (see for example [5], chapter 5, theorem 5.4.1).

The following lemma is a corollary of the lemma on the partition of the unity. See, for example, [1], chapter 3, Theorem 3.14.

LEMMA 2.1. Let K ⊆ ℝ^{N} be compact and r > 0 . There exists a function η ∈ C_{c}^{∞}(ℝ^{N}) so that η(x) = 1 for every x ∈ K , η(x) = 0

for every x ∈ ℝ^{N}\ K_{r}^{∘} and 0 ≤ η(x) ≤ 1 in ℝ^{N} .

DEFINITION 2.1. 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 1. It is easy to see that \overline{f} is lower semicontinuous; \overline{f} is the greatest lower semicontinuous function majorized by f . Moreover

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

If X satisfies the first axiom of countability, then, for every x ∈ X

\overline{f}(x) = inf\{lim inf_{n} f(x_{n}) | (x_{n}) converges to x in X\} . (2)

2.1. Γ-convergence

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

DEFINITION 2.2. 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) ,

Γ- lim sup_{r→r_{0}} f(r, x) = sup_{U∈U(x)} lim sup_{r→r_{0}} 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 say that f(r, ·) Γconverges to g as r → r_{0} or that g is the Γ-limit of f(r, ·) as r → r_{0} and we write g(x) = Γ- lim_{r→r_{0}} f(r, x) .

REMARK 2. A simple consequence of the previous definition is that the Γ-limits are less than or equal to the pointwise ones.

If D = ℕ^{∗} and r_{0} = +∞ , we will write f_{n}(x) instead of f(n, x) . It is very easy to prove that if (f_{n}) and (g_{n}) are two sequences

of functions from X to ℝ^{∗} with 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) .

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 .

PROPOSITION 2.1. Let (f_{n}) be a sequence of functions from X to ℝ^{∗} and g : X → ℝ^{∗} be a function. Assume that X satisfies the first axiom of countability. Then, for every x ∈ X , g(x) = Γ- lim_{n} f_{n}(x) if and only if the following two conditions are satisfied:

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

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

The same statement holds for the lower (upper) Γ-limit, with lim inf_{n} f_{n}(x_{n}) instead of lim_{n} f_{n}(x_{n}) in 2 ( lim sup_{n} f_{n}(x_{n}) instead of lim inf_{n} f_{n}(x_{n}) in 1 and instead of lim_{n} f_{n}(x_{n}) in 2).

Proof. See [6], 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 2.2. 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 2.3. Let D ⊆ ℝ , f : D × X → ℝ^{∗} be a function, r_{0} ∈ ℝ^{∗} be an accumulation point of D and g : X → ℝ^{∗} be a function. Assume that X satisfies the first axiom of countability.

Then, for every x ∈ X , Γ- lim inf_{r→r_{0}} f(r, x) = min{Γ- lim_{n} f(r_{n}, x)} ( Γ- lim sup_{r→r_{0}} f(r, x) = max{Γ- lim_{n} f(r_{n}, x)} ), where the minimum (maximum) is taken over the sequences (r_{n}) converging to r_{0} in D and such that Γ- lim_{n} f(r_{n}, x) exists.

Proof. See [12], section 2, corollary 2.9.

2.2. Box-counting dimensions and Minkowski contents

Here Ω is a non-empty bounded subset of ℝ^{N} .

DEFINITION 2.3. For all δ > 0 define N_{δ}(Ω) to be the smallest number n ∈ ℕ^{∗} so that Ω can be covered by n subsets of ℝ^{N}, each of them having diameter less then or equal to δ .

The lower and upper box-counting dimensions of Ω are defined as

\underline{dim_{B}}Ω = lim inf_{δ→0} \frac{ln N_{δ}(Ω)}{− ln δ} , \overline{dim_{B}}Ω = lim sup_{δ→0} \frac{ln N_{δ}(Ω)}{− ln 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 .

REMARK 3. In [9], § 3.4 it is proved that always dim Ω ≤ \underline{dim_{B}}Ω .

DEFINITION 2.4. Let n ∈ ℕ^{∗} ; the n-th entropy number of Ω is

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

PROPOSITION 2.4. 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} .

Proof. See [10], section 2.4, corollary 2.10.

PROPOSITION 2.5. 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 + \underline{dim_{B}}B .

If also A is bounded, then \overline{dim_{B}}(A × B) ≤ \overline{dim_{B}}A + \overline{dim_{B}}B . Proof. See [9], chapter 7, Propositions 7.2, 7.3, 7.5.

REMARK 4. Let Ω′ = Ω×[0, 1] ⊆ ℝ^{N+1} . By proposition 2.5 dim Ω′ = 1+dim Ω and \overline{dim_{B}}Ω′ ≤ 1+\overline{dim_{B}}Ω ; we prove the opposite inequality.

Let δ > 0 ; at least N_{δ}(Ω) subsets of ℝ^{N} with diameter less than or equal to δ are needed to cover Ω . Then, for n ∈ ℕ , at least N_{δ}(Ω) subsets of ℝ^{N+1} with diameter less than or equal to δ are needed to cover Ω×[3nδ, 3(n+1)δ] . It follows that 4δN_{δ}(Ω′) ≥ N_{δ}(Ω) and by definition, \overline{dim_{B}}Ω′ ≥ 1 + \overline{dim_{B}}Ω .

DEFINITION 2.5. Let s ∈ [0, N] . The s-dimensional upper and lower Minkowski contents of Ω are defined as

\overline{M^{(s)}}(Ω) = lim sup_{r→0}(2r)^{s−N}L^{N}(Ω_{r}) , \underline{M^{(s)}}(Ω) = lim inf_{r→0}(2r)^{s−N}L^{N}(Ω_{r}).

PROPOSITION 2.6. We have

\underline{dim_{B}}Ω = inf\{s | \underline{M^{(s)}}(Ω) = 0\} = sup\{s | \underline{M^{(s)}}(Ω) = +∞\} ; (3)
\overline{dim_{B}}Ω = inf\{s | \overline{M^{(s)}}(Ω) = 0\} = sup\{s | \overline{M^{(s)}}(Ω) = +∞\} . (4)

Proof. See [15], chapter 5, section 5.5.

3. Quadratic forms and 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 always suppose that K_{r} ⊆ Ω for r = 1 . 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} .

On C(K) we will consider the norm ∥u∥_{C(K)} = max_{x∈K} |u(x)| .

DEFINITION 3.1. For any α ∈ [0, +∞[ and for every r ∈]0, 1] 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. Let 0 ≤ α < β , u ∈ C(\overline{Ω}) and let r ∈]0, 1] . Then 0 ≤ f_{r}^{(β)}(u) ≤ f_{r}^{(α)}(u) and then 0 ≤ f_{−}^{(β)}(u) ≤ f_{−}^{(α)}(u) and 0 ≤ f_{+}^{(β)}(u) ≤ f_{+}^{(α)}(u) .

DEFINITION 3.2. We call lower and upper capacity dimension of K the numbers

\underline{dim_{C}}K = inf\{α ∈ [0, +∞[ | f_{−}^{(α)}(u) = 0 ∀u ∈ C(\overline{Ω})\}
\overline{dim_{C}}K = inf\{α ∈ [0, +∞[ | 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 6. In the previous definition we can replace C(\overline{Ω}) with any dense subset Y ⊆ C(\overline{Ω}) .

We prove it for the upper capacity dimension. Let d = inf{α ∈ [0, +∞[ | f^{(α)}(u) = 0 ∀u ∈ Y \} , where Y 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 ∈ Y , for a given α ∈ [0, +∞[ . Given u ∈ C(\overline{Ω}) \ Y , for U ∈ U(u) there exists v ∈U^{∘} ⋂ Y and then f_{+}^{(α)}(v) = 0 . It follows lim sup_{r→0} inf_{w∈V} f(w) = 0 for every V ∈ U(v) and then even for V = U . Then f_{+}^{(α)}(u) = 0 .

PROPOSITION 3.1. We have \underline{dim_{C}}K ≤ \underline{dim_{B}}K and \overline{dim_{C}}K ≤ \overline{dim_{B}}K .

Proof. Let α ∈]\underline{dim_{B}}K, +∞[ and u ∈ C^{1}(\overline{Ω}) .

For all r ∈]0, 1] ,

0 ≤ f_{r}^{(α)}(u) ≤ \frac{L^{N}(K_{r})}{r^{N−α}} ∑_{i=1}^{N} ∥\frac{∂u}{∂x_{i}} ∥_{L^{∞}(Ω)}^{2} .  (5)

By Remark 2 and (3) we have f_{−}^{(α)}(u) = 0 .

By Remark 6, \underline{dim_{C}}K ≤ \underline{dim_{B}}K .

The other inequality can be proved in the same way. □

REMARK 7. 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 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

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

The claim follows by letting r → 0 (the same argument runs for f_{−}^{(β)}(u)).

In particular, if f_{+}^{(α)}(u) < +∞ (f_{−}^{(α)}(u) < +∞) ∀u ∈ Y , with Y dense in C(\overline{Ω}) , by Remark 6 we obtain \overline{dim_{C}}K ≤ α (\underline{dim_{C}}K ≤ α).

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

REMARK 8. Let H ⊆ K be compact. Let, for every α ∈ [0, N] and r > 0 , g_{r}^{(α)}, g_{−}^{(α)}, g_{+}^{(α)} be defined as f_{r}^{(α)}, f_{−}^{(α)}, f_{+}^{(α)} with H inside of K ; we have g_{r}^{(α)}(u) ≤ f_{r}^{(α)}(u) 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 .

Now we prove that f_{−}^{(α)}(u) and f_{+}^{(α)}(u) depend only on the restriction of u to K .

LEMMA 3.1. Let u, v ∈ C(\overline{Ω}) with u(x) = v(x) on K and let (v_{n}) be a sequence in C(\overline{Ω}) such that lim_{n} ∥v_{n} − v∥_{L^{∞}(Ω)} = 0 .

For every sequence (r_{n}) converging to 0 in ]0, 1] there exists a sequence (u_{n}) in C(\overline{Ω}) so that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 and u_{n}(x) = v_{n}(x) for every n ∈ ℕ^{∗} and x ∈ K_{r_{n}} .

Proof. Let for every n ∈ ℕ^{∗} η_{n} ∈ C^{∞}(ℝ^{N}) be so that for every x ∈ ℝ^{N} 0 ≤ η_{n}(x) ≤ 1 , η_{n}(x) = 1 if x ∈ K_{r_{n}} and η_{n}(x) = 0 if x ∉ K_{2r_{n}} .

We set u_{n}(x) = η_{n}(x)v_{n}(x) + (1 − η_{n}(x))u(x) .

Let x ∈ Ω and n ∈ ℕ^{∗} ; if x ∉ K_{2r_{n}} then u_{n}(x) = u(x) . If x ∈ K_{r_{n}} then u_{n}(x) = v_{n}(x) . If x ∈ K_{2r_{n}} then there exists y ∈ K such that ∥x − y∥ < 3r_{n} ; it follows

|u(x) − u_{n}(x)| ≤ |u(x) − v_{n}(x)| ≤ |u(x) − u(y)| + |v(y) − v(x)|+

|v(x) − v_{n}(x)| ≤ ω(u, 3r_{n}) + ω(v, 3r_{n}) + ∥v_{n} − v∥_{L^{∞}(Ω)} . □

PROPOSITION 3.2. Let u, v ∈ C(\overline{Ω}) and α ∈ [0, N] . If u(x) = v(x) for every x ∈ K , then f_{−}^{(α)}(u) = f_{−}^{(α)}(v) and f_{+}^{(α)}(u) = f_{+}^{(α)}(v) .

Proof. By Corollary 2.3 there exists a sequence (r_{n}) converging to 0 in ]0, 1] so that f_{−}^{(α)}(v) = Γ- lim_{n} f_{r_{n}}^{(α)}(v) .

By Proposition 2.1 there exists a sequence (v_{n}) in C(\overline{Ω}) so that lim_{n} ∥v_{n} − v∥_{L^{∞}(Ω)} = 0 and f_{−}^{(α)}(v) = lim_{n} f_{r_{n}}^{(α)}(v_{n}) .

By the previous lemma there exists a sequence (u_{n}) in C(\overline{Ω}) so that lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 and f_{r_{n}}^{(α)}(u_{n}) = f_{r_{n}}^{(α)}(v_{n}) for every n ∈ ℕ^{∗} . Then

f_{−}^{(α)}(v) = lim_{n} f_{r_{n}}^{(α)}(v_{n}) = lim_{n} f_{r_{n}}^{(α)}(u_{n}) ≥ Γ- lim inf_{n} f_{r_{n}}^{(α)}(u) ≥ f_{−}^{(α)}(u) .

By exchanging u and v we obtain the opposite inequality.

The proof of the second equality is similar. □

DEFINITION 3.3. For every α ∈ [0, N] we define E_{−}^{(α)}(K) = {u ∈ C(K) | f_{−}^{(α)}(u) < +∞} . On this set we define the function

∥u∥_{E_{−}^{(α)}(K)} = ∥u∥_{C(K)} + \sqrt{f_{−}^{(α)}(u)} .

We also define E_{+}^{(α)}(K) and ∥·∥_{E_{+}^{(α)}(K)} in the same way, with f_{+}^{(α)}

instead of f_{−}^{(α)} . If f_{+}^{(α)} = f_{−}^{(α)} then E_{+}^{(α)}(K) = E_{−}^{(α)}(K) = E^{(α)}(K) .

For the rest of this section we only consider the functional f_{−}^{(α)} and the space E_{−}^{(α)}(K) but the same considerations apply to f_{+}^{(α)} and E_{+}^{(α)}(K) .

REMARK 9. It is easy to see that for every r ∈]0, 1] , f_{r}^{(α)} is a non-negative quadratic form (see [6], Chapter 11, Definition 11.7 and Proposition 11.9).

By Theorem 11.10 in [6], Chapter 11 and Corollary 2.3, f_{−}^{(α)} is a non-negative quadratic form on C(K) and then f_{−}^{(α)}(u + v) ≤ f_{−}^{(α)}(u + v) + f_{−}^{(α)}(u − v) = 2f_{−}^{(α)}(u) + 2f_{−}^{(α)}(v) ∀u, v ∈ C(K) .

It follows that E_{−}^{(α)}(K) is a vector space. Moreover \sqrt{f_{−}^{(α)}} is a seminorm and then ∥ · ∥_{E_{−}^{(α)}(K)} is a norm.

DEFINITION 3.4. We call E_{−}^{(α)} the bilinear form associated to f_{−}^{(α)} .

PROPOSITION 3.3. For every α ∈ [0, N] , (E_{−}^{(α)}(K), ∥ · ∥_{E_{−}^{(α)}(K)}) is

a Banach space.

Proof. Let (u_{n}) be a Cauchy sequence in E_{−}^{(α)}(K) ; (u_{n}) is a Cauchy sequence in C(K) and then there exists u ∈ C(K) so that lim_{n} ∥u_{n} − u∥_{C(K)} = 0 .

By Proposition 6.8 in chapter 6 of [6] f_{−}^{α} is lower semicontinuous with respect to the uniform convergence and since the sequence (f_{−}^{(α)}(u_{n})) is bounded, we have u ∈ E_{−}^{(α)}(K) . Let ε > 0 and let n_{ε} ∈ ℕ^{∗} be so that f_{−}^{(α)}(u_{n} − u_{m}) < ε for every m, n ∈ ℕ^{∗} with m > n_{ε} , n > n_{ε} ; then, if n > n_{ε} , again by the semicontinuity of f_{−}^{(α)} , f_{−}^{(α)}(u_{n} − u) ≤ lim inf_{m} f_{−}^{(α)}(u_{n} − u_{m}) ≤ ε . □

REMARK 10. Let α ∈ [0, N] . If \underline{M^{(α)}}(K) < +∞ , by the Stone Weierstrass theorem, Remark 2 and (5), E_{−}^{(α)}(K) is dense in C(K) . In particular, if \underline{M^{(α)}}(K) = 0 , then E_{−}^{(α)}(K) = C(K) .

REMARK 11. Let α ∈ [0, N] , u ∈ C(\overline{Ω}) and for every x ∈ \overline{Ω} v(x) = max\{min\{u(x), 1\}, 0\} , then v ∈ C(\overline{Ω}) and for every r ∈]0, 1]

r^{N−α}f_{r}^{(α)}(v) = ∫_{K_{r}\u^{−1}(ℝ\[0,1])} ∥∇u(x)∥^{2}dx ≤ r^{N−α}f_{r}^{(α)}(u) .

It follows f_{−}^{(α)}(v) ≤ f_{−}^{(α)}(u) .

REMARK 12. Let u_{1}, u_{2} ∈ E_{−}^{(α)}(K) have disjoint supports A, B ⊆ K and let δ > 0 be so that ∥x − y∥ ≥ δ any time u_{1}(x)u_{2}(y) ≠ 0 .

Let v_{1}(x) = u_{1}(x) if x ∈ K , v_{1}(x) = 0 if x ∈ ℝ^{N}\ A_{δ/3}^{∘}

and v_{2}(x) = u_{2}(x) if x ∈ K , v_{2}(x) = 0 if x ∈ ℝ^{N}\ B_{δ/3}^{∘} . Since

v_{1}, v_{2} ∈ C(K ⋃(\overline{Ω}\ K_{δ/3}^{∘})) we can extend v_{1}, v_{2} to w_{1}, w_{2} ∈ C(ℝ^{N})

with disjoint supports in \overline{Ω} . It follows that f_{r}^{(α)}(w_{1}+w_{2}) = f_{r}^{(α)}(w_{1})+ f_{r}^{(α)}(w_{2}) for every α ∈ [0, N] and r ∈]0, δ/6[ and then, by Proposition 6.17 in [6] and Proposition 3.2, E_{−}^{(α)}(u_{1}, u_{2}) ≥ 0 . By replacing u_{1} with −u_{1} , we obtain E_{−}^{(α)}(u_{1}, u_{2}) = 0 .

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 Ω = J^{−1}(Ω′) .

In the next proposition we study relations between f_{−}^{(α)}, f_{+}^{(α)} and g_{−}^{(α)}, g_{+}^{(α)} , where g_{−}^{(α)}, g_{+}^{(α)}, g_{r}^{(α)} are defined as f_{−}^{(α)}, f_{+}^{(α)}, f_{r}^{(α)} with H and Ω′ instead of K and Ω . We set c_{M} = L^{M}(B_{ℝ^{M}} (0, 1)) .

PROPOSITION 3.4. Let α ∈]0, N] ; then g_{−}^{(α)}(v) ≤ c_{M}f_{−}^{(α)}(v ∘ J) and 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 and \overline{dim_{C}}H ≤ \overline{dim_{C}}K.

Proof. We only prove the assertion for g_{+}^{(α)} .

Let v ∈ C(\overline{Ω′}) ; by Corollary 2.3 there exists a sequence (r_{n}) converging to 0 in ]0, 1] such that

g_{+}^{(α)}(v) = Γ- lim_{n} g_{r_{n}}^{(α)}(v) ;  (6)

moreover f_{+}^{(α)}(v ∘ J) ≥ Γ- lim sup_{n} f_{r_{n}}^{(α)}(v ∘ J) and then there exists 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 sup_{n} f_{r_{n}}^{(α)}(u_{n}) .  (7)

Let π : ℝ^{N+M} → ℝ^{N} π(x, y) = x ; then lim_{n} ∥u_{n} ∘ π − v ∘ J ∘ π∥_{L^{∞}(Ω′)} = 0 and v ∘ J ∘ π = v on H . By Lemma 3.1 there exists a sequence (v_{n}) in C(\overline{Ω′}) so that lim_{n} ∥v_{n} − v∥_{L^{∞}(Ω)} = 0 and g_{r_{n}}^{(α)}(v_{n}) = g_{r_{n}}^{(α)}(u_{n} ∘ π) for every n ∈ ℕ^{∗} . Then, for every n ∈ ℕ^{∗} ,

c_{M}f_{r_{n}}^{(α)}(u_{n}) = \frac{1}{r_{n}^{N+M−α}} ∫_{B_{ℝ^{M}} (0,r_{n})} ∫_{K_{rn}} ∑_{i=1}^{N} \frac{∂u_{n}}{∂x_{i}} (x)^{2}dxdy ≥

\frac{1}{r_{n}^{N+M−α}} ∫_{H_{rn}} ∥∇(u_{n} ∘ π)(x, y)∥^{2}dxdy = g_{r_{n}}^{(α)}(v_{n}) .

The result follows from (7), (6) and the above inequality. □

3.1. Dirichlet forms on K

Now we suppose that the compact set K is the support of a Radon measure μ .

Let α ∈ [0, N] and suppose that \underline{M^{(α)}}(K) < +∞ ; then we can construct a Dirichlet form Σ_{−}^{(α)} on K (we refer to [2], chapter 4, definitions 4.1, 4.2, 4.3 for the definition of Dirichlet form).

Let

L^{2}(K, μ) = {u : K → ℝ^{∗} | u is μ-measurable and ∫_{K} u(x)^{2}dμ < +∞}

with its usual Hilbert norm ∥ · ∥_{L^{2}(K,μ)} .

We set h_{−}^{(α)} : L^{2}(K, μ) → ℝ^{∗} , h_{−}^{(α)}(u) = f_{−}^{(α)}(u) if u ∈ C(K) and h_{−}^{(α)}(u) = +∞ otherwise. Then we take its relaxed functional \overline{h}_{−}^{(α)} with respect to the L^{2}(K, μ)-convergence (we recall that f_{−}^{(α)} is lower semicontinuous with respect to the uniform convergence). Then \overline{h}_{−}^{(α)} is a quadratic form; we call Σ_{−}^{(α)} the bilinear form associated to \overline{h}_{−}^{(α)} and H_{−}^{(α)}(K) = \{u ∈ L^{2}(K, μ) | \overline{h}_{−}^{(α)}(u) < +∞\} , equipped with the scalar product (u, v)_{H_{−}^{(α)}(K)} = (u, v)_{L^{2}(K,μ)} + Σ_{−}^{(α)}(u, v) .

One can prove that H_{−}^{(α)}(K) is a Hilbert space (and so Σ_{−}^{(α)} is closed) in the same way proposition 3.3 was proved.

Since E_{−}^{(α)}(K) ⊆ H_{−}^{(α)}(K) , by remark 10, C(K) ⋂ H_{−}^{(α)}(K) is dense in C(K) with respect to the uniform convergence.

Let u ∈ L^{2}(K, μ) and let (u_{n}) be a sequence in C(K) so that lim_{n} ∥u_{n} − u∥_{L^{2}(K,μ)} = 0 and lim_{n} f_{−}^{(α)}(u_{n}) = \overline{h}_{−}^{(α)}(u) . We set

v(x) = max\{min\{u(x), 1\}, 0\} and prove that \overline{h}_{−}^{(α)}(v) ≤ \overline{h}_{−}^{(α)}(u) .

Let (u_{n_{k}}) be a subsequence that converges μ-almost everywhere to u ; we set v_{k}(x) = max{min{u_{n_{k}}(x), 1}, 0} . Then v_{k} ∈ C(K) for every k ∈ ℕ^{∗} and it is easy to see that (v_{k}) converges to v μalmost everywhere. Since |v_{k}(x)−v(x)|^{2} ≤ 1 in K , by the dominated convergence theorem, lim_{k} ∥v_{k} − v∥_{L^{2}(K,μ)} = 0 . By remark 11

\overline{h}_{−}^{(α)}(v) ≤ lim inf_{k} f_{−}^{(α)}(v_{k}) ≤ lim inf_{k} f_{−}^{(α)}(u_{n_{k}}) = lim_{n} f_{−}^{(α)}(u_{n}) = \overline{h}_{−}^{(α)}(u) .

Now, let u, v ∈ H_{−}^{(α)}(K) have disjoint support and let (u_{n}) , (v_{n}) be sequences in E_{−}^{(α)}(K) converging to u and v , respectively and so that \overline{h}_{−}^{(α)}(u) = lim_{n} f_{−}^{(α)}(u_{n}) , \overline{h}_{−}^{(α)}(v) = lim_{n} f_{−}^{(α)}(v_{n}) . It is not restrictive to suppose that u_{n} and v_{n} have disjoint support for every n ∈ ℕ^{∗} . Then \overline{h}_{−}^{(α)}(u + v) ≤ lim inf_{n} \overline{h}_{−}^{(α)}(u_{n} + v_{n}) and by Remark 12,

Σ_{−}^{(α)}(u, v) ≤ \frac{1}{2} lim inf_{n}(f_{−}^{(α)}(u_{n} + v_{n}) − f_{−}^{(α)}(u_{n}) − f_{−}^{(α)}(v_{n})) = 0 .

By replacing u with −u we have Σ_{−}^{(α)}(u, v) = 0 .

Then Σ_{−}^{(α)} is a conservative, local Dirichlet form.

It would be important to know if \overline{h}_{−}^{(α)}(u) = f_{−}^{(α)}(u) for every u ∈ E_{−}^{(α)}(K) .

PROPOSITION 3.5. Let α ∈ [0, N] with \underline{M^{(α)}}(K) < +∞ . Suppose that there exist N′ ∈ ℕ^{∗} and N′ linear operators T_{1}^{(α)}, T_{2}^{(α)}, ..., T_{N′}^{(α)} : E_{−}^{(α)}(K) → L^{2}(K, μ) so that the following conditions hold:

1. there exists c ∈]0, +∞[ such that

f_{−}^{(α)}(u) = c ∫_{K} ∑_{i=1}^{N′}(T_{i}^{(α)}u)(x)^{2}dμ ∀u ∈ E_{−}^{(α)}(K) ;

2. there exists a vector subspace F_{−}^{(α)}(K) of E_{−}^{(α)}(K) that is dense in L^{2}(K, μ) and so that for every i ∈ ℕ^{∗} i ≤ N′ and for every u ∈ E_{−}^{(α)}(K) ,

∫_{K}(T_{i}^{(α)}u)(x)φ(x)dμ = − ∫_{K} u(x)(T_{i}^{(α)}φ)(x)dμ ∀φ ∈ F_{−}^{(α)}(K) .

Then f_{−}^{(α)}(u) = \overline{h}_{−}^{(α)}(u) for every u ∈ E_{−}^{(α)}(K) .

Proof. Let u ∈ E_{−}^{(α)}(K) and let (u_{n}) be a sequence in E_{−}^{(α)}(K) so that lim_{n} ∥u_{n} − u∥_{L^{2}(K,μ)} = 0 . Let ε > 0 and let φ_{1}, φ_{2}, ..., φ_{N′} ∈ F_{−}^{(α)}(K) be so that N′∥T_{i}^{(α)}u − φ_{i}∥_{L^{2}(K,μ)}∥T_{i}^{(α)}u∥_{L^{2}(K,μ)} < ε for every i ∈ ℕ^{∗} i ≤ N′ . Moreover we take n_{ε} ∈ ℕ^{∗} so that N′∥u_{n} − u∥_{L^{2}(K,μ)}∥T_{i}^{(α)}φ_{i}∥_{L^{2}(K,μ)} < ε for all i, n ∈ ℕ^{∗} with i ≤ N′ , n > n_{ε} .

Then, for n > n_{ε} , we have

\frac{1}{c}f_{−}^{(α)}(u) ≤ | ∑_{i=1}^{N′}(T_{i}^{(α)}u, T_{i}^{(α)}u − φ_{i})_{L^{2}(K,μ)}| + | ∑_{i=1}^{N′}(T_{i}^{(α)}u, φ_{i})_{L^{2}(K,μ)}| ≤

| ∑_{i=1}^{N′}(u, T_{i}^{(α)}φ_{i})_{L^{2}(K,μ)}| + ∑_{i=1}^{N′} ∥T_{i}^{(α)}u∥_{L^{2}(K,μ)}∥T_{i}^{(α)}u − φ_{i}∥_{L^{2}(K,μ)} <

| ∑_{i=1}^{N′}(T_{i}^{(α)}u_{n}, φ_{i})_{L^{2}(K,μ)}| + ∑_{i=1}^{N′} ∥u_{n} − u∥_{L^{2}(K,μ)}∥T_{i}^{(α)}φ_{i}∥_{L^{2}(K,μ)} + ε <

\sqrt{∑_{i=1}^{N′} ∥T_{i}^{(α)}u_{n}∥_{L^{2}(K,μ)}^{2}}\sqrt{∥φ_{i}∥_{L^{2}(K,μ)}^{2}} + 2ε ≤

\sqrt{\frac{1}{c}f_{−}^{(α)}(u_{n})}\sqrt{∑_{i=1}^{N′} (\frac{ε}{N′∥T_{i}^{(α)}u∥_{L^{2}(K,μ)}} + ∥T_{i}^{(α)}u∥_{L^{2}(K,μ)})^{2}} + 2ε .

The conclusion follows by letting n → +∞ and ε → 0 . □

REMARK 13. Let α ∈ [0, N] and u ∈ H_{−}^{(α)}(K) : if f_{−}^{(α)} satisfies the hipotheses of the previous proposition, there exists a sequence (u_{n}) in E_{−}^{(α)}(K) ⊆ H_{−}^{(α)}(K) ⋂ C(K) so that lim_{n} ∥u_{n} − u∥_{L^{2}(K,μ)} = 0 and

lim_{n} \overline{h}_{−}^{(α)}(u_{n}) = \overline{h}_{−}^{(α)}(u) .  (8)

Since H_{−}^{(α)}(K) is reflexive, there exists a weakly convergent subsequence (u_{n_{k}}) of (u_{n}) . Suppose u_{n_{k}} ⇀ v in H_{−}^{(α)}(K) ; then u_{n_{k}} ⇀ v in L^{2}(K, μ) ; it follows v = u and by (8), lim_{n} ∥u_{n} − u∥_{H_{−}^{(α)}(K)} = 0 .

Then H_{−}^{(α)}(K) ⋂ C(K) is dense in H_{−}^{(α)}(K) and Σ_{−}^{(α)} is regular.

4. examples

In this section we give a representation for f^{(α)} (and f_{−}^{(α)}, f_{+}^{(α)}) when K has some additional properties. The representations we find satisfy the hypotheses of Proposition 3.5.

We will use many times the following lemma. It follows from the density of C_{c}^{∞}(U) in L^{2}(U) ad the definition of sobolev space H^{1}(U) .

LEMMA 4.1. Let U ⊆ ℝ^{N} be open and u ∈ L^{2}(U) . Let us consider the linear functionals f_{1}, f_{2}, ..., f_{N} : C_{c}^{∞}(U) → ℝ defined by

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

Let us suppose that for every i ∈ ℕ^{∗} i ≤ N sup{| < f_{i}, φ > | | φ ∈ C_{c}^{∞}(U) , ∥φ∥_{L^{2}(U)} ≤ 1} < +∞ . Then u ∈ H^{1}(U) .

4.1. Sets with non-empty interior

We refer to [3], section IX.2 for the definition of open set of class C^{1} .

THEOREM 4.1. 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} .  (9)

It follows E^{(N)}(K) = C(K) ⋂ H^{1}(K^{∘}) and H^{(N)}(K) = H^{1}(K^{∘}) . 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; then (f_{r_{n}}^{(N)}) converges to f pointwise. Indeed, if u ∈ C(\overline{Ω}) and f(u) = +∞ , then f_{r_{n}}^{(N)}(u) = +∞ ∀n ∈ ℕ^{∗} ; otherwise

u ∈ H^{1}(K_{r_{0}}^{∘}) ⋂ C(\overline{Ω}) for some r_{0} > 0 and by the Lebesgue theorem, lim_{n} |f_{r_{n}}^{(N)}(u) − f(u)| = 0 , since L^{N}(∂K) = 0 .

By Proposition 5.7 in [6], chapter 5, f^{(N)} = \overline{f} . Let g : C(\overline{Ω}) → ℝ^{∗} be as in (9): by applaying (2) we prove that g = \overline{f} .

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_{n}}^{∘}) .

Since K is compact, u_{n} → u in L^{2}(K^{∘}) and then, by an argument similar to the one used in the proof of Proposition 3.5 (set

c = 1 , μ = L^{N}|_{K} , N′ = N , E_{−}^{(N)}(K) = H^{1}(K^{∘}) , F^{(N)}(K) = C_{c}^{∞}(K^{∘})

and T_{i}^{(N)}u = \frac{∂u}{∂x_{i}}) it is easy to prove that g(u) ≤ lim inf_{n} g(u_{n}) = lim inf_{n} f(u_{n}) .

We construct a sequence (u_{n}) in C(\overline{Ω}) so that g(u) = lim_{n} f(u_{n}) . If f(u) < +∞ then this condition is satisfied with u_{n} = u ∀n ∈ ℕ^{∗}. 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}) = g(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 [3]).

Indeed, by using the definition of regular open set, Lemma IX.2 in [3] and the lemma on the partition of the unity, it is possible to construct a sequence (v_{n}) in C(\overline{Ω}) ⋂ H^{1}(Ω) and a sequence (δ_{n}) in ]0, 1[ so that v_{n}|_{K} = u|_{K} ∀n ∈ ℕ^{∗} , lim_{n} δ_{n} = 0 and lim_{n} ∥v_{n} − u∥_{L^{∞}(K_{δn})} = 0 .

Let, for every n ∈ ℕ^{∗} , σ_{n} = δ_{n}/2 : by Lemma 2.1 there exists a function η_{n} ∈ C^{∞}(ℝ^{N}) so that η_{n}(x) = 1 if x ∈ K_{σ_{n}} , η_{n}(x) = 0

if x ∈ ℝ^{N}\ K_{δ_{n}}^{∘} and 0 ≤ η_{n}(x) ≤ 1 ∀x ∈ ℝ^{N} . We set u_{n}(x) =

η_{n}(x)v_{n}(x)+(1−η_{n}(x))u(x) . Of course u_{n} ∈ C(\overline{Ω}) for every n ∈ ℕ^{∗} .

Let n ∈ ℕ^{∗} ; if x ∈ Ω\ K_{δ_{n}}^{∘} then u_{n}(x) = u(x) . Otherwise |u_{n}(x) − u(x)| ≤ |v_{n}(x) − u(x)| . It follows lim_{n} ∥u_{n} − u∥_{L^{∞}(Ω)} = 0 .

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}) = g(u_{n}) = g(u) . Then g(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 and f(u_{n}) < +∞ ∀n ∈ ℕ^{∗} . Let φ_{1}, φ_{2}, ..., φ_{N} ∈ C_{c}^{∞}(K^{∘})

be so that ∥φ_{i}∥_{L^{2}(K^{∘})} ≤ 1 and let n_{1} ∈ ℕ^{∗} be so that ∥u_{n} −

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

For n > n_{1} , by the same arguments as before, we have

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

By letting n → +∞ and by Lemma 4.1, lim_{n} f(u_{n}) = +∞ . □

COROLLARY 4.2. If K^{∘}≠ ∅ then dim_{C} K = N .

THEOREM 4.3. Let M, J, H, Ω′, g_{−}^{(α)}, g_{+}^{(α)}, c_{M} be as in Proposition

3.4. If K = \overline{K^{∘}} , K^{∘} is non-empty and of class C^{1} (in ℝ^{N}), 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, 1] . 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 v∘J ∈ H^{1}(K^{∘}) and v_{n} ∈ H^{1}(H_{r_{n}}^{∘}) for all n ∈ ℕ^{∗} . Let ε > 0 ; there exist φ_{1}, φ_{2}, ..., φ_{N} ∈ C_{c}^{∞}(K^{∘}) so that for every

i ∈ ℕ^{∗} i ≤ N , c_{M} \sqrt{N}∥φ_{i} − \frac{∂(v∘J)}{∂x_{i}}∥_{L^{2}(K^{∘})} < ε .

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} .

By an argument similar to the one used in the proof of Proposition 3.5, we have

c_{M}f^{(N)}(v ∘ J) < c_{M} ∑_{i=1}^{N} |∫_{K^{∘}} v_{n}(x, 0)\frac{∂φ_{i}}{∂x_{i}} (x)dx| + (1 + \sqrt{f^{(N)}(v ∘ J)})ε =

\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| + (1 + \sqrt{f^{(N)}(v ∘ J)})ε

for every n ∈ ℕ^{∗} with n > n′_{ε} .

Let δ > 0 be so that ω(v, δ) < ε(3c_{M}L^{N}(K^{∘}) ∑_{i=1}^{n} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})})^{−1}.

There exists n′′_{ε} ∈ ℕ^{∗} so that r_{n} < δ and ∥v_{n} − v∥_{L^{∞}(Ω′)} <

ε(3c_{M}L^{N}(K^{∘}) ∑_{i=1}^{n} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})})^{−1} for every n ∈ ℕ^{∗} n > n′′_{ε} . For

n ∈ ℕ^{∗} n > n′′_{ε} and for every x ∈ K , y ∈ B_{ℝ^{M}} (0, r_{n}) we have

|v_{n}(x, 0) − v_{n}(x, y)| < \frac{ε}{c_{M}L^{N}(K^{∘}) ∑_{i=1}^{n} ∥\frac{∂φ_{i}}{∂x_{i}} ∥_{L^{∞}(K^{∘})}} .

Let n_{ε} = max{n′_{ε}, n′′_{ε}} ; 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) − 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 ≤

\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 ≤ (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}} <

\sqrt{g_{r_{n}}^{(N)}(v_{n})}\sqrt{c_{M} ∑_{i=1}^{N} (ε + ∥\frac{∂(v ∘ J)}{∂x_{i}} ∥_{L^{2}(K^{∘})})^{2}} + (2 + \sqrt{f^{(N)}(v ∘ J)})ε .

If n → +∞ and 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 all 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^{∗} be as before, with regard to ε = 1 . By the same arguments as before, we can show that

∑_{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 . (10)

terzo.tex; 20/09/2002; 16:09; p.17

By Lemma 4.1 and (10), lim_{n} g_{r_{n}}^{(N)}(v_{n}) = +∞ .

By Proposition 2.2 c_{M}f^{(N)}(v ∘ J) ≤ g_{−}^{(N)}(v) .

By Proposition 3.4 c_{M}f^{(N)}(v ∘ J) ≥ g_{+}^{(N)}(v) . □

COROLLARY 4.4. If K^{∘}≠ ∅ then dim_{C} H = N .

4.2. Smooth curves in ℝ^{2}

Let us recall some basic facts. We call curve in ℝ^{2} a continuous function γ : [a, b] → ℝ^{2} , γ(t) = (x(t), y(t)) .

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] .

THEOREM 4.5. Let γ ∈ C([a, b]) be a simple curve in ℝ^{2} .

Let us suppose that there exist n ∈ ℕ^{∗} and n + 2 real numbers c_{0} = a < c_{1} < ... < c_{n} < c_{n+1} = b so that γ is regular on [c_{i}, c_{i+1}] and γ ∈ C^{3}([c_{i}, c_{i+1}]) for every i ∈ ℕ i ≤ n .

If K = γ([a, b]) , 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}(a, b) \\ +∞ & otherwise \end{matrix}

and dim_{C} K = 1 .

Proof. A proof of this proposition can be found in [12], section 4, Proposition 4.6 and remark 16.

In a subsequent paper we will give a different proof for some continuous curves for which a tangent vector is not defined, such as for example, the Koch curve. □

4.3. Totally disconnected compact sets

LEMMA 4.2. 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 disjoint subsets C_{1}, C_{2}, ..., C_{n} of ℝ^{N} satisfying the following conditions:

a) |C_{i}| < ε ∀i ∈ ℕ^{∗} i ≤ n ,

b) K_{r} = ⋃_{i=1}^{n} C_{i} .

Proof. See [12], section 4, Lemma 4.4.

PROPOSITION 4.6. Let a :]0, +∞[→ [0, +∞[ be a function and for every r ∈]0, 1] , 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}

If K is totally disconnected then Γ- lim_{r→0} f_{r}(u) = 0 for every u ∈ C(\overline{Ω}) . It follows dim_{C} K = 0 and for every α ∈ [0, N] , E^{(α)}(K) = C(K) , H^{(α)}(K) = L^{2}(K, μ) .

Proof. Let u ∈ C(\overline{Ω}) and δ > 0 . Let r > 0 and C_{1}, C_{2}, ..., C_{n} ⊆ Ω be the sets given by condition 2 of Lemma 4.2. Since these sets are compact and disjoint, there exists σ ∈]0, δ/4[ such that inf{∥x − y∥ | x ∈ C_{i} , y ∈ C_{j}} > 3σ for every i, j ∈ ℕ^{∗} i < j ≤ n .

By Lemma 2.1 there exists a function η_{δ} : ℝ^{N} → [0, 1] so that η_{δ} ∈ C^{∞}(ℝ^{N}) , η_{δ}(x) = 1 if x ∈ K_{r} and η_{δ}(x) = 0 if x ∈ ℝ^{N} \K_{r+σ} .

Let for every i ∈ ℕ^{∗} i ≤ n , y_{i} be a point of C_{i} ; we set v : Ω → ℝ

v(x) = η_{δ}(x) ∑_{i=1}^{n} u(y_{i})χ_{(C_{i})_{σ}}(x) + (1 − η_{δ}(x))u(x)) .

It is easy to see that v is continuous and ∥u−v∥_{L^{∞}(Ω)} < ω(u, 2δ) . Moreover v is constant on the connected components of K_{r} ; then f_{s}(u) = 0 for every s ∈]0, r] and (take ε ≥ ω(u, 2δ))

lim sup_{r→0} inf\{f_{r}(v) | v ∈ C(\overline{Ω}) ∥u − v∥_{L^{∞}(Ω)} < ε\} ≤ 0 . (11)

Since (11) holds for every ε > 0, we have Γ- lim sup_{r→0} f_{r}(u) = 0. □

4.4. An attempt about fractal sets

In this section we consider the cartesian product between a totally disconnected Cantor-type set C ⊆ ℝ and the closed unit interval. We will see that the upper (lower) capacity dimension of the set we are going to construct equals its upper (lower) box dimension. In some cases the upper capacity dimension is grater than the Hausdorff one.

4.4.1. Construction of the set and of a measure supported on it Let d ∈]0, 1[ ; we construct a set C ⊆ ℝ and a set K ⊆ ℝ^{2} so that dim C = d and dim K = 1 + d .

Let (m_{n}) be a sequence of integer numbers with m_{n} ≥ 2 for every n ∈ ℕ^{∗} and let (t_{n}) and (d_{n}) be two sequences of real numbers defined in the following way: t_{0} = 1 and for every n ∈ ℕ^{∗} , m_{n}t_{n}^{d} = t_{n−1}^{d} , (m_{n} − 1)d_{n} = t_{n−1} − m_{n}t_{n} .

Moreover we set, for every n ∈ ℕ^{∗} , I_{n} = {1, 2, ..., m_{n}} and

f_{j}^{(n)}(x) = \frac{t_{n}x + (j − 1)(t_{n} + d_{n})}{t_{n−1}} ∀j ∈ I_{n} ∀x ∈ ℝ .

We refer to [11] for the results below mentioned.

For every n ∈ ℕ^{∗} let p_{n} : ∏_{k=1}^{n} I_{k} → ℝ , p_{n}(i_{1}, i_{2}, ..., i_{n}) = f_{i_{1}i_{2}...i_{n}}(0) , where f_{i_{1}i_{2}...i_{n}} = f_{i_{1}}^{(1)} ∘ f_{i_{2}}^{(2)} ∘ ⋯ ∘ f_{i_{n}}^{(n)} .

Let I = ∏_{n=1}^{+∞} I_{n} ; we define a function p : I → ℝ ,

p(i) = lim_{n} p_{n}(i_{1}, i_{2}, ..., i_{n}) .  (12)

The limit in (12) exists. We set C = p(I) .

We consider the discrete topology on each factor I_{n} and the product topology on I . The function p is continuous and hence C is compact. It is easy to see that C ⊆ [0, 1] and 0, 1 ∈ C .

For every n ∈ ℕ^{∗} t_{n} is the lenght of each interval in the pre-Cantor set ⋃_{(i_{1},i_{2},...,i_{n})∈∏_{k=1}^{n} I_{k}} f_{i_{1}i_{2}...i_{n}}([0, 1]) and d_{n} is the minimum distance between two consecutive intervals.

For every n ∈ ℕ^{∗} we define a Radon probability measure τ_{n} on I_{n} by τ_{n}(A) = ♯A/m_{n} and a Radon probability measure τ on I by τ = ×_{n=1}^{+∞}τ_{n} .

Now we define a Radon probability measure μ on C by μ = p_{♯}τ , that is μ(A) = τ(p^{−1}(A)) for every Borel set A ⊆ ℝ (μ is the image measure of τ under p). It is clear that C is the support of μ .

Let ν be a Radon probability measure with bounded support and let, for every n ∈ ℕ^{∗} , ν_{n} be the measure defined by

ν_{n} = \frac{1}{∏_{k=1}^{n} m_{k}} ∑_{i_{1}=1}^{m_{1}} ∑_{i_{2}=1}^{m_{2}} ... ∑_{i_{n}=1}^{m_{n}} f_{i_{1}i_{2}...i_{n}♯}ν ;

then the sequence ν_{n} weakly converges to μ ; that is

lim_{n} ∫_{−∞}^{+∞} u(x)dν_{n} = ∫_{−∞}^{+∞} u(x)dμ ∀u ∈ C_{c}(ℝ) .  (13)

Moreover, for every i ∈ I and n ∈ ℕ^{∗} ,

μ([p_{n}(i_{1}, i_{2}, ..., i_{n}), p_{n}(i_{1}, i_{2}, ..., i_{n}) + t_{n}]) = t_{n}^{d} .  (14)

In [8], chapter 1, Theorem 1.15, it is proved that dim C = d . In [11] it is proved that the set C is a d-set if and only if the sequence (m_{n}) is bounded (see for example [14] chapter 2 or [18] chapter 1, for the definition of d-set). It is well known that for every bounded d-set in ℝ^{N} the Hausdorff and the box dimensions are equal to d .

Let us suppose that ε ∈]0, 1 − d[ and the sequence (m_{n}) satisfies the following condition: m_{1} = 2 and for all 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} ;  (15)

we prove that \overline{dim_{B}}C > d .

Let us suppose that dim_{B} C = d ; by proposition 2.4 there exists n_{ε} ∈ ℕ^{∗} so that

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

In [11] it is proved that

ε_{m_{n}q_{n}/2}(C) (\frac{1}{2}m_{n}q_{n})^{\frac{1}{d}} = \frac{2^{1− \frac{1}{d}} (\frac{1}{2}m_{n} − 1) + (\frac{1}{2}m_{n})^{\frac{1}{d}}}{2(m_{n} − 1)} ∀n ∈ ℕ^{∗} .

Then, for all n > n_{ε} we have

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

\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)}} > 4 .

Now we define K ⊆ ℝ^{2} as K = C × [0, 1] . Then dim K = 1 + d and if the sequence (m_{n}) is bounded, dim_{B} K = 1+d . If for example, the sequence (m_{n}) satisfies the condition (15), then by remark 4, \overline{dim_{B}}K > 1 + d .

4.4.2. A functional on C(K) DEFINITION 4.1. For every x ∈ ℝ we call J_{x} : ℝ → ℝ^{2} the inclusion J_{x}(y) = (x, y) .

REMARK 14. Let u ∈ C_{c}(ℝ^{2}) and define f_{u} : ℝ → ℝ^{∗}

f_{u}(x) = \{ \begin{matrix}∫_{0}^{1} (\frac{d}{dy} u(J_{x}(y)))^{2} dy & if u ∘ J_{x} ∈ H^{1}(0, 1) \\ +∞ & otherwise \end{matrix} .

We prove that f_{u} is μ-measurable. Let us consider the functions g : ℝ → C([0, 1]) and E : C([0, 1]) → ℝ^{∗} defined by g(x)(y) = u(x, y) and E(v) = ∫_{0}^{1} v′(t)^{2}dt if v ∈ H^{1}(0, 1) , E(v) = +∞ otherwise; on C([0, 1]) we consider the usual norm ∥ · ∥_{L^{∞}(0,1)} .

By arguing as in Proposition 3.5 and using Lemma 4.1 one can prove that E is lower semicontinuous.

It is easy to see that g is continuous and f_{u} = E ∘ g . It follows that f_{u} is lower semicontinuous and then it is μ-measurable.

By Remark 14 we can justify the following definition.

DEFINITION 4.2. 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(x) & if u ∘ J_{x} ∈ H^{1}(0, 1) \\ & for μ-almost every x ∈ C \\ +∞ & otherwise \end{matrix} .

Now we show some properties of this functional.

LEMMA 4.3. Let K ⊆ ℝ^{N} K ≠ ∅ be compact and let λ be a Radon measure supported on K .

If H ⊆ K is compact with λ(H) = 0 , then, for every u ∈ L^{2}(K, λ) and for every ε > 0 there exists a function φ ∈ C^{∞}(ℝ^{N}) so that ∥u − φ∥_{L^{2}(K,λ)} < ε and the support of φ does not intersect H .

Proof. Let u ∈ L^{2}(K; λ) and ε > 0 ; we take w ∈ C(ℝ^{N}) so that 3∥u − w∥_{L^{2}(K,λ)} < ε (see, for example [17], chapter 3, Theorem 3.14).

Let Ω ⊆ ℝ^{N} be an open bounded set so that K ⊆ Ω ; by the Stone Weierstrass theorem there exists a real polynomial q such that 3\sqrt{λ(K)}∥q − v∥_{L^{∞}(Ω)} < ε .

Since λ is a Radon measure and λ(H) = 0 there exists V ⊆ Ω open such that H ⊆ V and 9∥q∥_{L^{∞}(Ω)}^{2}λ(V) < ε^{2} . Let δ > 0 be so that H_{2δ} ⊆ V . By Lemma 2.1 there exists a function η : ℝ^{N} → [0, 1] so that η ∈ C^{∞}(ℝ^{N}) , η(x) = 1 if x ∈ H_{δ} and η(x) = 0 if x ∈ ℝ^{N} \ H_{2δ} .

Then the function φ defined by φ(x) = (1 − η(x))q(x) has the desired properties. □

LEMMA 4.4. 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(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(E\F) = 0 we set

ν(F) = ∫_{E} g(x)dμ ;

ν is a Radon measure because μ is Borel regular and ν is finite.

By the Lebesgue - Besicovitch theorem (see for example, [7], section 1.7.1, Theorem 1) there exists a set B ⊆ C so that μ(B) = 0 and

lim_{r→0} \frac{ν([x − r, x + r])}{μ([x − r, x + r])} = g(x) ∀x ∈ C \ B .  (16)

We set D = A ⋃ B so that μ(D) = 0 .

Now, let ε > 0 and let δ_{1} > 0 be so that ω(u, δ_{1}) < ε .

By (16), 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([x − r, x + r])}| < ε .

By a corollary of the Besicovitch coverig theorem (see for example, [7], section 1.5.2, Corollary 1) there exists a sequence of disjoint intervals of the form [x_{n} − r_{n}, x_{n} + r_{n}] , so that x_{n} ∈ C \ D , 0 < 3r_{n} < r_{x_{n}} for every n ∈ ℕ^{∗} and

μ((C \ D) \ ( ⋃_{n=1}^{+∞}[x_{n} − r_{n}, x_{n} + r_{n}])) = 0 .

Then ∑_{n=1}^{+∞} μ([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 m ∈ ℕ^{∗} so that ∑_{n=m+1}^{+∞} μ([x_{n} − r_{n}, x_{n} + r_{n}]) < ε and ∑_{n=m+1}^{+∞} ν([x_{n} − r_{n}, x_{n} + r_{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 and M_{n} = max{1, g(x_{n})} . Since μ is a Radon measure there exist s′_{n}, s′′_{n} ∈]0, r_{x_{n}}/3[⋂]0, δ_{2}[ so that

μ(]x_{n} − r_{n} − s′_{n}, x_{n} + r_{n} + s′′_{n}[) − μ([x_{n} − r_{n}, x_{n} + r_{n}]) < \frac{ε}{mM_{n}} .

Since lim_{k} t_{k} = 0 , there exist h, k ∈ ℕ^{∗} and r′_{n} ∈ [r_{n}, r_{n}+s′_{n}[ , r′′_{n} ∈ [r_{n}, r_{n} + s′′_{n}[ so 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( ⋃_{n=1}^{h}[q_{n}, q_{n} + t_{k}]) ≤ μ( ⋃_{n=m+1}^{+∞}[x_{n} − r_{n}, x_{n} + r_{n}]) < ε

and the same inequality holds with ν instead of μ .

Let for every n ∈ ℕ^{∗} n ≤ h , c_{n} = sup{c ∈ ℝ | c ≤ g(x) for μ-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 \ D) so that 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 distances 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 Lemma 2.1, for every n ∈ ℕ^{∗} n ≤ m there exists a function ψ_{n} ∈ C_{c}^{∞}(ℝ^{2}) so that for every (x, y) ∈ ℝ^{2} , 0 ≤ ψ_{n}(x, y) ≤ 1 , ψ_{n}(x, y) = 1 if x ∈ [x_{n} − r′_{n}, x_{n} + r′′_{n}] and y ∈ [0, 1] , ψ_{n}(x, y) = 0 if (x, y) ∉ ([x_{n} − r′_{n}, x_{n} + r′′_{n}] × [0, 1])_{δ_{3}} . Moreover we take, for every l ∈ ℕ^{∗} l ≤ h , a function η_{l} ∈ C_{c}^{∞}(ℝ^{2}) with the same properties with respect to [q_{l}, q_{l} + t_{k}] × [0, 1] and δ_{3} .

We set ξ(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 polynomial p on ℝ^{2} such that ∥u − p∥_{L^{∞}(Ω)} < ε .

The function v is defined as

v(x, y) = p(x, y)ξ(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 (n ≤ h) then |x − x_{n}| < r_{x_{n}} < δ_{1}/4 (|x − z_{n}| ≤ t_{k} < δ_{1}/12) and by the Hölder’s inequality |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 .

The remaining cases can be treated in the same way and the result is ∥u − v∥_{L^{∞}(Ω)} < 4ε .

Construction of energies on compact subsets of ℝ^{N} 25

Let us estimate |f(v) − f(u)| . It is equal to

|∑_{n=1}^{m} μ([x_{n} − r′_{n}, x_{n} + r′′_{n}])∥φ_{x_{n}}∥_{L^{2}(0,1)}^{2}+

∑_{n=1}^{h} μ([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})| <

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([x_{n} − r′_{n}, x_{n} + r′′_{n}])(ε^{2} + 2ε\sqrt{g(x_{n})})+

|∑_{n=1}^{m} μ([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([q_{n}, q_{n} + t_{k}])(ε^{2} + 2ε\sqrt{g(z_{n})}) + ∑_{n=1}^{h} μ([q_{n}, q_{n} + t_{k}])g(z_{n}) + ε <

ε^{2}μ( ⋃_{n=1}^{m}[x_{n} − r′_{n}, x_{n} + r′′_{n}]) + 2ε ∑_{n=1}^{m} μ([x_{n} − r′_{n}, x_{n} + r′′_{n}])\sqrt{g(x_{n})}+

∑_{n=1}^{m}(μ([x_{n} − r′_{n}, x_{n} + r′′_{n}]) − μ([x_{n} − r_{n}, x_{n} + r_{n}]))g(x_{n})+

|∑_{n=1}^{m} μ([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}μ( ⋃_{n=1}^{h}[q_{n}, q_{n} + t_{k}]) + 2ε ∑_{n=1}^{h} μ([q_{n}, q_{n} + t_{k}])\sqrt{g(z_{n})}+

∑_{n=1}^{h} μ([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)| < 2ε ∑_{i=1}^{a} μ([x_{n′_{i}} − r′_{n′_{i}}, x_{n′_{i}} + r′′_{n′_{i}}]) + ∑_{n=1}^{h} μ([q_{n}, q_{n} + t_{k}])c_{n}+

2ε ∑_{i=1}^{m−a} μ([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}}+ε ∑_{n=1}^{h} μ([q_{n}, q_{n}+t_{k}])+

∑_{n=1}^{m} μ([x_{n} − r_{n}, x_{n} + r_{n}]) |g(x_{n}) − \frac{ν([x_{n} − r_{n}, x_{n} + r_{n}])}{μ([x_{n} − r_{n}, x_{n} + r_{n}])}| +

2ε ∑_{i=1}^{b} μ([q_{h′_{i}}, q_{h′_{i}} + t_{k}]) + 2ε ∑_{i=1}^{h−b} μ([q_{h′′_{i}}, q_{h′′_{i}} + t_{k}])g(z_{h′′_{i}}) + ε + ε^{2} + ε^{3}

By estimating this quantity we obtain

|f(u) − f(v)| < (2f(u) + 6 + 9ε + ε^{2})ε . □

REMARK 15. Let M_{I} ∈ ℕ \ {0, 1} and q ∈]0, M_{I}^{−1}[ . If we set d = − log_{q} M_{I} and m_{n} = M_{I} for every n ∈ ℕ^{∗} , we have

f_{i}^{(n)}(x) = qx+(i−1)\frac{1 − q}{M_{I} − 1} ∀n ∈ ℕ^{∗} ∀i ∈ I_{n} = \{1, 2, ..., M_{I}\} ∀x ∈ ℝ .

For simplicity we set I′ = I_{n} and f_{i} = f_{i}^{(n)} for every n ∈ ℕ^{∗} and i ∈ I′ . Then C is invariant with respect to the family 𝔉 = {f_{1}, f_{2}, ..., f_{M_{I}} } ; that is C = 𝔉(C) = ⋃_{i=1}^{M_{I}} f_{i}(C) .

One can prove that even μ is invariant for 𝔉 , that is μ = M_{I}^{−1} ∑_{i=1}^{M_{I}} f_{i♯}μ (see [13], section 4).

Let M_{J} ∈ ℕ \ {0, 1} and J = {1, 2, ..., M_{J}} . If we set

g_{j}(x) = \frac{x + j − 1}{M_{J}} ∀j ∈ J ∀x ∈ ℝ

and G = {g_{1}, g_{2}, ..., g_{M_{J}} } , than [0, 1] is invariant with respect to G .

Let for every i ∈ I′ , j ∈ J , x, y ∈ ℝ h_{i,j}(x, y) = (f_{i}(x), g_{j}(y)) , then K is invariant for {h_{i,j} | i ∈ I′ , j ∈ J} .

Let u ∈ C^{1}(\overline{Ω}) ; by theorem 1.19 in [15], Chapter 1,

∑_{i=1}^{M_{I}} ∑_{j=1}^{M_{J}} f(u ∘ h_{i,j}) = \frac{1}{M_{J}} ∑_{i=1}^{M_{I}} ∑_{j=1}^{M_{J}} ∫_{C} ∫ \frac{j}{M_{J}} ; \frac{j−1}{M_{J}} (\frac{∂u}{∂y}(f_{i}(x), y))^{2} dydμ(x) =

\frac{1}{M_{J}} ∑_{i=1}^{M_{I}} ∑_{j=1}^{M_{J}} ∫_{f_{i}(C)} ∫ \frac{j}{M_{J}} ; \frac{j−1}{M_{J}} (\frac{∂u}{∂y}(f_{i}(x), y))^{2} dydf_{i♯}μ(x) =

\frac{M_{I}}{M_{J}} ∑_{i=1}^{M_{I}} ∑_{j=1}^{M_{J}} ∫_{f_{i}(C)} ∫ \frac{j}{M_{J}} ; \frac{j−1}{M_{J}} (\frac{∂u}{∂y}(f_{i}(x), y))^{2} dydμ(x) = \frac{M_{I}}{M_{J}} f(u) .

By Lemma 4.4 this equality holds for every u ∈ C(\overline{Ω}) . Note that in this case f is invariant even if K is self-affine but

not self-similar.

4.4.3. Computation of the upper and lower Γ-limits Now we study relations between f and f_{−}^{(α)}, f_{+}^{(α)} .

LEMMA 4.5. 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 makes sense. H = C × {0, 1} ; H is in some sense the border of K . ν is the Dirac delta on ℝ , centered on 0 . Ω_{1}, Ω_{2} ⊆ ℝ^{2} are open sets so 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} ; by

Lemma 4.3 E is dense in L^{2}(K, μ × L^{1}) .

Finally ε > 0 is given and η : ℝ^{2} → [0, 1] is a continuous function so that η(x, y) = 0 if (x, y) ∈ ℝ^{2} \Ω_{2} and η(x, y) = 1 if (x, y) ∈ \overline{Ω}_{1} .

We first suppose f(u) < +∞ . In this case w ∈ L^{2}(K, μ × L^{1}) ; let φ ∈ E be so that ∥w − φ∥_{L^{2}(K,μ×L^{1})} < ε min\{1, \sqrt{1/f(u)}\} ( ∥w − φ∥_{L^{2}(K,μ×L^{1})} < ε if f(u) = 0 ).

Let σ > 0 be so that 3ω(u, σ)∥\frac{∂φ}{∂y}∥_{L^{∞}(Ω)} < ε . With these assumptions we have

f(u) = |(w, w)_{L^{2}(K,μ×L^{1})}| ≤ ∥w∥_{L^{2}(K,μ×L^{1})}∥w − φ∥_{L^{2}(K,μ×L^{1})}+

|(w, φ)_{L^{2}(K,μ×L^{1})}| < |∫_{C} ∫_{0}^{1} η(x, y)u(x, y)\frac{∂φ}{∂y}(x, y)dydμ(x)| + ε .

The function x → ∫_{0}^{1} η(x, y)u(x, y)\frac{∂φ}{∂y}(x, y)dy belongs to C_{c}(ℝ) and then, there exists n′_{ε} ∈ ℕ^{∗} such that, for every n ∈ ℕ^{∗} with n > n′_{ε} , we have

|∫_{C} ∫_{0}^{1} η(x, y)u(x, y)\frac{∂φ}{∂y}(x, y)dydμ(x) − \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} η(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} η(x, y)φ(x, y)^{2}dydμ(x) − \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} η(x, y)φ(x, y)^{2}dydf_{i_{1}i_{2}...i_{n}♯}ν(x)| < ε .  (17)

If n > n′_{ε} , then f(u) < 2ε+

\frac{1}{∏_{k=1}^{n} m_{k}} ∑_{i_{1}=1}^{m_{1}} ... ∑_{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| .

Now, let δ > 0 with 3δ∥\frac{∂φ}{∂y}∥_{L^{∞}(Ω)} < ε 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| < \frac{ε}{3} .

It follows f(u) < \frac{7}{3}ε+

\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| .

Let (i_{1}, i_{2}, ..., i_{n}) ∈ ∏_{k=1}^{n} I_{k} ; if y ∈ [0, 1] and −r ≤ x − p_{n}(i_{1}, i_{2}, ..., i_{n}) ≤ +t_{n}+r , then ∥(p_{n}(i_{1}, i_{2}, ..., i_{n}), y)−(x, y)∥ ≤ t_{n}+r < σ and |v(p_{n}(i_{1}, i_{2}, ..., i_{n}), y) − v(x, y)| < ε/∥\frac{∂φ}{∂y}∥_{L^{∞}(Ω)} . Then

|\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+

− ∫_{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| < 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} \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]

and ψ(x, y) = φ(p_{n}(i_{1}, i_{2}, ..., i_{n}), y) if −r ≤ x − p_{n}(i_{1}, i_{2}, ..., i_{n}) ≤ t_{n} + r ; then, if n > n′′_{ε} ,

f(u) < \frac{10}{3}ε + \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 + \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(x) + ε} ,

where the last inequality is due to (17).

Now, recalling that ∥w − φ∥_{L^{2}(K,μ×L^{1})} < ε ,

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, by letting r → 0 , δ → 0 and ε → 0 , we obtain the desired result.

Now we suppose f(u) = +∞ . For simplicity we set g(α, r, δ, u) =

\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 φ ∈ E be so that ∥φ∥_{L^{2}(K,μ×L^{1})} ≤ 1 and let σ, δ, n′_{ε}, n′′_{ε}, r, n, v be as in the previous case, with respect to ε = 1 and 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(x)| ≤ \frac{7}{3} + \sqrt{2 sup_{δ>0} lim inf_{r→0} g(α, r, δ, u)} .

To conclude we have to prove that M =

sup \{∫_{C} ∫_{0}^{1} u(x, y)\frac{∂φ}{∂y}(x, y)dydμ(x) | φ ∈ E, ∥φ∥_{L^{2}(K,μ×L^{1})} ≤ 1\} = ∞.

Let D = {x ∈ C | u ∘ J_{x} ∉ H^{1}(0, 1)} ; we consider two different cases. \underline{1. μ(D) = 0 .} Suppose that M < +∞ . For every φ ∈ E with ∥φ∥_{L^{2}(K,μ×L^{1})} ≤ 1 ,

|∫_{C} ∫_{0}^{1} w(x, y)φ(x, y)dydμ(x)| = |∫_{C} ∫_{0}^{1} u(x, y)\frac{∂φ}{∂y}(x, y)dydμ(x)| ≤ M.

Since E is dense in L^{2}(K, μ × L^{1}) , it follows \sqrt{f(u)} =

sup {∫_{C} ∫_{0}^{1} w(x, y)φ(x, y)dydμ(x) | φ ∈ E , ∥φ∥_{L^{2}(K,μ×L^{1})} ≤ 1} ≤ M.

\underline{2. μ(D) > 0 .} Let R ∈]0, +∞[ ; by Lemma 4.1, for every x ∈ D there exists ψ_{x} ∈ C_{c}^{∞}(]0, 1[) so that ∥ψ_{x}∥_{L^{2}(0,1)} ≤ 1 and ∫_{0}^{1} u(x, y)ψ′_{x}(y)dy > \frac{8R}{μ(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)μ(D) > 4R for every t ∈ [x − δ_{x}, x + δ_{x}] .

By a corollary of the Besicovitch covering theorem (see [7], section 1.5.2, Corollary 1), there exist a sequence (x_{n}) in D and a sequence (r_{n}) in ]0, 1] so that r_{n} ≤ δ_{x_{n}} ∀n ∈ ℕ^{∗} , μ(D) = ∑_{n=1}^{+∞} μ(D ⋂[x_{n} − r_{n}, x_{n} + r_{n}]) and these intervals are disjoint.

Then there exists n ∈ ℕ^{∗} such that ∑_{k=1}^{n} μ(D ⋂[x_{k} − r_{k}, x_{k} + r_{k}]) > μ(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 ; since μ is a Radon measure there exist s′_{k} ∈]x_{k} − r_{k} − δ′, x_{k} − r_{k}] and s′′_{k} ∈]x_{k} + r_{k}, x_{k} + r_{k} + δ′] so that μ([s′_{k}, s′′_{k}]) − μ([x_{k} − r_{k}, x_{k} + r_{k}]) < R/(n∥u∥_{L^{∞}(Ω)}∥ψ′_{x_{k}}∥_{L^{∞}(0,1)}) .

Let k ∈ ℕ^{∗} k ≤ n ; and η_{k} : ℝ → [0, 1] a function so that η_{k} ∈ C^{∞}(ℝ) , η_{k}(x) = 1 if x ∈ [x_{k} − r_{k}, x_{k} + r_{k}] and η_{k}(x) = 0 if x ∈ ℝ\]s′_{k}, s′′_{k}[ . We set φ_{k}(x, y) = η_{k}(x)ψ_{x_{k}}(y) and φ = ∑_{k=1}^{n} φ_{k} ; it is clear that φ ∈ E and since the supports of the functions φ_{k} are disjoint, it follows ∥φ∥_{L^{2}(K,μ×L^{1})} ≤ 1 . Finally

∫_{C} ∫_{0}^{1} u(x, y)\frac{∂φ}{∂y}(x, y)dydμ(x) = ∑_{k=1}^{n} (∫_{x_{k}−r_{k}}^{x_{k}+r_{k}} ∫_{0}^{1} u(x, y)ψ′_{x_{k}}(y)dydμ(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(x)) ≥

\frac{4R}{μ(D)} ∑_{k=1}^{n} μ([x_{k} − r_{k}, x_{k} + r_{k}])+

− ∑_{k=1}^{n} ∥u∥_{L^{∞}(Ω)}∥ψ′_{x_{k}}∥_{L^{∞}(0,1)}(μ([s′_{k}, x_{k} − r_{k}]) + μ([x_{k} + r_{k}, s′′_{k}])) >

2R − 2n\frac{R}{2n} = R □

REMARK 16. 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

lim_{r→0} \frac{L^{2}(K_{r})}{(t_{n} + 2r) ∏_{k=1}^{n} m_{k}} = 1 .  (18)

Let α ∈ [0, 2] : by (18) 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}  (19)

and

\overline{M^{(α)}}(K) = 2^{α−2} lim sup_{r→0} \frac{t_{n} + 2r}{r^{2−α}} ∏_{k=1}^{n} m_{k} .  (20)

THEOREM 4.7. Let α ∈ [0, 2] and u ∈ C(\overline{Ω}) . Then:

1. f_{−}^{(α)}(u) = 2^{2−α}\underline{M^{(α)}}(K)f(u) ;

2. f_{+}^{(α)}(u) = 2^{2−α}\overline{M^{(α)}}(K)f(u) ;

Whenever these products make sense.

It follows \underline{dim_{C}}K = \underline{dim_{B}}K and \overline{dim_{C}}K = \overline{dim_{B}}K .

Proof. For every r ∈]0, d_{1}/2[ we set n = n(r) so that d_{n+1} ≤ 2r < d_{n} .

If \underline{M^{(α)}}(K) = +∞ (\overline{M^{(α)}}(K) = +∞) and f(u) > 0 , the conclusion easily follows from Lemma 4.5.

Let us suppose 0 < \underline{M^{(α)}}(K) < +∞ and 0 < \overline{M^{(α)}}(K) < +∞ . Let ε > 0 ; by Lemma 4.5 there exists δ > 0 and r_{0} > 0 such that

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\}  (21)

for every r ∈]0, r_{0}[ .

By (19), 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 (21) and (20), 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\} .

If (u_{k}) is a sequence in C(\overline{Ω}) so that lim_{k} ∥u_{k} − u∥_{L^{∞}(Ω)} = 0 ,

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 2.2 is satisfied. To prove the opposite inequalities we first suppose u ∈ C^{1}(\overline{Ω}) . 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} be the minimum of their mutual distance. Let

K(r) = [−r, 1+r]× ⋃_{i_{1}=1}^{m_{1}} ... ⋃_{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]

and let K(r, δ_{r}) = K(r)_{δ_{r}} be its closed δ_{r}-neighborhood.

By using the Lemma 2.1 it is possible to construct a function w_{r} ∈ C^{1}(\overline{Ω}) so that:

1. 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] ;

2. w_{r}(x, y) = u(x, y) ∀(x, y) ∈ Ω \ K(r, δ_{r}) ;

3. lim_{r→0} ∥u − w_{r}∥_{L^{∞}(Ω)} = 0 .

Now we set the following notation: (r_{k}) and (R_{k}) are two sequences converging to 0 in ]0, d_{1}/2[ ; (r_{k}) is so 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}) . For every r ∈]0, d_{1}/2[ ,

A_{n} = [0, 1] × ⋃_{i_{1}=1}^{m_{1}} ... ⋃_{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] ;

for every k ∈ ℕ^{∗} , v_{k} = w_{r_{k}} , V_{k} = w_{R_{k}} and

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})).

Finally Ω_{1}, Ω_{2} ⊆ ℝ^{2} are open and K ⊆ Ω_{1} ⊆ \overline{Ω_{1}} ⊆ Ω_{2} ⊆ \overline{Ω_{2}} ⊆ Ω ; ν is the Dirac delta on ℝ , centered on 0 ; M = ∥\frac{∂u}{∂y}∥_{L^{∞}(Ω)}^{2} and ε > 0 is given.

The sequences (c_{k}) and (C_{k}) tend to 0 ; indeed, for every k ∈ ℕ^{∗}

c_{k} ≤ 2Mr_{k} \frac{t_{n} + 2r_{k}}{r_{k}^{2−α}} ∏_{h=1}^{n} m_{h} .

By (19), lim_{k} c_{k} = 0 . With the same argument, using (20) instead of (19), one proves that lim_{k} C_{k} = 0 .

By Lemma 2.1 there exists a continuous function η : ℝ^{2} → [0, 1] so that η(x, y) = 0 if (x, y) ∈ ℝ^{2} \ Ω_{2} and η(x, y) = 1 if (x, y) ∈ \overline{Ω}_{1} .

The function x → ∫_{0}^{1} η(x, y)\frac{∂u}{∂y}(x, y)^{2}dy belongs to C_{c}(ℝ) and then, by (13), there exists k_{ε} ∈ ℕ^{∗} so that, for every k ∈ ℕ^{∗} with k > k_{ε} we have

| ∑_{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} η(x, y)\frac{∂u}{∂y}(x, y)^{2}df_{i_{1}i_{2}...i_{k}♯}ν(x)+

− ∫_{C} ∫_{0}^{1} η(x, y)\frac{∂u}{∂y}(x, y)^{2}dydμ(x)| < ε .  (22)

Now we prove statement 1 for u ∈ C^{1}(\overline{Ω}) . Let k ∈ ℕ^{∗} be so that n = n(r_{k}) > k_{ε} ; 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) ≤

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 (19), 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 1 follows by statement 1 of Proposition 2.2.

Let us prove statement 2 for u ∈ C^{1}(\overline{Ω}) . Let k ∈ ℕ^{∗} ; we have f_{R_{k}}^{(α)}(V_{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}} ∫_{C} ∫_{0}^{1} η(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 (22), we have

f_{R_{k}}^{(α)}(V_{k}) < \frac{t_{n} + 2R_{k}}{R_{k}^{2−α}} ( ∏_{h=1}^{n} m_{h})f(u) + ε + C_{k} .

By (20) lim sup_{k} f_{R_{k}}^{(α)}(V_{k}) ≤ 2^{2−α}\overline{M^{(α)}}(K)f(u) + ε . If ε → 0 , by statement 2 (b) of Proposition 2.2 we have 2.

Now we consider u ∈ C(\overline{Ω}) with f(u) < +∞ .

Let us prove 1. By Lemma 4.4 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}) .

By a standard diagonal argument it is possible to construct a sequence (k_{n}) of integer numbers so that lim_{n} r_{k_{n}}^{(n)} = 0 , lim_{n} ∥w_{k_{n}}^{(n)} −

u∥_{L^{∞}(Ω)} = 0 and lim_{n} f_{r_{kn}^{(n)}}^{(α)}(w_{k_{n}}^{(n)}) = 2^{2−α}\underline{M^{(α)}}(K)f(u) . Again by

Proposition 2.2, statement 1 follows.

Let us prove 2. By Lemma 4.4 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} ∈ ℕ^{∗} so that ∥V_{k}^{(n)} − v_{n}∥_{L^{∞}(Ω)} < n^{−1} 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 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})} − 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 2.2, 2 follows. The cases \underline{M^{(α)}}(K) = 0 and \overline{M^{(α)}}(K) = 0 can be treated in the

same way. □

REMARK 17. 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 . It may happen, for example, if (m_{n}) satisfies the condition (15).

Acknowledgements

I wish to thank Prof. S. Mortola who proposed me the problem and had many helpful discussions with me on this subject and Proff. D. Ferrario, G. Gatti and R. Peirone who suggested me some techniques used in the proof of the results.

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