On the relaxation of a class of functionals defined on Riemannian distances

Andrea Davini

Dipartimento di Matematica Università di Pisa via Buonarroti 2, 56127 Pisa (Italy) e-mail: davini@dm.unipi.it

November 1, 2002

Abstract

In this paper we study the relaxation of a class of functionals defined on distances induced by isotropic Riemannian metrics on an open subset of ℝ^{N}. We prove that isotropic Riemannian metrics are dense in Finsler ones and we show that the relaxed functionals admit a specific integral representation.

Keywords: Riemannian and Finsler metrics, relaxation, Gamma convergence

1 Introduction

In this paper we study an integral functional of the form

F(d_{a}) := ∫_{Ω} F (x, a(x)) dx,  (1)

defined on the family I of distances d_{a} induced by isotropic, continuous Riemannian metrics through the formula

d_{a}(x, y) := inf {L_{a}(γ) : γ ∈ Lip([0, 1]; Ω), γ(0) = x, γ(1) = y }  (2)

for every (x, y) ∈ Ω × Ω, where the length functional L_{a} is defined as follows

L_{a}(γ) := ∫_{0}^{1} a(γ(t))|γ̇(t)| dt.  (3)

Here a varies on the family of positive continuous functions from Ω to the interval [α, β], where α and β are fixed positive constants. Distances of this type have already been studied in [6, 3] and, in a more geometric framework, in [8]. The set I can be seen as a subspace of the space of Finslerian distances D (see Section 2) endowed with the metrizable topology given by the uniform convergence on compact subset of Ω × Ω. It has been proved in [6] that the convergence of a sequence (d_{n})_{n∈ℕ} to d in this topology is equivalent to the Γ-convergence of the associated length functionals L_{d_{n}} to L_{d} with respect to the uniform convergence of curves (see Section 2 for definitions). The main problem arising in our study is that I is not closed with respect to this topology. Indeed, one can build sequences of continuous metrics (a_{n})_{n∈ℕ} which develop an oscillatory behavior in such a way that the induced distances converge to an element d which do not belong to I (see [1]). Therefore, it is natural to consider the relaxed functional of (1), namely

\overline{F}(d) := inf\{lim inf_{n} F(d_{n}) : d_{n} ⟶^{D} d, (d_{n})_{n∈ℕ} ⊂ I\},  (4)

defined for every d belonging to the closure of I, where we have denoted by ⟶^{D} the convergence with respect to the topology of D.

In this paper we prove that the space I is dense in D and, under suitable assumptions on the integrand F in (1), that the relaxed functional (4), which is therefore defined on the whole D, has the following integral representation:

\overline{F}(d) = ∫_{Ω} F (x, Λ_{d}(x)) dx,  (5)

where Λ_{d}(x) := sup_{|ξ|=1} φ_{d}(x, ξ) and φ_{d} is the Finslerian metric associated to d by derivation (see Section 2).

We conclude this introduction with some considerations. It is clear by the definition that the relaxed functional \overline{F} is lower semicontinuous. Moreover, it can be shown that it is the greatest among all lower semicontinuous ones which are bounded from above by F on I (see [4] for various results on this topic). Therefore, in order to prove our relaxation result, we have to show first that the functional (5) is lower semicontinuous. The proof of this issue is just a technical adaptation of the arguments described in [5]. To prove the maximality of (5), we will approximate each d ∈ D by means of a sequence of suitably chosen distances d_{n} ∈ I, namely such that

lim sup_{n} ∫_{Ω} F (x, Λ_{d_{n}}(x)) dx ≤ ∫_{Ω} F (x, Λ_{d}(x)) dx.

Then, by a standard argument (see Section 4), the maximality of (5) follows. Indeed, finding such an approximating sequence is a delicate matter. In fact, one should define the Riemannian metrics a_{n} in such a way to have Γ-convergence of the relative length functionals L_{a_{n}} to L_{φ_{d}} and this problem is not trivial even in the simplified situation of an isotropic Riemannian metric φ_{d}, i.e. such that φ_{d} = b(x)|ξ| where b is a Borel function from Ω to [α, β]. It is clear, in fact, that this convergence strongly relies upon the convergence of the approximating metrics on curves, which is much finer than convergence almost everywhere in Ω. Moreover we do not have much information on the properties of the metric φ_{d}; we only know it is Borel measurable and such that the associated length functional L_{φ_{d}} is lower semicontinuous with respect to the uniform convergence of curves (see Section 2). In the general case of a non-isotropic metric the situation is obviously more delicate. The key idea of our proof is that it is sufficient to control the convergence of the approximating distances only on a fixed countable and dense subset of Ω × Ω (Lemma 3.7). Therefore, when we define the Riemannian metrics, we have only to control the value of the associated distance d_{n} on the first n points of the countable, dense subset. This will be done by approximating the Finsler metric φ_{d} along geodesics (or, more precisely, quasi-geodesics, see (20)). The problem of the density of (smooth) isotropic, Riemannian metrics in Finsler ones has already been studied. The question was raised in [6], and partially answered in [3] under the additional assumption that φ_{d} is lower semicontinuous in the first variable. We remark that our proof does not require any assumption on the Finsler metric and therefore completely answers to the question. Indeed, as pointed out in [3], once we have the density result for continuous and isotropic Riemannian metrics, the analogous result for smooth ones is easily recovered via a regularization argument (see Remark 4.4).

We conclude the paper by showing that every Finsler distance d ∈ D can indeed be seen as generated by a suitable Borel measurable, isotropic Riemannian metric a : \overline{Ω} → [α, β] (according to definition (28), see Proposition 4.8). In other words, by allowing the isotropic metric a to vary in a somehow “uncontrolled” way, one can recover all the possible anisotropies of φ_{d}.

The paper is organized as follows: in Section 2 we recall the main notation used in the sequel and some results on Finsler metrics, Section 3 contains some preliminary lemmas and in Section 4 we prove our main results.

Acknowledgements.- The author wish to thank Giuseppe Buttazzo for having suggested the problem and for several useful discussions on the subject.

2 Notation and preliminaries on Finsler metrics

We write here a list of symbols used throughout this paper.

Ω  an open subset of ℝ^{N}
𝕊^{N−1}  the unitary sphere of ℝ^{N}
B_{r}(x)  the open ball in ℝ^{N} of radius r centred in x
I  the closed interval [0, 1]
L^{N}  the N−dimensional Lebesgue measure
H^{N}  the N−dimensional Hausdorff measure
|u|  the Euclidean norm of the vector u ∈ ℝ^{N}
χ_{E}  the characteristic function of the set E
argmin(P) the set of minimizers of the problem (P)

In this paper the letter N denotes an integer number greater or equal to 2. We will say that a set ω is well contained in Ω and we will write ω ⊂⊂ Ω to mean that its closure \overline{ω} is contained in Ω. With the word curve or path we will always indicate a Lipschitz function from the interval I := [0, 1] to an open subset Ω of ℝ^{N}. Any curve γ is always supposed to be parametrized by constant speed, i.e. in such a way that |γ̇(t)| is constant for L^{1}-a.e. t ∈ I. We will say that a sequence of curves (γ_{n})_{n∈ℕ} (uniformly) converges to a curve γ to mean that sup_{t∈I} |γ_{n}(t)−γ(t)| tends to zero as n goes to infinity. We will denote by L_{x,y} the family of curves γ which join x to y, i.e. such that γ(0) = x and γ(1) = y. We remark that if a sequence of curves (γ_{n})_{n∈ℕ} ⊂ L_{x,y} is such that sup_{n} ∫_{0}^{1} |γ̇(t)|dt < +∞ then, since they are all parametrized by constant speed, we have that their first derivative is bounded from above. Therefore, by applying Ascoli-Arzelà theorem, we can find a curve γ ∈ L_{x,y} such that a subsequence (γ_{n_{i}})_{i∈ℕ} converges to γ. This argument will be widely used throughout the paper with no further explanation.

The function F : Ω × [α, β] → ℝ appearing in the integrand of (1) is assumed to be continuous and to fulfill the following conditions:

(i) the function F (x, ·) is convex and nondecreasing for L^{N}-a.e. x ∈ Ω;

(ii) ∫_{Ω} F (x, β) dx < +∞.

(6)

We recall the notion of Γ-convergence. Let (X, τ) be a topological space satisfying the first axiom of countability at the point x ∈ X. A sequence of functionals F_{n} : X → \overline{ℝ} is said to Γ-converge at x if

Γ − lim inf F_{n}(x) = Γ − lim sup F_{n}(x),

where

\{ \begin{matrix}Γ − lim inf F_{n}(x) := inf \{ lim inf_{n} F_{n}(x_{n}) : x_{n} ⟶^{τ} x \} \\ Γ − lim sup F_{n}(x) := inf \{ lim sup_{n} F_{n}(x_{n}) : x_{n} ⟶^{τ} x \}. \end{matrix}

Definition 2.1. A Borel function φ : Ω × ℝ^{N} → [0, +∞) is said to be a Finsler metric on the open set Ω ⊂ ℝ^{N} if the function φ(x, ·) is positively 1-homogeneous for every x ∈ Ω and convex for L^{N}-a.e. x ∈ Ω.

Given a Finsler metric, we can define a distance d_{φ} on Ω through the formula

d_{φ} (x, y) := inf {L_{φ} (γ) | γ ∈ L_{x,y} } ,  (7)

where the Finslerian length functional L_{φ} is defined by

L_{φ}(γ) := ∫_{0}^{1} φ(γ(t), γ̇(t))dt.

A distance deriving from a Finsler metric through (7) is said to be of Finsler type. We will say that a distance d is locally equivalent to the Euclidean one if, for every x ∈ Ω, there exists an open neighborhood U_{x} and some positive constants c_{x}, C_{x} such that c_{x}|x − y| ≤ d(x, y) ≤ C_{x}|x − y| for every y ∈ U_{x}. We will say that a distance function is of geodesic type if it satisfies the following identity:

d(x, y) = inf {L_{d}(γ) | γ ∈ L_{x,y}}  for every (x, y) ∈ Ω × Ω,  (8)

where L_{d} (γ) denotes the classical d-length of γ, obtained as the supremum of the d-lengths of inscribed polygonal curves:

L_{d}(γ) := sup { ∑_{i} d(γ(t_{i}), γ(t_{i+1})) : 0 = t_{0} < t_{1} < .. < t_{r} = 1, r ∈ ℕ} .  (9)

It can be easily shown by the definition that

Proposition 2.2. The length functional L_{d} is lower semicontinuous with respect to the uniform convergence of paths, namely if (γ_{n})_{n∈ℕ} converges to γ then

L_{d}(γ) ≤ lim inf_{n→+∞} L_{d}(γ_{n}).

If the distance d is locally equivalent to the Euclidean one, then it can be proved (cf. [8]) that the length functional L_{d} admits the integral representation

L_{d}(γ) = ∫_{0}^{1} φ_{d} (γ (t) , γ̇(t)) dt

for every curve γ, where φ_{d} is the Finsler metric associated to d by derivation, namely

φ_{d} (x, ξ) := lim sup_{t→0^{+}} \frac{d (x, x + tξ)}{t} (x, ξ) ∈ Ω × ℝ^{N} .

Denote by d_{Ω}(x, y) the Euclidean geodesic distance in Ω, that is d_{Ω} := d_{a} according to (2), with a identically equal to 1. We remark that d_{Ω} locally coincides with the Euclidean distance. We fix two positive constants α, β with β > α and we set

M := {φ Finsler metric on Ω : α |ξ| ≤ φ (x, ξ) ≤ β |ξ|} .

Then we define the family D of distances on Ω generated by the metrics M, namely D := {d_{φ} | φ ∈ M }. Obviously the set I, made up by distances d_{a} defined by (2) with a : Ω → [α, β] continuous, is trivially included in D identifying a(x) with the metric a(x)|ξ|. It is also evident that α d_{Ω} ≤ d ≤ β d_{Ω} for every d ∈ D, so such distances are locally equivalent to the Euclidean one. Moreover one can easily show the following result.

Proposition 2.3. Let d := d_{φ} for some φ ∈ M. Then L_{d}(γ) ≤ L_{φ}(γ) for every curve γ. In particular, d is a distance of geodesic type according to definition (8).

Remark 2.4. The inequality in the previous proposition may be strict. For example, take Ω := [−1, 1]×[−1, 1], Γ := {0}×[−1, 1] and a(x) := χ_{Ω}(x)+χ_{Γ}(x). Then d_{a}(x, y) = |x−y|. If now we take γ(t) := (0, −1/2)(1−t)+(0, 1/2)t, it is easily seen that L_{d_{a}}(γ) = 1 < 2 = L_{a}(γ).

We endow D with the topology given by the uniform convergence on compact subset of Ω × Ω. We will write d_{n} ⟶^{D} d to mean that the sequence (d_{n})_{n∈ℕ} ⊂ D converges to d ∈ D with respect to this topology. It has been proved [6, Theorem 3.1] that this convergence is equivalent to the Γ-convergence of the relative length functionals with respect to the uniform convergence of paths. Moreover, we have the following result (compare to [5, Proposition 4] and [6, Theorem 3.1]):

Proposition 2.5. Let Ω be an open subset of ℝ^{N} such that d_{Ω}(x, y) ≤ C_{r}|x − y| for every x and y in Ω ∩ B_{r}(0) and every r > 0, where C_{r} is some positive constant which depends on r. Then D is a metrizable compact space.

Throughout this paper we will always work with sets Ω which satisfy the condition stated in the proposition above. Therefore we will always assume that D is compact. In particular, this holds whenever Ω has a locally Lipschitz boundary.

Given a distance d ∈ D, we define for every x ∈ Ω

Λ_{d}(x) := sup_{|ξ|=1} φ_{d}(x, ξ),  (10)

which represents, with analogy to the Riemannian case φ_{d}(x, ξ) = B(x)ξ · ξ with B(x) a symmetric and positive definite matrix, the largest “eigenvalue” of φ_{d}(x, ·) at the point x. We notice that Λ_{d}(x) is a Lebesgue measurable function. Indeed, if (ξ_{n})_{n∈ℕ} is a dense sequence in 𝕊^{N−1}, we have that Λ_{d}(x) coincides with the Borel measurable function sup_{n} φ_{d}(x, ξ_{n}) on Ω \ E, where E is the set of points where φ_{d}(x, ·) is not continuous. We know that φ_{d}(x, ·) is convex for almost every x by definition of Finsler metric, therefore E is L^{N}-negligible and the claim follows.

3 Preliminary results

In this section we prepare the tools which will be used in the proof of our relaxation results.

We recall that the function F : Ω × [α, β] → ℝ is continuous and fulfills conditions (6). We have

Lemma 3.1. Let (φ_{n})_{n∈ℕ} ⊂ M such that d_{φ_{n}} ⟶^{D} d for some d ∈ D. Then, for every bounded Borel set ω ⊂⊂ Ω and every ξ ∈ 𝕊^{N−1}, we have

∫_{ω} F (x, φ_{d}(x, ξ)) dx ≤ lim inf_{n→∞} ∫_{ω} F (x, φ_{n}(x, ξ)) dx .

Proof : Let ω be a bounded Borel set well contained in Ω. Choose a bounded open set A ⊂⊂ Ω that contains ω. Arguing as in the proof of [5, Proposition 9], for every fixed ξ ∈ 𝕊^{N−1} it is possible to find a subsequence of (φ_{n})_{n∈ℕ} and a sequence of positive numbers t_{n} → 0 such that, for a.e. x ∈ A,

F (x, φ (x, ξ)) = lim_{n→∞} χ_{A_{n}}(x)F (x, \frac{d_{φ_{n}} (x, x + t_{n}ξ)}{t_{n}}) ,  (11)

where A_{n} := {x ∈ A | dist(x, ∂A) > t_{n}}. Now, integrating (11) over ω and applying the dominated convergence theorem, we get:

∫_{ω} F (x, φ (x, ξ)) dx = lim_{n→∞} ∫_{ω} χ_{A_{n}}(x)F (x, \frac{d_{φ_{n}} (x, x + t_{n}ξ)}{t_{n}}) dx.  (12)

Since d_{φ_{n}} (x, x + t_{n}ξ) is less than or equal to the (Finslerian) length of the straight line segment joining x and x + t_{n}ξ, we have

d_{φ_{n}} (x, x + t_{n}ξ) ≤ ∫_{0}^{1} φ_{n} (x + st_{n}ξ, t_{n}ξ) ds.

By the monotonicity and convexity of the function F (x, ·) for a.e. x we get, by using Jensen inequality, that for a.e. x ∈ A,

F (x, \frac{d_{φ_{n}} (x, x + t_{n}ξ)}{t_{n}}) ≤ ∫_{0}^{1} F (x, φ_{n} (x + st_{n}ξ, ξ))ds.  (13)

Combining (12) and (13), we obtain

∫_{ω} F (x, φ (x, ξ))dx ≤ lim inf_{n→∞} ∫_{Ω} χ_{A_{n}∩ω}(x) ∫_{0}^{1} F (x, φ_{n} (x + st_{n}ξ, ξ))ds dx

= lim inf_{n→∞} ∫_{0}^{1} ∫_{Ω} χ_{A_{n}∩ω} (x − st_{n}ξ) F (x − st_{n}ξ, φ_{n} (x, ξ))dx ds

= lim inf_{n→∞} ∫_{ω} F (x, φ_{n} (x, ξ))dx .

The following two lemmas are analogous to [5, Lemma10, Lemma 11] and may be proved in the same way, up to some technical adaptations.

Lemma 3.2. Let φ ∈ M be a continuous Finsler metric. Then, for every bounded open set A ⊂⊂ Ω and for every ε > 0, there exists δ > 0 such that

∫_{D_{i}^{δ}∩A} F (x, Λ_{φ}(x)) dx ≤ sup_{|ξ|=1} ∫_{D_{i}^{δ}∩A} [F (x, φ(x, ξ)) + ε] dx for all i ∈ ℤ^{N},

where we have set D_{i}^{δ} := Ω ∩ (i + [−δ, δ)^{N} ).

Lemma 3.3. Let φ ∈ M such that φ(x, ·) is convex for every x ∈ Ω. Then for every bounded open set A ⊂⊂ Ω and for every ε > 0 there exists a compact set K_{ε} ⊂ A such that L^{N}(A \ K_{ε}) < ε and φ is continuous on K_{ε} × ℝ^{N}.

By using the previous lemmas we can prove the following

Proposition 3.4. Let φ ∈ M. Assume that, for a sequence (μ_{n})_{n∈ℕ} of nonnegative Borel measures on Ω, the following property holds:

sup_{|ξ|=1} ∫_{ω} F (x, φ(x, ξ)) dx ≤ lim inf_{n→∞} μ_{n} (ω) for every Borel set ω ⊂⊂ Ω .

Then

∫_{Ω} F (x, Λ_{φ}(x)) dx ≤ lim inf_{n→∞} μ_{n} (Ω) .  (14)

Proof : Let (Ω_{l})_{l∈ℕ} be a sequence of bounded open sets well contained in Ω such that \overline{Ω}_{l} ⊂ Ω_{l+1} and Ω = ⋃_{l∈ℕ} Ω_{l}. We first remark that it is sufficient to prove that (14) holds for Ω := Ω_{l} for every l ∈ ℕ. Then, the claim is easily obtained by adapting the proof given in [5, Proposition 12] and by using Lemmas 3.1, 3.2 and 3.3.

Next, we show some results on Finsler metrics. We start by the following

Proposition 3.5. Let φ ∈ M and d := d_{φ}. Then

(i) φ_{d}(x, ξ) ≤ φ(x, ξ) for a.e. x ∈ Ω and for every ξ ∈ ℝ^{N}. In particular Λ_{d}(x) ≤ sup_{|ξ|=1} φ(x, ξ) for a.e. x ∈ Ω;

(ii) if φ(x, ξ) := a(x)|ξ| with a : Ω → [α, β] lower semicontinuous, then φ_{d}(x, ξ) ≥ a(x)|ξ| for every (x, ξ) ∈ Ω × ℝ^{N}. In particular a(x) = Λ_{d}(x) for a.e. x ∈ Ω.

Proof : Let us fix a ξ ∈ 𝕊^{N−1}. For every x ∈ Ω let us define the curve γ_{x}(t) := x + tξ. Then by Proposition 2.3 we have that

L_{d}(γ_{x}) := ∫_{0}^{1} φ_{d}(γ_{x}, ξ) dt ≤ ∫_{0}^{1} φ(γ_{x}, ξ) dt =: L_{φ}(γ_{x}).

Therefore we deduce that φ_{d}(x, ξ) ≤ φ(x, ξ) for a.e. x ∈ Ω. Then we can take a dense sequence (ξ_{n})_{n∈ℕ} in 𝕊^{N−1} and repeat the argument above for each ξ_{n}. Recalling that the functions φ_{d}(x, ·) and φ(x, ·) are continuous and 1-homogeneous for a.e. x ∈ Ω, we eventually get, by the density of (ξ_{n})_{n∈ℕ}, that φ_{d}(x, ξ) ≤ φ(x, ξ) for a.e. x ∈ Ω and for every ξ ∈ ℝ^{N}. In particular we get

Λ_{d}(x) ≤ sup_{|ξ|=1} φ(x, ξ)  a.e. in Ω.  (15)

Let us now take φ(x, ξ) := a(x)|ξ| with a lower semicontinuous. Then we have, by the lower semicontinuity, that a(x) = sup_{r>0} (inf_{B_{r}(x)} a). Therefore for every fixed x ∈ Ω and for every ε > 0 there exists r_{ε} > 0 such that B_{r_{ε}}(x) ⊂ Ω and a(y) ≥ a(x) − ε for every y ∈ B_{r_{ε}}(x). Let us fix a ξ ∈ 𝕊^{N−1} and take 0 < t < αr_{ε}/(2β). Choose a d-minimizing sequence (γ_{n})_{n∈ℕ} ⊂ L_{x,x+tξ} such that L_{a}(γ_{n}) ≤ d(x, x + tξ) + αr_{ε}/2 for every n. Then the curves γ_{n} lie within B_{r_{ε}}(x). In fact for every n and for every s ≤ 1:

α|γ_{n}(s) − x| ≤ ∫_{0}^{s} a(γ)|γ̇_{n}|dτ ≤ d(x, x + tξ) + αr_{ε}/2 ≤ βd_{Ω}(x, x + tξ) + αr_{ε}/2 < αr_{ε},

where we have used the fact that d_{Ω}(x, y) = |x − y| if y ∈ B_{r_{ε}}(x). Then we have for every n

L_{a}(γ_{n}) := ∫_{0}^{1} a(γ_{n})|γ̇_{n}| dτ ≥ (a(x) − ε) ∫_{0}^{1} |γ̇_{n}| dτ ≥ (a(x) − ε)t

and letting n go to infinity we obtain

\frac{d(x, x + tξ)}{t} ≥ a(x) − ε.  (16)

By passing to the limsup in (16) as t → 0 and since ε > 0, x ∈ Ω and ξ ∈ 𝕊^{N−1} were arbitrary we obtain

φ_{d}(x, ξ) ≥ a(x)  for every (x, ξ) ∈ Ω × 𝕊^{N−1}  (17)

and the claim follows by the 1-homogeneity of φ_{d}(x, ·). In particular, by taking the sup of the left-hand side of (17) over all ξ ∈ 𝕊^{N−1} and by using (15) we get that Λ_{d}(x) = a(x) for a.e. x ∈ Ω.

Remark 3.6. If a and b are two continuous isotropic metrics which give rise to the same distance function d through (2), then a(x) = b(x) for every x in Ω. In fact, by point (ii) of the stated lemma, we have that the previous equality holds almost everywhere, and therefore everywhere by the continuity of the metrics. In particular, this shows that the functional (1) is well definite.

The key idea used in the proof of the density result is stated in the following

Lemma 3.7. Let (d_{n})_{n∈ℕ} be a sequence contained in D which converges pointwise to some

d ∈ D on a dense subset of Ω × Ω. Then d_{n} ⟶^{D} d.

Proof : By the compactness of D, we already know that there is a subsequence (d_{n_{k}})_{k∈ℕ}

such that d_{n_{k}} ⟶^{D} δ for some δ ∈ D. By the pointwise convergence we get that δ(x, y) = d(x, y) on a dense subset of Ω × Ω and therefore δ coincides with d since they are both continuous functions. If the whole sequence did not converge uniformly (on compact subset of Ω × Ω) to d, by the compactness of D there would exists a subsequence which converges to some δ ∈ D with δ ≠ d. By arguing as above, this would lead to a contradiction.

The next result shows that the monotone convergence of metrics implies the convergence of the induced distances.

Lemma 3.8. Let (φ_{n})_{n∈ℕ} be a sequence in M such that for every (x, ξ) ∈ Ω × ℝ^{N} φ_{n}(x, ξ) converge increasingly (resp. decreasingly) to φ(x, ξ) for some φ ∈ M. Then d_{φ_{n}} ⟶^{D} d_{φ}.

Proof : By Lemma 3.7 it is sufficient to prove that (d_{φ_{n}})_{n∈ℕ} converges pointwise to d_{φ}. We start by considering the case of an increasing sequence of metrics. By the monotonicity of φ_{n} we obviously have that L_{φ}(γ) ≥ L_{φ_{n}}(γ) ≥ L_{φ_{n−1}}(γ) for every curve γ and therefore (d_{φ_{n}}(x, y))_{n∈ℕ} is an increasing sequence and d_{φ}(x, y) ≥ sup_{n} d_{φ_{n}}(x, y) for every (x, y) ∈ Ω × Ω. To prove the reverse inequality, let us take a sequence of curves (γ_{n})_{n∈ℕ} ⊂ L_{x,y} such that L_{φ_{n}}(γ_{n}) ≤ d_{φ_{n}}(x, y) + 1/n. Since the functionals L_{φ_{n}} are equi-coercive (in fact L_{φ_{n}}(γ) ≥ α ∫_{0}^{1} |γ̇|dt for every n), we may find a subsequence (γ_{n_{i}})_{i∈ℕ} which converges uniformly to some curve γ ∈ L_{x,y}. Now, by [7, Remark 5.5], we know that the functionals L_{φ_{n_{i}}} Γ-converge to L_{φ} with respect to the uniform convergence of path and therefore we have

d_{φ}(x, y) ≤ L_{φ}(γ) ≤ lim inf_{i→+∞} L_{φ_{n_{i}}} (γ_{n_{i}}) ≤ lim inf_{i→+∞} d_{φ_{n_{i}}} (x, y) = sup_{n} d_{φ_{n}}(x, y).

Since (x, y) ∈ Ω × Ω was arbitrary the claim follows. The proof in the case of a decreasing sequence of metrics is even simpler. In fact, by monotonicity we get d_{φ}(x, y) ≤ inf_{n} d_{φ_{n}}(x, y) for every (x, y) ∈ Ω × Ω. To show the reverse inequality, take a curve γ ∈ L_{x,y}. By the monotone convergence theorem and by the definition of d_{φ_{n}}(x, y) we have

L_{φ}(γ) = inf_{n} L_{φ_{n}}(γ) ≥ inf_{n} d_{φ_{n}}(x, y),

and the claim easily follows by taking the infimum over all curves in L_{x,y}.

We end this section with the proof of two lemmas which will be useful in the sequel.

Lemma 3.9. Let {γ^{i} | γ^{i} ∈ L_{x_{i},y_{i}}, i ≤ n} be a finite collection of curves such that

d(x_{i}, y_{i}) ≤ L_{d}(γ^{i}) ≤ d(x_{i}, y_{i}) + \frac{1}{n}  (18)

for some fixed points (x_{i}, y_{i}) ∈ Ω × Ω and for some n ∈ ℕ. Then it is possible to find a family of curves {γ̃^{i} | γ̃^{i} ∈ L_{x_{i},y_{i}}, i ≤ n} still satisfying (18) and such that (i) γ̃^{i} is injective for every i ≤ n;

(ii) γ̃^{i}(I) ∩ γ̃^{j}(I) is a (possibly void) disjoint finite union of closed arcs for every 1 ≤ i ≤ j ≤ n.

Proof : Let N be a 1-rectifiable closed set such that N ⊃ ∪_{i≤n}γ^{i}. First we remark that for every i ≤ n the set

R_{i} := argmin{L_{d}(γ) | γ ∈ L_{x_{i},y_{i}}, γ(I) ⊂ N}

is non-void. Indeed, the class of curves on which we minimize L_{d} is non-void, as it contains γ^{i}, and closed with respect to the uniform convergence of curves, as N is closed, therefore it contains an accumulation point γ̃^{i} of a minimizing sequence. Such a curve is of minimal d-length by the lower semicontinuity of L_{d} and so it belongs to R_{i}. Moreover, it is injective and satisfies (18) by minimality. The proof of the lemma is by induction on n. For n = 1 the claim is satisfied by choosing a γ̃^{1} which belongs to R_{1}. Let us then suppose the claim satisfied up to n − 1 and let us prove it for n. By induction we may find curves γ̃^{i} ∈ R_{i} for i ≤ n − 1 such to satisfy the claim. Let us choose a curve σ in R_{n}. For every j ≤ n − 1 let us set t_{j} := min{t ∈ I | σ(t) ∈ γ̃^{j}(I) } and T_{j} := max{t ∈ I | σ(t) ∈ γ̃^{j}(I) }. Up to reordering the curves γ̃^{j}, we can suppose that t_{1} = min{t_{j} | j ≤ n − 1 }. Then we define τ_{1} ∈ L_{x_{n},y_{n}} to be the curve obtained by moving from σ(0) to σ(t_{1}) along σ, from σ(t_{1}) to σ(T_{1}) along γ̃_{1} and from σ(T_{1}) to σ(1) along σ again. Remark that, by minimality, γ̃^{1} is a path which connects σ(t_{1}) to σ(T_{1}) in the shortest way among all those contained in N and so we have not increased the length, i.e. L_{d}(τ_{1}) ≤ L_{d}(σ) and τ_{1} ∈ R_{n}. Moreover τ_{1}([0, T_{1}]) ∩ γ̃_{i}(I) is a disjoint finite union of closed arcs for every 1 ≤ i ≤ n − 1. Then we set σ := τ_{1} |_{[T_{1},1]} and we repeat the argument above to obtain a τ_{2} : [T_{1}, 1] → N. By iterating this procedure we eventually find a finite number of curves {τ_{h} | 1 ≤ h ≤ M} for some M < n. Then we define

γ̃^{n}(t) := \{ \begin{matrix}τ_{1}(t) & if t ∈ [0, T_{1}] \\ τ_{h}(t) & if t ∈ [T_{h−1}, T_{h}] and 1 < h < M \\ τ_{M}(t) & if t ∈ [T_{M−1}, 1]. \end{matrix}

By what previously observed, we have that γ̃^{n} still belongs to R_{n} and is therefore injective by minimality. Moreover, it is such that γ̃^{n}(I) ∩ γ̃^{i}(I) is a disjoint finite union of closed arcs for every i ≤ n − 1 by construction. The claim is thus proved.

Lemma 3.10. Let γ be an injective curve, Γ := γ((0, 1)) ⊂ Ω and a : Ω → [α, β] a Borel function. Then there exists a sequence of continuous functions σ_{k} : Γ → [α, β] such that σ_{k}(x) converge to a(x) for H^{1}-a.e. x ∈ Γ. Moreover, for every ε > 0 there exists a Borel subset B_{ε} ⊂ Γ such that H^{1}(Γ \ B_{ε}) < ε and σ_{k} converge uniformly to a on B_{ε}.

Proof : The function a∘γ : (0, 1) → [α, β] is Borel measurable, therefore there exists a sequence (f_{k})_{k∈ℕ} of continuous functions f_{k} : (0, 1) → [α, β] such that f_{k}(t) converges to a∘γ(t) for a.e. t ∈ (0, 1). Moreover, by Severini-Egoroff’s theorem [9, Section 1.2, Theorem 3], for every ε > 0 there exist an infinitesimal sequence (δ_{k})_{k∈ℕ} and a Borel set E_{ε} such that H^{1}((0, 1) \ E_{ε}) < ε and |f_{k}(t) − a∘γ(t)| < δ_{k} for every t ∈ E_{ε}. The claim then follows by setting σ_{k}(x) := f_{k}(γ^{−1}(x)) and B_{ε} := γ(E_{ε}).

4 Main results

Our main result is stated as follows.

Theorem 4.1. Let F be the functional defined on I by (1), where F : Ω × [α, β] → ℝ is continuous and satisfies conditions (6). Then its relaxed functional (4) has the following integral representation:

\overline{F}(d) = ∫_{Ω} F (x, Λ_{d}(x)) dx  (19)

for all d ∈ D.

The proof of the theorem above is based on the following two results which we state separately.

Theorem 4.2. If d_{n} ⟶^{D} d, then lim inf_{n→+∞} ∫_{Ω} F (x, Λ_{d_{n}}(x)) dx ≥ ∫_{Ω} F (x, Λ_{d}(x)) dx.

Theorem 4.3. The family I of distances induced by continuous and isotropic Riemannian metrics is dense in D. Moreover, for every d ∈ D we can choose a sequence (d_{n})_{n∈ℕ} ⊂ I such that d_{n} ⟶^{D} d and

lim sup_{n→+∞} ∫_{Ω} F (x, Λ_{d_{n}}(x)) dx ≤ ∫_{Ω} F (x, Λ_{d}(x)) dx.

Remark 4.4. The class of distances induced by smooth isotropic Riemannian metrics is dense in I. Therefore, by the theorem just stated, smooth isotropic Riemannian metrics are dense in the class of Finsler metrics. In fact, let us take a distance d in I. Then d = d_{a} for some continuous metric a : Ω → [α, β]. We may extend a to the whole ℝ^{n} by setting a identically equal to α outside Ω. Then, by taking a sequence of convolution kernels ρ_{n}, we define the sequence of smooth isotropic metrics a_{n} : Ω → [α, β] by regularization, i.e. a_{n}(x) := ρ_{n} ∗ a(x), and we call d_{n} the induced distances. Since the functions a_{n} converge to a uniformly on compact subset of Ω × Ω, it can be easily shown that the length functionals L_{a_{n}} Γ-converge to L_{a} with respect to the uniform convergence of curves. Then, by [6, Theorem 3.1], we have that d_{n} ⟶^{D} d (this could also have been proved directly by using the equi-coercivity of the length functionals to show that the above convergence of distances is pointwise and then applying Lemma 3.7).

Once Theorem 4.2 and Theorem 4.3 are proved, the proof of Theorem 4.1 will trivially follows. In fact, Theorem 4.2 gives that the functional (19) is lower semicontinuous with respect to the uniform convergence of distances, and Theorem 4.3 implies it is the greatest lower semicontinuous functional defined on D which is bounded from above by F on I. In fact, let G be another candidate and let d ∈ D. Choose a sequence (d_{n})_{n∈ℕ} ⊂ I as in the statement of Theorem 4.3. We have

G(d) ≤ lim inf_{n→+∞} G(d_{n}) ≤ lim inf_{n→+∞} F(d_{n}) ≤ lim sup_{n→+∞} F(d_{n}) ≤ ∫_{Ω} F (x, Λ_{d}(x)) dx,

hence the claim. We remark that by Proposition 3.5 the functional (19) actually coincides with F on I.

Let us then start by proving Theorem 4.2.

Proof of Theorem 4.2: By applying Lemma 3.1 with φ_{n} := φ_{d_{n}}, we obtain

sup_{|ξ|=1} ∫_{ω} F (x, φ_{d}(x, ξ)) dx ≤ lim inf_{n→∞} ∫_{ω} F (x, Λ_{d_{n}}(x)) dx.

The claim then follows by applying Proposition 3.4 with μ_{n}(ω) := ∫_{ω} F (x, Λ_{d_{n}}(x)) dx.

Remark 4.5. The proof above still works for slightly more general functionals. Indeed, it is sufficient that there exists a sequence of continuous functions F_{k} : Ω × [α, β] → ℝ which satisfy conditions (6) and such that F (x, ξ) = sup_{k} F_{k}(x, ξ) for L^{N}-a.e. x ∈ Ω and for every ξ ∈ ℝ^{N}. In fact, one can apply the argument above to each F_{k} to get

∫_{ω} F_{k} (x, Λ_{d}(x)) dx ≤ lim inf_{n→∞} ∫_{ω} F (x, Λ_{d_{n}}(x)) dx,

and the claim immediately follows by taking the supremum over k of the left-hand side term and by the monotone convergence theorem.

We now come to the proof of Theorem 4.3. The proof is essentially divided in two steps: first, we approximate a given d ∈ D with distances induced by a sequence of Borel measurable and isotropic Riemannian metrics, then we approximate each distance of the sequence by means of distances in I.

Proposition 4.6. Let d ∈ D. Then there exists a sequence of Borel measurable isotropic metrics a_{n} : Ω → [α, β] such that

(i) d_{a_{n}} ⟶^{D} d;

(ii) a_{n}(x) = Λ_{d}(x) for a.e. x ∈ Ω.

Proof : By Lemma 3.7, it is sufficient to define the functions a_{n} in such a way that the generated distances d_{a_{n}} converges pointwise to d on a dense subset of Ω × Ω. Let us start then by setting S := ℚ^{N} ∩ Ω. Obviously S × S is dense in Ω × Ω and countable, so we write S × S := {(x_{i}, y_{i}) | i ∈ ℕ}. For each (x_{i}, y_{i}) we take a d-minimizing sequence (γ_{n}^{i})_{n∈ℕ} ⊂ L_{x_{i},y_{i}}, i.e. such that

d(x_{i}, y_{i}) ≤ L_{d}(γ_{n}^{i}) ≤ d(x_{i}, y_{i}) + \frac{1}{n}.  (20)

By Lemma 3.9, the curves γ_{n}^{i} can be chosen in such a way to satisfy conditions (i) and (ii) of the mentioned lemma (this assumption is not really needed here, but will be important in the proof of Theorem 4.3). By condition (ii), each non-empty set γ_{n}^{i}(I) ∩ γ_{n}^{j}(I) is a disjoint finite union of closed arcs. Let us denote by T_{n} the finite set given by the extreme points of such arcs for every 1 ≤ i ≤ j ≤ n and set N_{n} := ∪_{i≤n}γ_{n}^{i}(I). Let Σ_{n} be a Borel H^{1}-negligible subset of N_{n} which contains the points where the 1-rectifiable set N_{n} is not differentiable (this is possible by the regularity of the measure H^{1} and by the differentiability property of rectifiable sets [10, Theorem 1.6, Theorems 3.8 and 3.14]). Then we define the function a_{n} : Ω → [α, β] by

a_{n}(x) := \{ \begin{matrix}Λ_{d}(x) & if x ∈ Ω \ N_{n} \\ α & if x ∈ Σ_{n} ∪ T_{n} \\ φ_{d}(x, ξ_{x}) & if x ∈ N_{n} \ (Σ_{n} ∪ T_{n}) \end{matrix}  (21)

where ξ_{x} is the unitary tangent to N_{n} at the point x. It is not difficult to prove that a_{n} is Borel-measurable. Moreover it is clear that a_{n} satisfies point (ii) of the Proposition.

We remark that, by [8, Corollary 2.7], we have that φ_{d}(x, ξ_{x}) = φ_{d}(x, −ξ_{x}) for H^{1}-a.e. x ∈ N_{n}. By possibly enlarging the set Σ_{n} we may suppose that this holds everywhere on N_{n} \ Σ_{n}. Moreover, if x = γ_{i}^{n}(t) and γ_{i}^{n} is differentiable in t, we have that γ̇_{i}^{n}(t) is parallel to ξ_{x} and therefore φ_{d}(γ_{i}^{n}(t), γ̇_{i}^{n}(t)) = φ_{d}(γ_{i}^{n}(t), ξ_{x})|γ̇_{i}^{n}(t)| = a_{n}(γ_{i}^{n}(t))|γ̇_{i}^{n}(t)|.

Let d_{a_{n}} be the distances generated by such functions a_{n}. In order to prove point (i), we show that the distances d_{a_{n}} converge pointwise to d on S ×S. We claim that for every i ≤ n we have

d(x_{i}, y_{i}) ≤ d_{a_{n}}(x_{i}, y_{i}) ≤ d(x_{i}, y_{i}) + \frac{1}{n}.

Let us fix an i ≤ n and let us prove the second inequality. By the above remark and by (20) we have

d_{a_{n}}(x_{i}, y_{i}) ≤ ∫_{0}^{1} a_{n}(γ_{n}^{i})|γ̇_{n}^{i}| dt = ∫_{0}^{1} φ_{d}(γ_{n}^{i}, γ̇_{n}^{i}) dt ≤ d(x_{i}, y_{i}) + \frac{1}{n}.

To prove the first inequality, choose a curve σ ∈ L_{x_{i},y_{i}} and for every i ≤ n set I_{i} := {t ∈ I | σ(t) ∈ γ_{n}^{i}(I)} and I_{0} := I \ ∪_{i≤n}I_{i}. We remark that the vector σ̇(t) is parallel to ξ_{σ(t)} a.e. on each I_{i} and so a_{n}(σ)|σ̇| = φ_{d}(σ, σ̇) a.e. on I_{i}. Therefore we have

L_{a_{n}}(σ) = ∫_{0}^{1} a_{n}(σ)|σ̇| dt = ∑_{i=1}^{n} ∫_{I_{i}} a_{n}(σ)|σ̇| dt + ∫_{I_{0}} a_{n}(σ)|σ̇| dt

≥ ∑_{i=1}^{n} ∫_{I_{i}} φ_{d}(σ, σ̇) dt + ∫_{I_{0}} φ_{d}(σ, σ̇) dt ≥ d(x_{i}, y_{i}),

where we have used the fact that a_{n}(σ)|σ̇| ≥ φ_{d}(σ, σ̇) on I_{0}. By passing to the infimum over all possible curves σ ∈ L_{x_{i},y_{i}} we get the claim.

Proof of Theorem 4.3. The proof is organized in two steps.

Step 1. We first remark that the closure of I contains the family of distances generated by lower semicontinuous isotropic Riemannian metrics. In fact, let b : Ω → [α, β] be a lower semicontinuous metric. It is well known that b(x) = sup_{n∈ℕ} ã_{n}(x) for suitable continuous functions ã_{n} (and we may as well suppose that α ≤ ã_{n} ≤ β by possibly replacing the function ã_{n} with ã_{n} ∨ α). Setting a_{n}(x) := sup_{i≤n} ã_{i}(x), we have that d_{a_{n}} ⟶^{D} d_{b} by Lemma 3.8. Moreover, by Proposition 3.5 we have that Λ_{d_{b}}(x) = b(x) and Λ_{d_{an}} (x) = a_{n}(x) almost everywhere on Ω and therefore, by the monotone convergence, we get that

lim sup_{n} ∫_{Ω} F (x, Λ_{d_{an}} (x)) dx = ∫_{Ω} F (x, Λ_{b}(x)) dx.

To prove the theorem, it is then sufficient to find a sequence of lower semicontinuous metrics b_{n} : Ω → [α, β] such that the generated distances d_{b_{n}} satisfy the claim of the theorem. Indeed, by combining the idea just described with a diagonal argument, the conclusion would follow at once.

Step 2. To get the desired approximation of the distance d ∈ D via lower semicontinuous isotropic metrics, it is enough to prove that, for every fixed n ∈ ℕ there exists a sequence of lower semicontinuous isotropic metrics b_{k} : Ω → [α, β] such that

(i) d(x_{i}, y_{i}) ≤ lim sup_{k→+∞} d_{b_{k}}(x_{i}, y_{i}) ≤ d(x_{i}, y_{i}) + \frac{1}{n} for every i ≤ n;

(ii) lim sup_{k→+∞} ∫_{Ω} F (x, b_{k}(x)) dx ≤ ∫_{Ω} F (x, a_{n}(x)) dx where a_{n} are the Borel isotropic metrics built in the proof of Proposition 4.6.

In fact the desired sequence of lower semicontinuous metrics is then obtained via a diagonal argument and taking into account that a_{n}(x) = Λ_{d}(x) almost everywhere on Ω by Proposition 4.6.

Keeping the notation used in the proof of Proposition 4.6, we observe that the set N_{n}\T_{n} is a finite, disjoint union of open arcs. Therefore, by applying Lemma 3.10 to each arc, we can find a sequence of continuous functions σ_{k} : N_{n} \ T_{n} → [α, β] which converge to a_{n} H^{1}- a.e. on N_{n} \ T_{n}. Let us set A_{k} := {x ∈ Ω | dist(x, N_{n}) < 1/k}. Let (Ω_{k})_{k∈ℕ} be a sequence of bounded open sets well contained in Ω such that \overline{Ω}_{k} ⊂ Ω_{k+1} and Ω = ⋃_{k∈ℕ} Ω_{k}. By Lusin’s theorem we may find a sequence of closed set K_{k} ⊂ Ω_{k} \ A_{k} such that a |_{K_{k}} is continuous and L^{n}((Ω_{k} \ A_{k}) \ K_{k}) < 1/k. Then we define b_{k} : Ω → [α, β] by

b_{k}(x) := \{ \begin{matrix}σ_{k}(x) & if x ∈ N_{n} \ T_{n} \\ α & if x ∈ T_{n} \\ a_{n}(x) & if x ∈ K_{k} \\ β & elsewhere. \end{matrix}  (22)

Notice that b_{k} is lower semicontinuous. Moreover we have

lim sup_{k→+∞} ∫_{Ω} F (x, b_{k}(x)) dx = lim sup_{k→+∞} (∫_{K_{k}} F (x, a_{n}(x)) dx + ∫_{Ω\K_{k}} F (x, β) dx) .  (23)

Recalling that F (x, β) is summable over Ω (condition (ii) of (6)), we have that the second integral in the right-hand side of (23) goes to zero. In fact

∫_{Ω\K_{k}} F (x, β) dx = ∫_{Ω\Ω_{k}} F (x, β) dx + ∫_{Ω_{k}\K_{k}} F (x, β) dx,  (24)

and the first and second term of the right-hand side of (24) go to zero, respectively by the dominated convergence theorem and the absolute continuity of the integral. Therefore

lim sup_{k→+∞} ∫_{Ω} F (x, b_{k}(x)) dx ≤ ∫_{Ω} F (x, a_{n}(x)) dx,

so point (ii) of the claim is satisfied. Let us show now that (i) holds. We start by proving the second inequality. For i ≤ n we have by definition

d_{b_{k}}(x_{i}, y_{i}) ≤ L_{b_{k}}(γ_{i}^{n}) = ∫_{0}^{1} σ_{k}(γ_{i}^{n})|γ̇_{i}^{n}| dt,

therefore by the dominated convergence theorem we get

lim sup_{k→+∞} d_{b_{k}}(x_{i}, y_{i}) ≤ lim sup_{k→+∞} ∫_{0}^{1} σ_{k}(γ_{i}^{n})|γ̇_{i}^{n}| dt = ∫_{0}^{1} a_{n}(γ_{i}^{n})|γ̇_{i}^{n}| dt
= ∫_{0}^{1} φ_{d}(γ_{i}^{n}, γ̇_{i}^{n}) dt ≤ d(x_{i}, y_{i}) + \frac{1}{n}.  (25)

To prove the first inequality let us take for every k ∈ ℕ a curve γ_{k} ∈ L_{x_{i},y_{i}} such that

d_{k}(x_{i}, y_{i}) ≤ L_{b_{k}}(γ_{k}) ≤ d_{k}(x_{i}, y_{i}) + \frac{1}{k} .  (26)

Once again, we remark that, by Lemma 3.9, it is not restrictive to suppose that such curves are injective. Since α ∫_{I} |γ̇_{k}| dt ≤ L_{b_{k}}(γ_{k}), by (26) and (25) we get that lim sup_{k} ∫_{I} |γ̇_{k}| dt < +∞. Let us choose an ε > 0. By applying Lemma 3.10 to each open arc of N_{n} \ T_{n}, we can find a Borel set B_{ε} ⊂ N_{n} \ T_{n} and an infinitesimal sequence of positive numbers (δ_{k})_{k∈ℕ} such that H^{1}(N_{n} \ B_{ε}) < ε and |σ_{k}(x) − a_{n}(x)| < δ_{k} for every x ∈ B_{ε}. Let us set I_{k} := {t ∈ I|γ_{k}(t) ∈ N_{n} \ B_{ε}}. Then b_{k}(γ_{k}) ≥ a_{n}(γ_{k}) − δ_{k} a.e. on I \ I_{k}. Let us write

L_{b_{k}}(γ_{k}) = ∫_{I_{k}} b_{k}(γ_{k})|γ̇_{k}|dt + ∫_{I\I_{k}} b_{k}(γ_{k})|γ̇_{k}|dt.

We remark that, as γ_{k}(I_{k}) ⊂ N_{n} \ B_{ε} for every k ∈ ℕ, by the Area-formula we have

∫_{I_{k}} |γ̇_{k}| dt = H^{1}(γ_{k}(I_{k})) ≤ H^{1}(N_{n} \ B_{ε}) < ε.

Taking into account this remark we get

∫_{I_{k}} b_{k}(γ_{k})|γ̇_{k}|dt = ∫_{I_{k}} a_{n}(γ_{k})|γ̇_{k}|dt + ∫_{I_{k}}(b_{k}(γ_{k}) − a_{n}(γ_{k}))|γ̇_{k}|dt
≥ ∫_{I_{k}} a_{n}(γ_{k})|γ̇_{k}|dt − (β − α)ε.

Then we have

L_{b_{k}}(γ_{k}) ≥ ∫_{0}^{1} a_{n}(γ_{k})|γ̇_{k}| dt − δ_{k} ∫_{I\I_{k}} |γ̇_{k}| dt − (β − α)ε

≥ d_{a_{n}}(x_{i}, y_{i}) − δ_{k} ∫_{0}^{1} |γ̇_{k}| dt − (β − α)ε

and therefore, as δ_{k} ∫_{0}^{1} |γ̇_{k}| dt goes to zero, we obtain

lim sup_{k→+∞} d_{b_{k}}(x_{i}, y_{i}) ≥ lim sup_{k→+∞} L_{b_{k}}(γ_{k}) ≥ d_{a_{n}}(x_{i}, y_{i}) − (β − α)ε.

The claim then follows since ε was arbitrary.

Remark 4.7. It should be noticed that the proof of Theorem 4.3 holds under very general assumptions on the function F, namely it is sufficient to take an F which is Borel measurable and satisfies assumption (ii) of (6), and such that the function F (x, ·) is non-decreasing for L^{N}-a.e. x ∈ Ω. This consideration, together with Remark 4.5, enables us to conclude that our relaxation result, namely Theorem 4.1, holds under the following milder conditions on F : Ω × [α, β] → ℝ:

(i) there exist a sequence of continuous functions F_{k} : Ω×[α, β] → ℝ satisfying conditions (6) and such that F (x, ξ) = sup_{k∈ℕ} F_{k}(x, ξ) for L^{N}-a.e. x ∈ Ω, for every ξ ∈ ℝ^{N};

(ii) ∫_{Ω} F (x, β) dx < +∞.

With a slight modification of the argument used in the proof of Proposition 4.6 we can prove the following result.

Proposition 4.8. Let d ∈ D. Then there exists a Borel function a : \overline{Ω} → [α, β] such that, for every (x, y) ∈ Ω × Ω,

d(x, y) = inf \{ ∫_{0}^{1} a(γ(t))|γ̇(t)| dt : γ ∈ Lip([0, 1]; \overline{Ω}), γ(0) = x, γ(1) = y \}.

In particular, if Ω := ℝ^{N}, for every d ∈ D there exists a Borel measurable, isotropic Riemannian metric a : ℝ^{N} → [α, β] such that d = d_{a} according to definition (2).

Proof : Let us first remark that one can think the distance d ∈ D to be defined on \overline{Ω} × \overline{Ω} by extending it continuously up to the boundary. Therefore the d-length of every path γ : I → \overline{Ω} is defined, according to definition (9). Let us define the metric derivative of the path γ at the point t ∈ I as

md_{d}(γ)(t) := lim_{h→0} \frac{d(γ(t + h), γ(t))}{h}.  (27)

It is well known (see [2] for instance) that the limit in (27) exists for L^{1}-a.e. t ∈ I and that

L_{d}(γ) = ∫_{0}^{1} md_{d}(γ)(t) dt.

Notice also that, if γ(t) ∈ Ω, then md_{d}(γ)(t) = φ_{d}(γ(t), γ̇(t)), as one can easily show comparing the definitions of φ_{d} and md_{d} and recalling that locally α|x − y| ≤ d(x, y) ≤ β|x − y|. Moreover, we observe that a Borel function a : \overline{Ω} → [α, β] induces a distance δ_{a} on \overline{Ω} through the formula

δ_{a}(x, y) := inf \{ ∫_{0}^{1} a(γ(t))|γ̇(t)| dt : γ ∈ Lip([0, 1]; \overline{Ω}), γ(0) = x, γ(1) = y \}  (28)

for every (x, y) ∈ \overline{Ω} × \overline{Ω}.

Comparing the definition of δ_{a} with the one of d_{a} given in (2), we see that the main difference relies upon the fact that the curves on which we minimize the length L_{a} are now allowed to lye in the closure of Ω, therefore δ_{a} depends also from the values assumed by a on the boundary of Ω. In particular, we remark that in general δ_{a}(x, y) ≤ d_{a}(x, y) for (x, y) ∈ Ω × Ω, and this inequality may be strict due to the fact that a is not continuous. For instance, take Ω := (−1, 1) × (−1, 1) and a(x) := χ\overline{Ω}(x) + χ_{Ω}(x). One can easily see that points near the boundary of Ω are closer with respect to δ_{a} since also the boundary of Ω can be used to connect points in definition (28).

Let us now set S := ℚ^{N} ∩ Ω and write S × S = {(x_{i}, y_{i}) | i ∈ ℕ}. By the lower semicontinuity of L_{d}, we have that for every i ∈ ℕ there exists a curve γ_{i} : I → \overline{Ω} such that L_{d}(γ_{i}) = d(x_{i}, y_{i}) (just take for γ_{i} an accumulation point of a d-minimizing sequence of curves in \overline{Ω} which connect x and y). Let N_{n} := ∪_{i≤n}γ_{i}(I) and Σ_{n} be an H^{1}-negligible Borel set which contains the non-differentiability points of N_{n}. Then define a_{n} : \overline{Ω} → [α, β] by

a_{n}(x) := \{ \begin{matrix}α & if x ∈ Σ_{n} \\ \frac{md_{d}(γ_{i})(t)}{|γ̇_{i}(t)|} & if x = γ_{i}(t) ∈ N_{n} \ Σ_{n} for some i ≤ n and some t ∈ I \\ β & elsewhere. \end{matrix}  (29)

It is easy to show that a_{n} is Borel measurable. Moreover, arguing as in the proof of Proposition 4.6, one can show that δ_{a_{n}}(x_{i}, y_{i}) = d(x_{i}, y_{i}) for every i ≤ n. Notice that N_{n} ⊂ N_{n+1} and, up to replacing Σ_{n+1} with Σ_{n+1} ∪ Σ_{n}, we can always suppose that Σ_{n} ⊂ Σ_{n+1}. Therefore (a_{n})_{n∈ℕ} is a decreasing sequence of metrics. Let a(x) := inf_{n∈ℕ} a_{n}(x). Then, arguing as in Lemma 3.8, we get that

δ_{a}(x_{i}, y_{i}) = lim_{n→+∞} δ_{a_{n}}(x_{i}, y_{i}) = d(x_{i}, y_{i})

for every i ∈ ℕ. This means that δ_{a} = d on a dense subset of Ω × Ω and hence δ_{a} coincides with d by continuity, which is the claim.

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