SMOOTH APPROXIMATION OF WEAK FINSLER METRICS

ANDREA DAVINI

Dipartimento di Matematica Applicata, Università di Pisa via S. Maria 26, 56126 Pisa, Italy

Abstract. Smooth Finsler metrics are a natural generalization of Riemannian ones and have been widely studied in the framework of differential geometry. The definition can be weakened by allowing the metric to be only Borel measurable. This generalization is necessary in view of applications, such as, for instance, optimization problems. In this paper we show that smooth Finsler metrics are dense in Borel ones, generalizing the results obtained in [15]. The case of degenerate Finsler distances is also discussed.

1. Introduction

A Finsler metric on a differential manifold M is a map φ : T M → [0, +∞) such that φ(x, ·) is convex and positively 1-homogeneous on T_{x}M for every x ∈ M. A non-symmetric distance on M can be associated to φ as follows:

d_{φ} (x, y) := inf {∫_{0}^{1} φ(γ(t), γ̇(t)) dt | γ ∈ Lip ([0, 1], M) , γ(0) = x, γ(1) = y } (1)

for x, y ∈ M, provided the class of admissible paths γ is non-void (for instance, assume M connected). Distances of this kind are usually known in literature as Finsler distances. Finsler metrics are a natural generalization of Riemannian ones. The smooth case has been largely investigated in the last century in the framework of differential geometry. The literature on the subject is wide; an introduction is supplied, for instance, by [3].

The above definitions still make sense under weaker regularity assumptions on the metric. In fact, we may allow φ to be only Borel-measurable. This generality is necessary in view of applications. In optimization problems, for instance, one is often interested in minimizing a cost functional which takes the form C(d_{φ}), where φ is to be chosen in some suitable class of Finsler metrics. Then, if one restricts himself to consider smooth or continuous metrics only, the problem may not have a solution: indeed, when attacking the problem via the direct method of the Calculus of Variations, it might happen that the minimizing sequences converge to distances deriving from Finsler metrics that are shown to be Borel-measurable only (see [1, 5, 8]). Note that problems of this kind fall into the framework of shape optimization: in fact, φ describes the geometric properties of the metric space (M, d_{φ}), and may possibly depend on some quantities one is allowed to vary. For example, if a(·) is the density of a viscous material that fills a region M of the space, the distance between points is expected to be proportional to the resistance opposed by the medium and may be described by d_{φ} with φ(x, ξ) := a(x)|ξ|. If now we let a(·) vary in a suitable family of functions satisfying integral and pointwise constraints, the corresponding shape optimization problem is that of finding a ”best” way to distribute a certain amount of this material on M. A problem of this kind has been considered, for instance, in [7].

Date: October 5, 2004.

1991 Mathematics Subject Classification. 41A99, 49J45, 58B20.

Measurable Finsler metrics have been also considered in [16, 17, 18], where this point of view was adopted to suitably generalize the notions of Riemannian and Finsler metrics for Lipschitz manifolds, namely topological manifolds with a countable basis whose changes of coordinates are Lipschitz maps. Lipschitz manifolds are a generalization of polyhedra, and were introduced to treat the case of manifolds with singularities, such as vertices, edges, conical points, even not isolated.

The perspective can be also reversed: instead of dealing with singularities carried by the manifold, one could treat the case of a sufficiently smooth manifold equipped with an irregular metric. For example, the problem concerning geodesics with obstacles can be studied as a problem about metrics with singularities. This topic is also of physical interest, since the irregularities of the metric may actually represent irregularities of the materials where the physical phenomena take place: think, for instance, to geometrical optics in non-homogeneous media.

With this regard, we underline the relation with the theory of Hamilton-Jacobi equations. In the study of equations of eikonal type, a central role is in fact played by a Finsler distance associated to the equation, the so called optical length function. The metric character of equations of this kind has been recognized and explored by several authors [20, 22, 23, 25]. The study of Finsler metrics in the weak measurable setting acquires therefore further interest in view of generalizations of the theory of viscosity solutions for Hamilton-Jacobi equations with discontinuous ingredients, a topic which is the object of growing attention [4, 10, 11, 12, 26].

In this paper we consider the space D_{α} of non-symmetric distances defined through (1), where M is now replaced by the closure of a connected open subset Ω of ℝ^{N}, and φ varies in the family M_{α} of Borel-measurable Finsler metrics that satisfy the following bounds for two fixed positive constants α and β:

α|ξ| ≤ φ(x, ξ) ≤ β|ξ| on \overline{Ω} × ℝ^{N}.

The aim is to show that continuous (smooth) Finsler metrics are dense in Borel ones. More precisely, we will show that any element of D_{α} is the uniform limit of a suitable sequence of distances derived through (1) from continuous (smooth) metrics belonging to M_{α} (Theorems 4.1 and 4.5). The hard side of the job corresponds to Theorem 4.1: indeed, once the density result is proved for continuous metrics, the analogous result for smooth ones is obtained through a standard mollification argument.

These results can be read as the counterpart of those obtained in [15], where the case of symmetric distances was considered, and analogous density theorems for continuous and smooth Riemannian metrics were obtained. As a matter of fact, the proofs exploit similar ideas; in particular, the key observation still corresponds to Lemma 3.1 (cf. [15, Lemma 3.4]), which allows to replace the uniform convergence of distances with a pointwise convergence on a fixed, countable subset of \overline{Ω} × \overline{Ω}. On the other hand, new arguments have to be introduced to overcome the difficulties produced by the non-symmetric character of distances here considered.

We also wish to underline the content of Theorem 4.2: it amounts to saying that any geodesic distance d, locally equivalent to the Euclidean one, can be obtained from a convex Finsler metric through (1), a fact which is not trivial at all: notice indeed that the Finsler metric φ_{d}(x, ξ), obtained from d by derivation (cf. formula (7) below), is proved to be convex in ξ for almost every x only.

The density results obtained for α > 0 are then extended to the case α = 0. The main difference between the two cases relies on the fact that, while D_{α} is closed as a metric space when α > 0, this is no longer true when α = 0. This fact is investigated in more detail in Section 5 through suitable, explicit examples. We remark that the family D_{0} includes distances for which the local equivalence with the Euclidean one fails to hold somewhere. The interest for this class of degenerate distances is motivated by the study of Hamilton-Jacobi equations of eikonal type in the critical case (see [12, 20]).

Finally, we remark that our results also holds if ℝ^{N} is replaced by a N-dimensional, differential manifold without boundary and of class C^{1}. Since all arguments exploit local properties, proofs can actually be rephrased by using local coordinates. We have preferred, however, to consider this more special case not to add further technicalities.

Our paper is organized as follows: Section 2 is devoted to notation and definitions. In particular, some results holding in classical metric spaces are generalized to cover the case of non-symmetric distances. In Section 3 we state some preliminary propositions, in Section 4 we prove our main results for the class of non-symmetric distances D_{α} when α > 0, and in Section 5 we examine the case α = 0.

2. Notation and definitions

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

Ω  open connected subset of ℝ^{N}
𝕊^{N−1}  (N − 1) – dimensional unitary sphere of ℝ^{N}
B_{r}(x), B_{r} open ball in ℝ^{N} of radius r centred in x and 0 respectively
I  closed interval [0, 1]
L^{k}  k-dimensional Lebesgue measure
H^{k}  k-dimensional Hausdorff measure
|x|  Euclidean norm of the vector x ∈ ℝ^{N}
ℝ_{+}  non-negative real numbers
𝒫(ℝ^{N})  family of all subsets of ℝ^{N}.

We will denote by N an integer number. A subset of ℝ^{N} is said to be negligible if its N-dimensional Lebesgue measure is null. When not otherwise specified, the word curve or path will always denote a Lipschitz function from the interval I := [0, 1] to \overline{Ω}. The family of all such curves will be denoted by Lip(I, \overline{Ω}), and is equipped with the metric of uniform convergence; in particular, we will say that the sequence (γ_{n})_{n} converges to γ to mean that sup_{t∈I} |γ_{n}(t) − γ(t)| tends to zero as n goes to infinity. Unless otherwise specified, 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 denote by Lip_{x,y} the family of curves γ which join x to y, i.e. such that γ(0) = x and γ(1) = y. The domain Ω will be always assumed to satisfy the following condition:

∀ r > 0 ∃ C_{r} ≥ 1 s.t. d_{Ω}(x, y) ≤ C_{r}|x − y|  ∀ x, y ∈ \overline{Ω} ∩ B_{r},  (Ω)

where d_{Ω} is the Euclidean geodesic distance in \overline{Ω}, i.e. the distance defined through formula (4) below with φ(x, ξ) = |ξ| identically in \overline{Ω} × ℝ^{N}. Note that (Ω) depends on the regularity of ∂Ω; when the latter can be (locally) expressed as a graph of a function h, (Ω) is equivalent to require that h is Lipschitz continuous.

For a measurable function f : I → ℝ^{N}, ∥f∥_{∞} stands for \sqrt{∑_{i=0}^{N} ∥f_{i}∥_{L^{∞}(I)}^{2}}, where f_{i} and ∥f_{i}∥_{L^{∞}(I)} denote the i-th component of f and the L^{∞}-norm of f_{i} respectively.

In the sequel, a function d defined on \overline{Ω} × \overline{Ω} will be said to be a distance on \overline{Ω} if the following conditions hold:

(i) d(x, x) = 0 for every x ∈ \overline{Ω}; (ii) d(x, y) ≤ d(x, z) + d(z, y) for every x, y, z ∈ \overline{Ω}.

With respect to the classical definition of distance, two conditions are not required: first, d may be non-symmetric, namely the identity d(x, y) = d(y, x) may fail to hold in \overline{Ω} × \overline{Ω}; second, d can possibly be degenerate, namely it might happen that d(x, y) = 0 for some x ≠ y. Degenerate distances, however, are only considered at the end of the paper, in Section 5.

We begin by extending to this setting some well known definitions and classical results for usual distances. We will omit the proofs of the next theorems, which can be derived by minor changes from those proposed, for instance, in [2] in the classical case.

First, let us define the d-length of γ, obtained as the supremum of the d-lengths of inscribed polygonal curves:

L_{d}(γ) := sup { ∑_{i=0}^{m−1} d(γ(t_{i}), γ(t_{i+1})) | 0 = t_{0} < t_{1} < .. < t_{m} = 1, m ∈ ℕ} . (2)

Definition 2.1 (Geodesic distance). We will say that d is a geodesic distance if it satisfies the following identity:

d(x, y) = inf \{L_{d}(γ) | γ ∈ Lip_{x,y}\} for every (x, y) ∈ \overline{Ω} × \overline{Ω}.

All distances considered in the sequel will fulfill the following hypotheses: (d1) d is geodesic; (d2) there exist two non-negative constants α and β such that

α|x − y| ≤ d(x, y) ≤ β|x − y| locally in Ω

(i.e. for every x_{0} ∈ Ω there exists an open ball B_{r}(x_{0}) ⊂ Ω such that the above inequality holds for every x, y ∈ B_{r}(x_{0})).

Any distance d which satisfies the above hypotheses induces on \overline{Ω} a topology which is equivalent to the Euclidean one when α > 0, and weaker, in general, if α = 0. Notice that α > 0 implies that d is non-degenerate. By applying to our framework a classical theorem due to Busemann (cf. [2, Theorem 4.3.1]), we obtain what follows.

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

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

Moreover, when α > 0, for every couple of points x, y in \overline{Ω} there exists a curve γ ∈ Lip_{x,y} which is a path of minimal d-length, i.e. such that L_{d}(γ) = d(x, y).

As in the symmetric case, we can define the metric derivative of a curve.

Definition 2.3 (Metric derivative). Given a curve γ, we define the metric derivative |γ̇|_{d}(t) of γ at the point t ∈ (0, 1) as

|γ̇|_{d}(t) := lim sup_{h→0^{+}} \frac{d(γ(t), γ(t + h))}{h}.  (3)

The d-length of a curve γ admits an integral representation in terms of its metric derivative.

Theorem 2.4. For every curve γ the lim sup at the right-hand side of (3) is actually a limit for L^{1}-a.e. t ∈ I. Moreover we have

L_{d}(γ) = ∫_{0}^{1} |γ̇|_{d}(t) dt.

We now make precise what we mean by weak Finsler metric.

Definition 2.5. A Borel-measurable function φ : \overline{Ω} × ℝ^{N} → ℝ_{+} is said to be a (weak) Finsler metric on \overline{Ω} if

(i) φ(x, ·) is positively 1-homogeneous for every x ∈ \overline{Ω}; (ii) φ(x, ·) is convex for L^{N}-a.e. x ∈ \overline{Ω}.

We will say that the metric φ is convex if (ii) holds for every x ∈ \overline{Ω}. We will say that φ is a continuous (resp. smooth) Finsler metric if φ(·, ξ) is continuous (resp. smooth) on \overline{Ω} for every ξ ∈ ℝ^{N}.

We now fix two non-negative constants α and β and we consider the following family of functions:

M_{α} := \{φ Finsler metrics on \overline{Ω} | α|ξ| ≤ φ(x, ξ) ≤ β|ξ| on \overline{Ω} × ℝ^{N}\} .

For each φ ∈ M_{α}, we can define a function d_{φ} on \overline{Ω} × \overline{Ω} through the formula

d_{φ} (x, y) := inf {𝕃_{φ} (γ) | γ ∈ Lip_{x,y} } ,  (4)

where the length functional 𝕃_{φ} is defined by

𝕃_{φ}(γ) := ∫_{0}^{1} φ(γ(t), γ̇(t)) dt.  (5)

The main properties of d_{φ} are summarized below.

Proposition 2.6. The function d_{φ} given by (4) is well defined on \overline{Ω}×\overline{Ω} and satisfies the following properties:

(i) 0 ≤ d_{φ}(x, y) ≤ d_{φ}(x, z) + d_{φ}(z, y) for all x, y, z ∈ \overline{Ω}; (ii) α|x − y| ≤ d_{φ}(x, y) ≤ β|x − y| locally in Ω.

Proof. Let γ be a curve. Since the map t ↦ (γ(t), γ̇(t)) is Lebesgue measurable on I, and φ is Borel measurable on \overline{Ω} × ℝ^{N}, their composition φ(γ(t), γ̇(t)) is Lebesgue measurable on I. Therefore the integral in (5) is well defined and so is d_{φ}. The remainder of the assertion is a simple consequence of the definitions. □

Next proposition clarifies the relation between the functional (5) and the (intrinsic) metric length functional. The proof can be derived by minor changes from that of Theorem 4.3 in [16].

Proposition 2.7. Let d := d_{φ} with φ ∈ M_{α}. Then, for any γ ∈ Lip(I, \overline{Ω}), we have that L_{d}(γ) ≤ 𝕃_{φ}(γ). In particular, d is a geodesic distance according to Definition 2.1. If α > 0, we have moreover:

L_{d}(γ) = inf \{lim inf_{n→+∞} 𝕃_{φ}(γ_{n}) | (γ_{n})_{n} converges to γ in Lip(I, \overline{Ω}) \} ,  (6)

namely L_{d} is the relaxed functional of 𝕃_{φ} on Lip(I, \overline{Ω}).

Remark 2.8. By Proposition 2.7, 𝕃_{φ} will coincide with L_{d} whenever 𝕃_{φ} is lower semicontinuous on Lip(I, \overline{Ω}). This happens, for instance, when φ is lower semicontinuous on \overline{Ω} × ℝ^{N} and φ(x, ·) is convex on ℝ^{N} for every x ∈ \overline{Ω} (cf. [6, Theorem 4.1.1]).

Let us denote by D_{α} the family of distances on \overline{Ω} generated by the metrics M_{α}, namely

D_{α} := \{d_{φ} distance on \overline{Ω} given by (4) | φ ∈ M_{α}\}.

We endow D_{α} with the metric given by the uniform convergence on compact subsets of \overline{Ω} × \overline{Ω}. The convergence of a sequence of distances (d_{n})_{n} to d with respect to this metric will be denoted by d_{n} ⟶^{D_{α}} d. When α > 0, this convergence is equivalent to the Γ-convergence of the relative length functionals L_{d_{n}} to L_{d} with respect to the uniform convergence of paths. This result has been proved in [8, Theorem 3.1] considering usual symmetric distances, but actually it still holds in the nonsymmetric case too and the proof is the same. Moreover, we have (cf. [8, Theorem 3.1]):

Theorem 2.9. Assume α > 0. Then the set D_{α}, endowed with the metric given by the uniform convergence of distances on \overline{Ω} × \overline{Ω}, is a metrizable compact space.

Let us stress that the interesting part of the result provided by Theorem 2.9 corresponds to the closed character of the space D_{α}, since the compactness trivially follows from Ascoli-Arzelà Theorem. The requirement that α is strictly positive is necessary, as we will see in Section 5.

In general, as condition (Ω) is satisfied, D_{α} can be seen as a subspace of Finsler distances on ℝ^{N}: indeed, given d ∈ D_{α}, a suitable Finsler distance \overline{d} on ℝ^{N} which extends d can be constructed as shown in [15, Remark 2.7]. Therefore, by possibly replacing d with \overline{d}, the following definition always makes sense:

φ_{d} (x, ξ) := lim sup_{h→0^{+}} (\frac{d (x, x + hξ)}{h} ∧ β|ξ|)  (x, ξ) ∈ \overline{Ω} × ℝ^{N}.  (7)

The function φ_{d} is a Finsler metric, as proved in [18]; moreover, for each n ∈ ℕ, there holds

|φ_{d}(x, ξ) − φ_{d}(x, η)| ≤ βC_{n}|ξ − η| for every x ∈ B_{n} ∩ \overline{Ω}, ξ, η ∈ ℝ^{N},

where C_{n} are the constants appearing in (Ω). We will refer to φ_{d} as the metric associated to d by derivation. In view of Proposition 2.6 and [15, Remark 2.10], for every curve γ the following holds:

|γ̇|_{d}(t) = lim sup_{h→0^{+}} \frac{d (γ(t), γ(t) + hγ̇(t))}{h} ≤ β|γ̇(t)| for L^{1}-a.e. t ∈ I,

therefore by Theorem 2.4

L_{d}(γ) = ∫_{0}^{1} φ_{d}(γ(t), γ̇(t)) dt for all γ ∈ Lip(I, \overline{Ω}),  (8)

i.e. L_{d} = 𝕃_{φ_{d}} on Lip(I, \overline{Ω}) (cf. [18, Theorem 2.5]). In particular, d = d_{φ_{d}} according to (4).

We recall the following definitions for set-valued maps (cf. [13]).

Definition 2.10. Let X be a subset of ℝ^{N} and C : X → 𝒫(ℝ^{N}) a set-valued map such that C(x) is compact for every x ∈ X. We will say that

(i) C(·) is Borel measurable on X if the set {x ∈ X : C(x) ∩ A ≠ ∅ } is Borelmeasurable in ℝ^{N} for every open subset A of ℝ^{N}.

(ii) C(·) is upper semicontinuous at x ∈ X if for all ε > 0, there exists δ > 0 such that

C(z) ⊂ C(x) + B_{ε} for all z ∈ B_{δ}(x) ∩ X.

(iii) C(·) is lower semicontinuous at x ∈ X if for all ε > 0, there exists δ > 0 such that

C(x) ⊂ C(z) + B_{ε} for all z ∈ B_{δ}(x) ∩ X.

(iv) C(·) is continuous at x ∈ X if it is both upper and lower semicontinuous at x.

The support function of a subset C of ℝ^{N} is the function σ_{C} defined by

σ_{C}(ξ) := sup{⟨ξ, p⟩ | p ∈ C } for every ξ ∈ ℝ^{N},

which is convex and positively 1-homogeneous on ℝ^{N}.

Given a Borel measurable set-valued map C : \overline{Ω} → 𝒫(ℝ^{N}) such that C(x) is compact and contains 0 for every x ∈ \overline{Ω}, a Finsler metric φ can be defined on \overline{Ω} × ℝ^{N} as follows:

φ(x, ξ) := σ_{C(x)}(ξ) for all (x, ξ) ∈ \overline{Ω} × ℝ^{N}.

Moreover, if C(·) is upper semicontinuous (resp. lower semicontinuous) on \overline{Ω}, then φ(·, ξ) is upper semicontinuous (resp. lower semicontinuous) on \overline{Ω} as well, for every ξ ∈ ℝ^{N}.

3. Some preliminary results

We collect in this section some results we will need in the proofs of our main theorems. To begin, we state a simple but important consequence of Ascoli-Arzelà Theorem.

Lemma 3.1. Let (d_{n})_{n} be a sequence contained in D_{α} which converges pointwise to some d ∈ D_{α} on a dense subset of \overline{Ω} × \overline{Ω}. Then d_{n} ⟶^{D_{α}} d.

A first application of Lemma 3.1 is provided in the proof of the following proposition.

Proposition 3.2. Assume α > 0. Let φ, φ_{n} ∈ M_{α} and d and d_{n} be the distances associated respectively to φ and φ_{n} through (4). Then d_{n} ⟶^{D_{α}} d in the following cases:

(i) (φ_{n})_{n} converges uniformly to φ on compact subsets of \overline{Ω} × ℝ^{N}; (ii) φ_{n} are lower semicontinuous in x, convex in ξ and converge increasingly to

φ pointwise on \overline{Ω} × ℝ^{N}; (iii) (φ_{n})_{n} converges decreasingly to φ pointwise on \overline{Ω} × ℝ^{N}.

Proof. By Lemma 3.1 it is sufficient to prove that (d_{n})_{n} converges pointwise to d. For any fixed (x, y) ∈ \overline{Ω} × \overline{Ω} set X := Lip_{x,y}. We want to prove that lim_{n} d_{n}(x, y) = d(x, y).

(i) For γ ∈ X and the uniform convergence of φ_{n} we have

𝕃_{φ}(γ) = lim_{n→+∞} 𝕃_{φ_{n}}(γ) ≥ lim sup_{n→+∞} d_{n}(x, y),

which entails d(x, y) ≥ lim sup_{n} d_{n}(x, y) by taking the infimum over all possible curves γ ∈ X. Let us now choose a sequence of curves (γ_{n})_{n} ⊂ X such that 𝕃_{φ_{n}}(γ_{n}) ≤ d_{n}(x, y) + 1/n. Since the curves γ_{n} are equi-Lipschitz continuous (as 𝕃_{φ_{n}}(γ_{n}) ≥ α ∫_{0}^{1} |γ̇_{n}(t)| dt and all curves are parametrized by constant speed), the uniform convergence of the metrics implies that lim sup_{n} |𝕃_{φ_{n}}(γ_{n}) − 𝕃_{φ}(γ_{n})| = 0, therefore

d(x, y) ≤ lim inf_{n→+∞} 𝕃_{φ}(γ_{n}) = lim inf_{n→+∞} 𝕃_{φ_{n}}(γ_{n}) = lim inf_{n→+∞} d_{n}(x, y),

hence the claim.

(ii) (𝕃_{φ_{n}})_{n} is an increasing sequence of lower semicontinuous functionals on X (cf. Remark 2.8), which converges pointwise to 𝕃_{φ} (i.e. lim_{n} 𝕃_{φ_{n}}(γ) = 𝕃_{φ}(γ) for each γ ∈ X). By [14, Remark 5.5] the functionals 𝕃_{φ_{n}} Γ-converge to 𝕃_{φ}. In particular, since the functionals 𝕃_{φ_{n}} are equi-coercive (as 𝕃_{φ_{n}}(γ) ≥ α ∫_{0}^{1} |γ̇|dt for every n), we have, by the crucial result of Γ-convergence [14, Theorem 7.4], that the sequence (inf_{X} 𝕃_{φ_{n}})_{n} converges to inf_{X} 𝕃_{φ}, that is

lim_{n→+∞} d_{n}(x, y) = lim_{n→+∞} inf_{γ∈X} 𝕃_{φ_{n}}(γ) = inf_{γ∈X} 𝕃_{φ}(γ) = d(x, y).

(iii) By monotonicity we get d(x, y) ≤ inf_{n} d_{n}(x, y). To show the reverse inequality, take a curve γ ∈ X. By the monotone convergence theorem and by the definition of d_{n}(x, y) we have

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

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

Remark 3.3. Notice that Proposition 3.2 (iii) holds for α = 0 too.

We end this brief section with two results which will be needed in the sequel. The first one is just a restatement of Theorem 1 in [19, Section 6.6].

Theorem 3.4. Suppose γ : I → ℝ^{N} is a Lipschitz continuous curve. Then for each ε > 0, there exists a C^{1} curve \overline{γ} : I → ℝ^{N} such that:

L^{1} (\{t ∈ I | \overline{γ}(t) ≠ γ(t) or ˙ \overline{γ}(t) ≠ γ̇(t) \}) ≤ ε.

In addition

∥ ˙\overline{γ}∥_{∞} ≤ c \left∥{ \atop γ̇}\right∥_{∞}

for some constant c depending only on N.

The second result is an easy application of Severini-Egoroff’s theorem. A proof is provided in [15].

Lemma 3.5. Let γ be an injective Lipschitz continuous curve, Γ := γ((0, 1)) ⊂ \overline{Ω} and a : \overline{Ω} → [α, β] a Borel function. Then there exists a sequence of continuous functions a_{k} : Γ → [α, β] such that a_{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 a_{k} converge uniformly to a on B_{ε}.

4. The density results for non-degenerate distances

Throughout this section, α is always assumed to be strictly positive. Our first density result is then stated as follows:

Theorem 4.1. Let d ∈ D_{α}. Then there exists a sequence (φ_{n})_{n} of continuous and convex Finsler metrics in M_{α} such that d_{φ_{n}} ⟶^{D_{α}} d, where d_{φ_{n}} is the distance associated to φ_{n} through (4).

When d is symmetric, Theorem 4.1 has been already proved in [15]. In this case, the approximating metrics φ_{n}(x, ξ) can be chosen of the form a_{n}(x)|ξ|. This accounts to saying that, for any x ∈ \overline{Ω}, each metric φ_{n}(x, ·) is the support function of the closed convex set C^{n}(x) := {p ∈ ℝ^{N} : |p| ≤ a_{n}(x) }. This remark suggests how to generalize the proof to non-symmetric distances: for each n ∈ ℕ, we set

φ_{n}(x, ξ) := σ_{C^{n}(x)}(ξ) for every (x, ξ) ∈ \overline{Ω} × ℝ^{N},

where σ_{K} denotes the support function of the set K ⊂ ℝ^{N}, and C^{n} : \overline{Ω} → 𝒫(ℝ^{N}) is a set-valued map which has to be suitably defined. Of course, since we want to approximate a non-symmetric distance d, the sets C^{n}(x) will be chosen nonsymmetric with respect to 0, in general. The proof of Theorem 4.1 then proceeds by following the same idea used in [15]: we define the approximating sequence of setvalued maps C^{n}(·) in such a way to control the convergence of the induced distances d_{φ_{n}} to d pointwise on a dense subset of \overline{Ω} × \overline{Ω} (in view of Lemma 3.1). To this aim, we set S := ℚ^{N} ∩ \overline{Ω}. Obviously S × S is countable and dense in \overline{Ω} × \overline{Ω}, so we write

S × S := {(x_{i}, y_{i}) | i ∈ ℕ}.

This notation is adopted in the remainder of the paper with no further mention.

Instead of providing a direct proof to Theorem 4.1, we prefer to break it into intermediate propositions, which will be proved separately. The first one can be read as a ”dual” formulation of [15, Theorem 4.7].

Theorem 4.2. Let d ∈ D_{α}. Then there exists a decreasing sequence of convex metrics ψ_{n} ∈ M_{α} such that

d_{ψ_{n}}(x_{i}, y_{i}) = d(x_{i}, y_{i}) for each i ≤ n.  (9)

Moreover, if we set ψ(x, ξ) := inf_{n} ψ_{n}(x, ξ) for every (x, ξ) ∈ \overline{Ω} × ℝ^{N}, then d = d_{ψ}, that is every Finsler distance is induced by a convex Finsler metric.

Proof. For each (x_{i}, y_{i}) ∈ S × S, let γ_{i} ∈ Lip_{x_{i},y_{i}} be a path of minimal d-length, i.e. L_{d}(γ_{i}) = d(x_{i}, y_{i}), and set Γ_{i} := γ_{i} ((0, 1)). For every x ∈ Γ_{i}, let {ξ_{1}^{i}(x), ξ_{2}^{i}(x), …, ξ_{N}^{i}(x)} be an orthonormal basis of ℝ^{N} such that ξ_{1}^{i}(γ_{i}(t)) = γ̇_{i}(t)/∥γ̇_{i}∥_{∞} for L^{1}-a.e t ∈ I (the curves γ_{i} are assumed to be parametrized by constant speed), and set a^{i}(x) := φ_{d}(x, ξ_{1}^{i}(x)). Such vectors can be chosen in such a way that the map x → ξ_{j}^{i}(x) is Borel-measurable on Γ_{i} for each 1 ≤ j ≤ N. For every x ∈ Γ_{i}, we denote by Q^{i}(x) the rectangle in ℝ^{N} defined by

Q^{i}(x) := {μ_{1}ξ_{1}^{i}(x) + ⋯ + μ_{N}ξ_{N}^{i}(x) | μ_{1} ∈ [−β, a^{i}(x)], μ_{2}, …, μ_{N} ∈ [−β, β] } .

We define the set-valued map K^{i}(·) on \overline{Ω} as follows:

K^{i}(x) := \{ \begin{matrix}Q^{i}(x) ∩ B_{β} & if x ∈ Γ_{i} \\ B_{α} & if x ∈ \{x_{i}, y_{i}\} \\ B_{β} & otherwise, \end{matrix}

and set C^{n}(x) := ∩_{i=1}^{n}K^{i}(x) for every x ∈ \overline{Ω}. Clearly, C^{n}(x) is a closed, convex set satisfying B_{α} ⊂ C_{n}(x) ⊂ B_{β} for every x ∈ \overline{Ω}. Moreover the set-valued map C^{n}(·) is Borel measurable on \overline{Ω}, as can be checked. The required metrics ψ_{n} are then defined as follows:

ψ_{n}(x, ξ) := σ_{C^{n}(x)}(ξ) for every (x, ξ) ∈ \overline{Ω} × ℝ^{N},

for each n ∈ ℕ. Such functions ψ_{n} are convex Finsler metrics belonging to M_{α}, as easily seen. In order to conclude, we need to show that (9) holds. This will be proved by showing that the curves γ_{i} are still optimal for d_{ψ_{n}}(x_{i}, y_{i}) for each 1 ≤ i ≤ n.

To this end, first recall that the tangent spaces of two 1-rectifiable sets M_{1}, M_{2} coincide at H^{1}-a.e. point belonging to their intersection (cf. [21]). In particular, we get

ξ_{i}^{1}(x) = ±ξ_{h}^{1}(x) for H^{1}-a.e. x ∈ Γ_{i} ∩ Γ_{h},

for every 1 ≤ i < h ≤ n. This in turn implies that, for any fixed 1 ≤ i ≤ n, the vector a^{i}(x)ξ_{1}^{i}(x) belongs to ∂C^{n}(x) for H^{1}-a.e. x ∈ Γ_{i}, and so that ψ_{n}(γ_{i}(t), γ̇_{i}(t)) = φ_{d}(γ_{i}(t), γ̇_{i}(t)) for L^{1}-a.e. t ∈ I. Consequently we get

d_{ψ_{n}}(x_{i}, y_{i}) ≤ ∫_{0}^{1} ψ_{n}(γ_{i}, γ̇_{i}) dt = ∫_{0}^{1} φ_{d}(γ_{i}, γ̇_{i}) dt = d(x_{i}, y_{i}).

To prove the reverse inequality, choose a curve γ ∈ Lip_{x_{i},y_{i}} and, for every 1 ≤ j < n, set I_{j+1} := {t ∈ I \ ∪_{h≤j} I_{h} | γ(t) ∈ γ_{j+1}(I)}, I_{1} := {t ∈ I | γ(t) ∈ γ_{1}(I)} and I_{0} := I \ ∪_{j≤n}I_{j}. By arguing as above, we deduce that, for each 1 ≤ j ≤ n, the vector γ̇(t) is parallel to ξ_{1}^{j}(γ(t)) for L^{1}-a.e. t ∈ I_{j}, hence ψ_{n}(γ, γ̇) ≥ φ_{d}(γ, γ̇) L^{1}-a.e. on I_{j} (depending on whether the two vectors are equally oriented or not). Note that this trivially holds for j = 0 too. Therefore we have

𝕃_{ψ_{n}}(γ) = ∫_{0}^{1} ψ_{n}(γ, γ̇) dt = ∑_{j=0}^{n} ∫_{I_{j}} ψ_{n}(γ, γ̇) dt ≥ ∑_{j=0}^{n} ∫_{I_{j}} φ_{d}(γ, γ̇) dt ≥ d(x_{i}, y_{i}).

By passing to the infimum over all possible curves γ ∈ Lip_{x_{i},y_{i}} we get the claim. To conclude the proof of the statement, set ψ(x, ξ) := inf_{n} ψ_{n}(x, ξ) for every (x, ξ) ∈ \overline{Ω} × ℝ^{N}. By applying Proposition 3.2 (iii), we obtain

d_{ψ}(x_{i}, y_{i}) = lim_{n→+∞} d_{ψ_{n}}(x_{i}, y_{i}) = d(x_{i}, y_{i}) for every i ∈ ℕ,

namely d_{ψ} coincide with d on S × S, hence everywhere in \overline{Ω} × \overline{Ω} by density. □

Next, we prove the following

Proposition 4.3. Let d ∈ D_{α}. Then, for every n ∈ ℕ and every ε > 0, there exists a lower semicontinuous, convex metric ϕ_{ε} ∈ M_{α} such that

|d_{ϕ_{ε}}(x_{i}, y_{i}) − d(x_{i}, y_{i})| < ε for all i ≤ n.

Proof. First, note that, if the set-valued map K^{i}(·) defined in the proof of Theorem 4.2 were continuous on Γ_{i}, the set-valued map C^{n}(·) would be lower semicontinuous on \overline{Ω} and a possible choice for ϕ_{ε} would simply be ψ_{n}. Hence, the idea is that of modifying the definition of K^{i}(·) in order to get a continuous map on Γ_{i} for each i. We adopt the notation used in the proof of Theorem 4.2. Let n ∈ ℕ be fixed, let γ_{i}, Γ_{i} and a^{i}(x) be defined as above for each i ∈ ℕ, and set N_{n} = ∪_{i=1}^{n}Γ_{i}. Let us choose λ > 0 and let 1 ≤ i ≤ n. Note that γ_{i} is injective by minimality, hence we can apply Lemma 3.5 to obtain a sequence of continuous functions a_{k}^{i} : Γ_{i} → [α, β], k ∈ ℕ, converging pointwise to a^{i} H^{1}-a.e. on Γ_{i} as k goes to infinity. Let \overline{γ}_{i} be the C^{1}-continuous curve obtained by applying Theorem 3.4 with γ := γ_{i} and ε := λ. For every x ∈ Γ_{i}, let \{\overline{ξ}_{1}^{i}(x), \overline{ξ}_{2}^{i}(x), …, \overline{ξ}_{N}^{i}(x)\} be an orthogonal basis of ℝ^{N} such that:

(i) \overline{ξ}_{1}^{i}(γ_{i}(t)) = ˙ \overline{γ}_{i}(t)/∥γ̇_{i}∥_{∞} for L^{1}-a.e t ∈ I; (ii) |\overline{ξ}_{j}^{i}(x)| = 1 for every 2 ≤ j ≤ N.

Such vectors can be chosen in such a way that the map x → \overline{ξ}_{j}^{i}(x) is continuous on Γ_{i} for each 1 ≤ j ≤ N. For each k ∈ ℕ and every x ∈ Γ_{i}, we denote by \overline{Q}_{k}^{i}(x) the rectangle in ℝ^{N} defined by

\overline{Q}_{k}^{i}(x) := \{μ_{1}\overline{ξ}_{1}^{i}(x) + ⋯ + μ_{N} \overline{ξ}_{N}^{i}(x) | μ_{1} ∈ [−β, a_{k}^{i}(x)], μ_{2}, …, μ_{N} ∈ [−β, β] \} .

We define the set-valued map \overline{K}_{k}^{i}(·) on \overline{Ω} as follows:

\overline{K}_{k}^{i}(x) := \{ \begin{matrix}\overline{Q}_{k}^{i}(x) ∩ B_{β} & if x ∈ Γ_{i} \\ B_{α} & if x ∈ \{x_{i}, y_{i}\} \\ B_{β} & otherwise. \end{matrix}

It is easy to check by definition that \overline{K}_{k}^{i}(·) is lower semicontinuous for each 1 ≤ i ≤ n, so the same is true for the set-valued map \overline{C}_{k}^{n}(·) defined by setting \overline{C}_{k}^{n}(x) := ∩_{i=1}^{n}\overline{K}_{k}^{i}(x) for every x ∈ \overline{Ω}. For each k ∈ ℕ, a convex, lower semicontinuous Finsler metric ϕ_{k} belonging to M_{α} may be then defined as follows:

ϕ_{k}(x, ξ) := σ_{\overline{C}_{k}^{n}(x)}(ξ) ∨ α|ξ| for every (x, ξ) ∈ \overline{Ω} × ℝ^{N}.

We claim that

lim sup_{k→+∞} |d_{ϕ_{k}}(x_{i}, y_{i}) − d(x_{i}, y_{i})| ≤ (nβL) λ for all 1 ≤ i ≤ n,  (10)

where L := sup_{1≤i≤n} ∥γ̇_{i}∥_{∞} This will be enough to conclude: indeed, it is sufficient to take λ < ε/(nβL) and ϕ_{ε} := ϕ_{k} with k suitably large.

Let us then prove (10). For 1 ≤ i ≤ n we have by definition

d_{ϕ_{k}}(x_{i}, y_{i}) ≤ 𝕃_{ϕ_{k}}(γ_{i}) = ∫_{0}^{1} ϕ_{k}(γ_{i}, γ̇_{i}) dt.

Let J_{i} := \{ t ∈ I | ˙ \overline{γ}_{i}(t) = γ̇_{i}(t) \} and recall that, by Theorem 3.4, L^{1}(I \ J_{i}) ≤ λ. An argument analogous to the one used in the proof of Theorem 4.2 shows that ϕ_{k}(γ_{i}, γ̇_{i}) = a_{k}^{i}(γ_{i})|γ̇_{i}| L^{1}-a.e. on J_{i}. By the dominated convergence theorem we then have

lim sup_{k→+∞} d_{ϕ_{k}}(x_{i}, y_{i}) ≤ lim sup_{k→+∞} (∫_{J_{i}} a_{k}^{i}(γ_{i})|γ̇_{i}| dt + ∫_{I\J_{i}} β |γ̇_{i}(t)| dt)

≤ ∫_{0}^{1} a^{i}(γ_{i})|γ̇_{i}| dt + βLλ = d(x_{i}, y_{i}) + βLλ.  (11)

Now, set Γ_{0} := ∪_{i=1}^{n}γ_{i}(I \ J_{i}), and remark that H^{1}(Γ_{0}) ≤ nLλ. Fix an index 1 ≤ i ≤ n. For each k ∈ ℕ, pick up a curve γ̃_{k} ∈ Lip_{x_{i},y_{i}} such that

𝕃_{ϕ_{k}}(γ̃_{k}) = d_{ϕ_{k}}(x_{i}, y_{i}).  (12)

Note that such curves are injective by minimality. Since α ∫_{I} | ˙γ̃_{k}| dt ≤ 𝕃_{ϕ_{k}}(γ̃_{k}), by (12) and (11) we get that lim sup_{k} ∫_{I} | ˙γ̃_{k}| dt < +∞. Let us choose an ε̃ > 0. By applying Lemma 3.5 to each open arc Γ_{i}, we can find a Borel set Bε̃ ⊂ N_{n} and an infinitesimal sequence of positive numbers (δ_{k})_{k∈ℕ} such that H^{1}(N_{n} \ Bε̃) < ε̃ and |a_{k}^{i}(x) − a^{i}(x)| < δ_{k} for every x ∈ Bε̃, 1 ≤ i ≤ n and k ∈ ℕ. Let us set I_{k} := {t ∈ I | γ̃_{k}(t) ∈ Γ_{0} ∪ (N_{n} \ Bε̃)}. Then we have

ϕ_{k}(γ̃_{k}, ˙ γ̃_{k}) ≥ ψ_{n}(γ̃_{k}, ˙ γ̃_{k}) − δ_{k}| ˙γ̃_{k}|  L^{1}-a.e. on I \ I_{k}.  (13)

Let us write

𝕃_{ϕ_{k}}(γ̃_{k}) = ∫_{I_{k}} ϕ_{k}(γ̃_{k}, ˙ γ̃_{k}) dt + ∫_{I\I_{k}} ϕ_{k}(γ̃_{k}, ˙ γ̃_{k}) dt.

As γ̃_{k}(I_{k}) ⊂ Γ_{0} ∪ (N_{n} \ Bε̃) for every k ∈ ℕ, by the Area-formula we have

∫_{I_{k}} | ˙γ̃_{k}| dt = H^{1}(γ̃_{k}(I_{k})) ≤ H^{1}(N_{n} \ Bε̃) + H^{1}(Γ_{0}) < ε̃+ nLλ.

Taking this remark into account we get

∫_{I_{k}} ϕ_{k}(γ̃_{k}, ˙ γ̃_{k})dt = ∫_{I_{k}} ψ_{n}(γ̃_{k}, ˙ γ̃_{k}) dt + ∫_{I_{k}} (ϕ_{k}(γ̃_{k}, ˙ γ̃_{k}) − ψ_{n}(γ̃_{k}, ˙ γ̃_{k})) dt
≥ ∫_{I_{k}} ψ_{n}(γ̃_{k}, ˙ γ̃_{k}) dt − β (ε̃+ nLλ) .  (14)

Inequalities (13) and (14) then yields

𝕃_{ϕ_{k}}(γ̃_{k}) ≥ ∫_{0}^{1} ψ_{n}(γ̃_{k}, ˙ γ̃_{k}) dt − δ_{k} ∫_{I\I_{k}} | ˙γ̃_{k}| dt − β (ε̃+ nLλ)

≥ d_{ψ_{n}}(x_{i}, y_{i}) − δ_{k} ∫_{0}^{1} | ˙γ̃_{k}| dt − β (ε̃+ nLλ)

and therefore, as δ_{k} ∫_{0}^{1} | ˙γ̃_{k}| dt goes to zero when k → +∞, we obtain

lim inf_{k→+∞} d_{ϕ_{k}}(x_{i}, y_{i}) = lim inf_{k→+∞} 𝕃_{ϕ_{k}}(γ̃_{k}) ≥ d_{ψ_{n}}(x_{i}, y_{i}) − β (ε̃+ nLλ) .

Since ε̃ was arbitrary and d_{ψ_{n}}(x_{i}, y_{i}) = d(x_{i}, y_{i}), the above inequality coupled with (11) gives (10), as claimed. □

We are now ready to give the

Proof of Theorem 4.1. First, we claim that, for every ε > 0, there exists a continuous, convex metric φ_{ε} ∈ M_{α} such that

|d_{φ_{ε}}(x_{i}, y_{i}) − d(x_{i}, y_{i})| < ε for all i ≤ n.  (15)

Indeed, Proposition 4.3 provides a convex, lower semicontinuous metric ϕ_{ε} satisfying (15). By setting ϕ_{ε}(x, ξ) := β|ξ| when x ∈ ℝ^{N} \ \overline{Ω}, ϕ_{ε} can be extended to ℝ^{N} × ℝ^{N}. Then Lemma 2.2.3 of [6] easily implies the existence of an increasing sequence (Φ_{n})_{n} of continuous, convex Finsler metrics on ℝ^{N} such that Φ_{n}|_{\overline{Ω}×ℝ^{N}} ∈ M_{α} for each n ∈ ℕ and ϕ_{ε}(x, ξ) = sup_{n∈ℕ} Φ_{n}(x, ξ) in \overline{Ω} × ℝ^{N} (cf. Proof of [5, Theorem 3.1]). By Proposition 3.2 (ii), the sequence of distances (d_{Φ_{n}})_{n} converges to d_{ϕ_{ε}} in D_{α}, hence (15) is proved by setting φ_{ε} := Φ_{n} for n sufficiently large. It is now clear how to conclude: for each n ∈ ℕ, define φ_{n} := φ_{ε_{n}} with ε_{n} := 1/n. The distances d_{φ_{n}} converge to d pointwise on S × S, which is dense in \overline{Ω} × \overline{Ω}, so the statement follows in view of Lemma 3.1. □

Remark 4.4. Note that the metrics ϕ_{n} obtained in the previous proof are actually defined and continuous in all ℝ^{N} × ℝ^{N}.

The result established in Theorem 4.1 can now be easily improved by requiring the approximating metrics to be smooth.

Theorem 4.5. Let d ∈ D_{α}. Then there exists a sequence (φ̃_{n})_{n} of smooth and convex Finlser metrics in M_{α} such that dφ̃_{n} ⟶^{D_{α}} d.

Proof. Theorem 4.1 provides a sequence (φ_{n})_{n} of continuous, convex metrics in M_{α} satisfying the claim. Such metrics are actually defined and continuous in ℝ^{N} × ℝ^{N} (cf. Remark 4.4). Take a sequence (ρ_{k})_{k} of convolution kernels in ℝ^{N} and, for each n ∈ ℕ, let (ρ_{k} ∗ φ_{n}) (x, ξ) := ∫_{ℝ^{N}} ρ_{k}(x − y)φ_{n}(y, ξ) dy. Clearly (ρ_{k} ∗ φ_{n})_{k} is a sequence of convex, smooth metrics in M_{α}, uniformly converging to φ_{n} on compact subsets of \overline{Ω} × ℝ^{N}. In view of Proposition 3.2 (i), the claim follows by setting φ̃_{n} := ρ_{k} ∗ φ_{n} for each n ∈ ℕ, with k := k(n) suitably large. □

We end this section by providing an example of Finsler metric which gives rise to a non-symmetric distance. The latter plays a central role in the study of Hamilton-Jacobi equations of eikonal type.

Example 4.6. Let Ω be an open subset of ℝ^{N} and let us consider the following problem

\{ \begin{matrix}H(Du) = n(x) & in Ω \\ u = 0 & on ∂Ω \end{matrix}  (16)

where n(x) is a continuous positive function defined on \overline{Ω}, and H is a non-negative convex function with H(0) = 0. Solutions and subsolutions to (16) are usually meant in the viscosity sense, since classical ones do not exist in general, as well known (see e.g. [23]). In particular, viscosity subsolutions of (16) are characterized by the property of being 1-Lipschitz continuous with respect to the distance D defined by

D(x, y) := inf \{∫_{0}^{1} φ(γ(t), γ̇(t)) dt | γ ∈ Lip_{x,y} ([0, 1], \overline{Ω}) \} for all x, y ∈ \overline{Ω},

where φ is the Finsler metric associated to the equation as follows:

φ(x, ξ) := sup\{⟨ξ, p⟩ | H(p) ≤ n(x)\} for every (x, ξ) ∈ \overline{Ω} × ℝ^{N}.

Moreover, the unique viscosity solution v to (16) is given by the following Lax formula

v(x) := min_{y∈∂Ω} D(y, x).

Note that, when the function H is non-symmetric, the distance D is, in general, non-symmetric as well. The distance D is called optical length function. This name originates from geometrical optics. Indeed, we may regard the differential equation in (16) as the equation which rules the propagation of light in a non-isotropic medium, where u(x) represents the time at which the light reaches the location x, and n(x) is the refractive index of the medium at x. The presence of the function H accounts for the fact that the speed of light depends on its direction of travel. According to Fermat’s principle, a ray of light moves from one source y to another point x along the path which is the shortest in time, i.e. in such a way to realize D(y, x).

When n is not continuous, the metric φ is not continuous either. This case is of clear interest: think, for instance, to propagation of light in a medium composed by two materials with different indexes of refraction. This motivates the study of first order Hamilton-Jacobi equations with measurable ingredients. A metric approach to this problem, relying on the analysis of the properties of the Finsler metric φ and of the related optical length function, can be found in [4, 11, 24].

5. The degenerate case

We want to extend the density results obtained in the previous section to the degenerate case, namely when α = 0. In fact, the analogous of Theorem 4.5 holds.

Theorem 5.1. Let d ∈ D_{0}. Then there exists a sequence (φ̃_{n})_{n} of smooth and convex Finsler metrics in M_{0} such that dφ̃_{n} ⟶^{D_{0}} d.

Proof. Let φ ∈ M_{0} such that d = d_{φ}. For each k ∈ ℕ, set φ_{k}(x, ξ) := φ(x, ξ) ∨ \frac{1}{k}|ξ| in \overline{Ω} × ℝ^{N}. As φ_{k} ∈ M_{1/k}, we can apply Theorem 4.5 to d_{φ_{k}} for each k ∈ ℕ, to obtain a smooth Finsler metric φ̃_{k} ∈ M_{1/k} such that

|d_{φ_{k}}(x, y) − dφ̃_{k}(x, y)| ≤ \frac{1}{k} for every x, y ∈ \overline{Ω}.

Now the claim easily follows, since the sequence (d_{φ_{k}})_{k} converges to d in D_{0} in view of Remark 3.3. □

As already remarked in Section 2, when α > 0, the space D_{α} is closed, that is, if (d_{n})_{n} is a sequence in D_{α} converging to d, then d belongs to D_{α}. Does the same property hold for α = 0 too? The point is to show that d is still of geodesic type. When α > 0, this is basically due to the fact that the corresponding length functionals are equi-coercive, namely L_{d_{n}}(γ) ≥ α ∫_{0}^{1} |γ̇(t)| dt for any curve γ and for each n ∈ ℕ, and this means that any sequence of curves (γ_{n})_{n} ⊂ Lip_{x,y} such that lim sup_{n} L_{d_{n}}(γ_{n}) < +∞ admits (at least) a cluster point γ ∈ Lip_{x,y}. In particular, for a suitable choice of (γ_{n})_{n} and of γ, that yields

L_{d}(γ) ≤ lim inf_{n→+∞} L_{d_{n}}(γ_{n}) = lim_{n→+∞} d_{n}(x, y) = d(x, y),

which obviously means that d is a geodesic distance. When α = 0, instead, it may happen that the Euclidean lengths of the curves (γ_{n})_{n} diverge, and two critical phenomena may basically occur in this case: either the curves go to infinity and disappear in the limit, or they stay bounded and converge to a non-rectifiable curve (hence, no longer Lipschitz continuous). In both cases, the limit distance might be not of geodesic type; hence, the answer to the question previously raised is no: D_{0} is not closed. Explicit examples of these possible situations are provided below.

Example 5.2. Let Ω := ℝ^{2}, and let us set Γ_{n} := {(x, y) | x ∈ {0, 1}, y ∈ [0, n] } ∪ [0, 1] × {n} for each n ∈ ℕ, Γ_{∞} := {(x, y) | x ∈ {0, 1}, y ∈ [0, +∞) }. For each n ∈ ℕ, we define the metric φ_{n} on ℝ^{2} × ℝ^{2} as follows:

φ_{n}(x, ξ) := \{ \begin{matrix}\frac{1}{4n^{2}} |ξ| & if x ∈ Γ_{n} and ξ ∈ ℝ^{2} \\ 2|ξ| & if x ∈ ℝ^{2} \ Γ_{n} and ξ ∈ ℝ^{2} \end{matrix}

Let x_{0} := (0, 0), y_{0} := (1, 0), and let d_{n} := d_{φ_{n}} be the distance defined on ℝ^{2} × ℝ^{2} via (4) for each n ∈ ℕ. Let us notice that d_{n}(x_{0}, y_{0}) = (2n + 1)/4n^{2}: in fact, the d_{n}-minimizing path connecting x_{0} to y_{0} is given by the polygonal arc Γ_{n}. Up to subsequences, (d_{n})_{n} converges (uniformly on compact subsets of ℝ^{2} × ℝ^{2}) to some distance d, by Ascoli-Arzelà Theorem. Now we have that d(x_{0}, y_{0}) = 0, while the d-metric length of any curve connecting x_{0} to y_{0} is at least 2. To see this, simply notice that φ_{d}(x, ξ) = 2|ξ| for every x ∈ ℝ^{2} \ Γ_{∞}. Hence d is not of geodesic type, though uniform limit of geodesic distances.

Example 5.3. Let Ω := (0, 1) × (0, 1) and let q = (1/2, 0), q′ = (1, 1/2). Our example relies upon the construction provided by Whitney in [27]. In this remarkable paper, Whitney recursively defines an arc A joining q to q′ of infinite length, and a function f, defined on \overline{Ω} and of class C^{1}, whose gradient is null on A, but f(q) = 0 and f(q′) = 1 (in particular, f is not constant on A).

Such an arc can be represented as the image of a continuous function γ : I → \overline{Ω} with γ(0) = q, γ(1) = q′, as explained by the author. Our goal is to define a sequence of Lipschitz curves uniformly converging to γ. This can be easily done by exploiting Whitney’s construction. Let us keep the same notation of [27] and assume we are at the n-th step of the iterative procedure leading to the definition of A, i.e. we have already defined squares Q_{i_{1}···i_{t}}, points q_{i_{1}···i_{t}}, q′_{i_{1}···i_{t}} and lines A_{j_{1}···j_{t}} (each i_{k} = 0, 1, 2, 3, each j_{k} = 0, 1, 2, 3, 4) for t ≤ n. Then, an arc A_{n} of finite length, joining q to q′, can be obtained by connecting each point q′_{i_{1}···i_{s}} to q_{i_{1}···i_{s}+1}, if i_{s} ≤ 2, and q′_{i_{1}···i_{s−1},3} to q_{i_{1}···i′_{s−1}} by means of a segment, and by gluing all these segments with all the lines A_{j_{1}···j_{t}}, t ≤ n. Each arc A_{n} may be represented as the image of a Lipschitz curve γ_{n}. Up to a reparametrization (not by constant speed, in particular), the curves γ_{n} uniformly converge to γ, as easily understood.

Let us now denote by ℓ_{n} the Euclidean length of the curve γ_{n}, and define the following sequence (a_{n})_{n} of Riemannian metrics:

a_{n}(x) := \{ \begin{matrix}\frac{1}{nℓ_{n}} & if x ∈ A_{n} \\ |Df(x)| & if x ∈ \overline{Ω} \ A_{n}. \end{matrix}

Let d_{n} the distances on \overline{Ω} × \overline{Ω} associated to a_{n}(x)|ξ| through (4). Obviously, d_{n}(q, q′) ≤ 1/n. Up to subsequences, (d_{n})_{n} converges to some distance d, by Ascoli-Arzelà Theorem, and obviously d(q, q′) = 0. Notice also that φ_{d}(x, ξ) = |Df(x)||ξ| for every x ∈ \overline{Ω} \ A. Now, let us take a curve ξ ∈ Lip_{q,q′}. We have

L_{d}(ξ) = ∫_{0}^{1} φ_{d}(ξ(t), ξ̇(t)) dt ≥ ∫_{0}^{1} |Df(ξ(t))| |ξ̇(t)| dt ≥ f(q′) − f(q) = 1,

so d is not of geodesic type.

Acknowledgements. − The idea of using support functions to generalize the results in [15] was suggested by Antonio Siconolfi in one of the many interesting discussions we had together during my sojourn in Rome in october 2003. I also wish to thank Fabio Camilli for having attracted my attention on Whitney’s example.

References

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Dipartimento di Matematica Applicata ”U. Dini”, Università di Pisa, via S. Maria 26, 56126 Pisa (Italy)

   E-mail address: davini@dm.unipi.it