INTEGRAL REPRESENTATION OF ABSTRACT FUNCTIONALS OF AUTONOMOUS TYPE

ANDREA DAVINI

Abstract. In this work we extend the results of [2] to a family of abstract functionals of autonomous type satisfying suitable locality and additivity properties, and general integral growth conditions of superlinear type. We single out a condition which is necessary and sufficient in order for a functional of this class to admit an integral representation, and sufficient as well to have an integral representation for its lower semicontinuous envelope. We also show that the integrand F (x, q) satisfies some nice regularity properties in the q–variable, in particular a convexity–type property along lines. By adapting to the case at issue the reparametrization techniques introduced in [17], we then prove that the family of integral functionals associated to integrands of this kind do meet the condition mentioned above, in particular it is closed by Γ–convergence.

1. Introduction

1.1. Description of the results. In [2], the authors prove an integral representation result for a class of lower semicontinuous functionals F defined on W^{1,p}(I; ℝ^{N})× I(I) (where p > 1, I is a bounded open interval and I(I) denotes the family of all open subintervals of I), satisfying suitable locality and additivity properties, and the following integral growth conditions

λ ∫_{a}^{b} |γ̇|^{p} ds ≤ F(γ, (a, b)) ≤ Λ ∫_{a}^{b} (1 + |γ̇|^{p}) ds  (1)

for every γ ∈ W^{1,p}(I; ℝ^{N}) and (a, b) ⊂ I, with Λ, λ fixed positive constants. It turns out that

F(γ, (a, b)) = ∫_{a}^{b} f(s, γ(s), γ̇(s)) ds for every γ ∈ W^{1,p}(I; ℝ^{N}) and (a, b) ⊂ I,

with f defined as

f(t, x, q) := lim sup_{h→0^{+}} \frac{1}{h} inf_{γ∈W^{1,p}(I;ℝ^{N})}\{F(γ, (t, t + h)) : γ(t) = x, γ(t + h) = x + h q \} .

Moreover, f(t, x, ·) is proved to be continuous on ℝ^{N}, and convex for almost every (t, x) ∈ I × ℝ^{N}, in agreement with previous results for similar functionals (cf. [16, 19]). As a consequence of their analysis, the authors provide an integral representation for the Γ–limit of a sequence of functionals defined as

F_{k}(γ, (a, b)) := ∫_{a}^{b} f_{k}(s, γ(s), γ̇(s)) ds for every γ ∈ W^{1,p}(I; ℝ^{N}) and (a, b) ⊂ I,

where f_{k} : I × ℝ^{N} × ℝ^{N} → ℝ are Borel functions such that

λ |q|^{p} ≤ f_{k}(t, x, q) ≤ Λ (1 + |q|^{p})  for every (t, x, q) ∈ I × ℝ^{N} × ℝ^{N}. (2)

In particular, by taking f_{k} = f for every k ∈ ℕ, one derives an integral representation result for the lower semicontinuous envelope of the map

W^{1,p}(I; ℝ^{N}) ∋ γ ↦ ∫_{a}^{b} f(s, γ, γ̇) ds,

where f(t, x, q) is a Borel function without continuity or convexity assumptions.

In this paper we extend the results of [2] to a wider class of abstract functionals satisfying, in place of (1), the following more general growth conditions:

∫_{a}^{b} α(|γ̇|) ds ≤ F(γ, (a, b)) ≤ ∫_{a}^{b} β(|γ̇|) ds  (3)

for every γ ∈ W_{loc}^{1,1}(ℝ; ℝ^{N}) and (a, b) ⊂ ℝ, where α, β are two superlinear functions from [0, +∞) to ℝ, which can be taken non–decreasing and convex as well without any loss of generality. On the other hand, we will be only concerned with functionals of autonomous type, i.e. such that

F(γ(·), h + (a, b)) = F(γ(· + h), (a, b))

for every γ ∈ W_{loc}^{1,1}(ℝ; ℝ^{N}), (a, b) ⊂ ℝ and h ∈ ℝ.

Our main motivation stems from the interest to treat integral functionals of the kind

𝕃^{t}(γ) := ∫_{0}^{t} L(γ, γ̇) ds,  γ ∈ W^{1,1}((0, t), ℝ^{N}), t > 0,  (4)

in presence of a Borel–measurable Lagrangian L : ℝ^{N} ×ℝ^{N} → ℝ, convex, superlinear and locally bounded in q, uniformly with respect to x. Such growth condition can be equivalently restated (cf. Lemma 2.3 in [17]) by saying that

α(|q|) ≤ L(x, q) ≤ β(|q|)  for every (x, q) ∈ ℝ^{N} × ℝ^{N},  (5)

for a suitable pair of superlinear functions α, β : [0, +∞) → ℝ, which, in general, do not grow as h^{p} when h → +∞, and might have different growths as well. The model examples of Lagrangians included in this class are the one of the form

L(x, q) = V (q) + n(x)

with V (·) convex and superlinear, and n(·) Borel–measurable and bounded. Through our analysis we are able to treat sequences of integral functionals of the form (4) associated to Borel–measurable Lagrangians L_{k}(x, q), convex in q, and satisfying (5) for a fixed pair of superlinear functions α, β, and to prove that their Γ–limits admit an integral representation of the same form.(^{1})

In particular, we get that the lower semicontinuous envelope of (4) admits an integral representation for some Borel–measurable \overline{L} : ℝ^{N} × ℝ^{N} → ℝ (cf. [6, 9, 10, 23, 24]). This result turns out to be very useful when one is interested in proving the local Lipschitz–continuity in (0, +∞) × ℝ^{N} of the value function

v(t, x) := inf {u(γ(0)) + ∫_{0}^{t} L(γ(s), γ̇(s)) ds : γ ∈ W^{1,1}([0, t], ℝ^{N}), γ(t) = x}

associated with a possibly discontinuous initial cost u : ℝ^{N} → [0, +∞], and with a discontinuous Lagrangian of the kind considered above (cf. [15, 17]). According to what proved in [15], v has the desired regularity if the infimum above is attained for every (t, x) ∈ (0, +∞) × ℝ^{N}. Yet, this need not be true in this case, no matter how regular u is. In fact, the lack of continuity of L does not guarantee that the associated action functional 𝕃^{t} is lower semicontinuous on W^{1,1}((0, t), ℝ^{N}) with respect to the suitable convergence which assures the existence of minimizers, which, in this case, turns out to be the uniform convergence (cf. Proposition 2.3). By classical results of Olech [25] and Ioffe [22], the latter is in fact assured if the Lagrangian is lower semicontinuous in x and convex in q. This difficulty can be overcome by looking for a relaxed formulation of the problem, consisting in replacing the functional 𝕃^{t} in the minimization formula above with its lower semicontinuous envelope \overline{𝕃}^{t}, but to conclude one needs to know that the latter admits an integral representation.

1Throughout the paper, W_{loc}^{1,1}(ℝ; ℝ^{N}) will be regarded as a metric space, endowed with the topology of uniform convergence on compact subsets of ℝ.

We now describe in a more detailed way the content of the paper. We will also underline the main technical difficulties and differences with respect to [2], which have made the extension of their results to the present setting non trivial at all.

We start by introducing, in Section 2, a family A of abstract functionals of autonomous type sharing the main properties enjoyed by (4) (see (F1)–(F4)). Following [2], we geodesically associate with each F ∈ A a sort of distance–function d_{F} on (0, +∞) × ℝ^{N} (see (8)), and a lenght–type functional L (see (10) with d_{F} in place of d).

When α and β have p–growth, we know by [2] that L is indeed the lower semicontinuous envelope of F and does admit an integral representation. A crucial step in the proof is showing that d_{F}((0, ·), (t, ·)) is continuous on ℝ^{N} × ℝ^{N}, for every t > 0. This fact basically relies on the possibility to derive suitable a priori L^{p}–estimates on the derivatives of quasi–optimal curves. The general assumptions made here on α and β do not allow us to use the same arguments, the main obstruction being given by the fact that α and β do not have, in general, the same kind of growth at infinity.

To face these difficulties, we resort to a metric–type approach. In Section 3.1 we consider a distance function d on (ℝ × ℝ^{N}) × (ℝ × ℝ^{N}) satisfying the same properties as d_{F}, and we introduce a condition on d, namely (*), under which the associated length–type functional L admits an integral representation. Moreover we show that the associated integrand is obtained by ”differentiating” the function d, and satisfies some nice regularity properties in the q–variable, such as continuity for every x ∈ ℝ^{N}, and convexity for almost every x ∈ ℝ^{N}. An interesting aspect of this study is having singled out a condition on the integrand (cf. Theorem 3.10-(v) and (L3) in Section 4), weaker than convexity in q, which is necessary in order to have lower semicontinuity of the associated action functional. This instance is specific of the autonomous case.

In Section 3.2 we show that, when F ∈ A is lower semicontinuous, condition (*) holding for d = d_{F} is necessary and sufficient in order for F to admit an integral representation (see Theorem 3.8). When F is any element of A, such condition turns out to be sufficient to get an integral representation result for its lower semicontinuous envelope (cf. Theorem 3.10), and necessary as well when α and β have the same kind of growth at infinity (cf. Remark 3.11).

In Section 4 we proceed to show that the function d associated to a functional of the form (4) does satisfy condition (*) whenever L is superlinear and locally bounded in q, uniformly with respect to x, and such to satisfy the convexity–type assumption (L3). This is accomplished in Section 4, see Theorem 4.2. The consequent extension to the case of Lagrangians locally bounded in (x, q) is easily derived via a localization argument (see Theorem 4.18). Most of the section is devoted to derive suitable a priori estimates on the Lipschitz constants of quasi–minimizers, which is the crucial step for the proof of Theorem 4.2. We note that such constants only depend on the kind of growth conditions assumed on L. In view of the results of Section 3, this allows us to recover compactness when dealing with sequences of locally equi–bounded discontinuous Lagrangians (cf. Theorem 4.19), thus showing that the family of integrands herein considered is stable with respect to Γ–convergence of the associated action functionals.

We end this introduction by briefly describing the main ideas exploited in the proofs of Section 4. We recall that we are interested in proving that the function defined through

d((0, y), (t, x)) := inf {∫_{0}^{t} L(γ, γ̇) ds : γ ∈ W^{1,1}((0, t), ℝ^{N}), γ(0^{+}) = y, γ(t^{−}) = x}

for every (y, t, x) ∈ ℝ^{N} × (0, +∞) × ℝ^{N} satisfies condition (*) when L is a discontinuous Lagrangian enjoying the properties mentioned above. This issue is strictly related to the study of the regularity of Lagrangian minimizers, when they exist (cf. [1, 3, 7, 9, 10, 13, 23, 24, 28, 29]). The desired property for d can be in fact recovered via a fairly easy argument (cf. proof of Theorem 4.4 in [15], or proof of Theorem 3.8). The same argument however works as soon as we provide some a priori estimates on the Lipschitz constants of quasi–minimizers for d((0, y), (t, x)), with some uniformity with respect to (y, t, x) ∈ ℝ^{N} × (0, +∞) × ℝ^{N}.

To this aim, the idea we have followed is that, in the above minimization formula, one needs not consider all possible absolutely continuous curves connecting y to x in time t; we can restrict to consider only those having an optimal parametrization, where optimal means that we are interested in making the action of L as small as possible. This has lead us to first consider a minimization problem with fixed support: we fix a Lipschitz curve γ : (0, ℓ) → ℝ^{N} parametrized by arc–length, the support, and we try to solve the following problem

min {∫_{0}^{t} L(ξ, ξ̇) ds : ξ ∈ [γ]_{t}}  (6)

for every t > 0, where [γ]_{t} denotes the family of absolutely continuous curves ξ : (0, t) → ℝ^{N} obtained through a reparametrization of γ (see Definition 4.8). Then we introduce the notion of a–Lagrangian parametrization on curves (cf. Definition 4.11). When L is convex in q, it reduces to requiring that the curve satisfies the DuBois–Raymond necessary condition for optimality, i.e.

L(γ(s), γ̇(s)) = ⟨γ̇(s), p⟩ − a for every p ∈ ∂_{q}L(γ(s), γ̇(s))

for some constant a ∈ ℝ and for almost every s ∈ (0, t). (^{2}) Then we consider the multifunction T_{γ}(·) defined on ℝ by

T_{γ}(a) := {t > 0 : [γ]^{♭}(a, t) is non empty } for every a ∈ ℝ,

where [γ]^{♭}(a, t) denotes the subset of [γ]_{t} consisting of a–Lagrangian bi–Lipschitz reparametrizations of γ, and we remark that the relation t ∈ T_{γ}(a) implies that problem (6) admits a solution in [γ]^{♭}(a, t). Our attention is then addressed to establish the main properties of the multifunction T_{γ}(·), with particular interest to its range ⋃_{a∈ℝ} T_{γ}(a) (see Proposition 4.12). When this coincides with (0, +∞), we conclude that problem (6) is solvable for every t > 0. In particular, (6) has a minimizer belonging to [γ]^{♭}(a, t) for some a ∈ ℝ, and its Lipschitz constant can be estimated by some κ_{a} ∈ ℝ depending on a and on the kind of growth conditions assumed on L only. However, our analysis reveals that the range of T_{γ}(·) may actually be a bounded interval of the form (0, T). In this instance, a solution to (6) exists if t ≤ T. For t > T, the minimum in (6) is only an infimum, in general; nevertheless, we prove that this value can be obtained by minimizing the action over the family of κ_{c_{γ}}–Lipschitzian reparametrizations of γ, where κ_{c_{γ}} is a positive constant that can be estimated in terms of the growth conditions assumed on L (see Theorem 4.14).

2 Here ∂_{q}L(x, q) denotes the subdifferential of the function L(x, ·) at q.

This information is used to get the sought a priori estimates on the Lipschitz constants of quasi–minimizers (see Lemma 4.16): since any absolutely continuous curve from (0, t) to ℝ^{N} belongs to [γ]_{t} for a suitable choice of the Lipschitz curve γ : (0, ℓ) → ℝ^{N} (cf. Lemma 4.10), a quasi–minimizer for d((0, y), (t, x)) can be always assumed to be κ_{a}–Lipschitz continuous, for some a ∈ ℝ. By using the superlinearity of L(x, ·), the constant a is last estimated with some uniformity with respect to (y, t, x).

The analysis outlined above relies on suitable reparametrization techniques which use in an essential way the convexity assumption (L3). The argument on which they are based was originally introduced in [20], then developed in [18] for a continuous and convex Lagrangian, and subsequently extended to the measurable case in [17]. In the present paper, we have replaced the convexity assumption of L in q with the weaker condition (L3). This gives rise to some technical difficulties (namely, the convex envelope of L(x, ·) does not agree with L(x, ·) any longer, cf Remark 3.18 in [17]), but the underlying idea, as well as many proofs, is the same.

2. Notation and standing assumptions

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

N  an integer number
B_{r}(x) the open ball in ℝ^{N} of radius r centered at x
B_{r}  the open ball in ℝ^{N} of radius r centered at 0
⟨ ·, ·⟩ the scalar product in ℝ^{N}
[u]  the integer part of u ∈ ℝ
ℝ_{+}  the set of nonnegative real numbers
P(ℝ_{+}) the space of all subsets of ℝ_{+}

Given a subset U of ℝ^{k}, we denote by \overline{U} its closure. If E is a Lebesgue measurable subset of ℝ^{k}, we denote by |E| its k–dimensional Lebesgue measure, and we say that E is negligible whenever |E| = 0. The characteristic function of E is denoted by χ_{E}. We say that a property holds almost everywhere (a.e. for short) on ℝ^{k} if it holds up to a negligible subset of ℝ^{k}. The Euclidean norm of u ∈ ℝ^{k} is denoted by |u|.

Given a measurable vector–valued function f : E → ℝ^{m}, we write ∥f∥_{∞} to mean

(∑_{i=1}^{k} ∥f_{i}∥_{L^{∞}(E)})^{1/2}, where f_{i} and ∥f_{i}∥_{L^{∞}(E)} denote the i-th component of f and the L^{∞}-norm of f_{i}, respectively. We will say that f is transversal to S ⊂ ℝ^{m} if |\{x ∈ E : f(x) ∈ S\}| = 0. The notation ∫ −_{E}f dx and ∫ −_{a}^{b}f ds stands for \frac{1}{|E|} ∫_{E} f dx

and \frac{1}{b−a} ∫_{a}^{b} f ds, respectively. Notice that ∫ −_{a}^{b}f ds = ∫ −_{b}^{a}f ds for any a ≠ b.

Let X ⊆ ℝ^{k} and B(X) the family of all Borel subsets of X. A multifunction Γ from X to compact subsets of ℝ is said to be Borel–measurable (cf. [8]) if

{x ∈ X : Γ(x) ∩ U ≠ ∅ } ∈ B for every open set U ⊆ ℝ.

We say that Γ is upper semicontinuous at x if, for any ε > 0, there exists δ > 0 such that

Γ(z) ⊆ Γ(x) + (−ε, ε) for all z ∈ B_{δ}(x) ∩ X.

When k = 1, we say that Γ is non–decreasing on X if

sup Γ(x) ≤ inf Γ(y) for every x, y ∈ X with x < y.

We say that Γ is non–increasing on X if the multifunction −Γ(·) is non–decreasing on X.

For a function g : ℝ^{k} → (−∞, +∞], we denote by dom(g) its effective domain; i.e., the subset of ℝ^{k} where g is finite valued. We will say that g is superlinear if

lim_{|x|→+∞} \frac{g(x)}{|x|} = +∞.

For a convex function f from ℝ^{k} to ℝ, we will denote by ∂f(x) the subdifferential of f at x, defined as

∂f(x) := {p ∈ ℝ^{k} : f(y) ≥ f(x) + ⟨p, y − x⟩ for every y ∈ ℝ^{k} }.

The set ∂f(x) is closed and convex. We furthermore have (see [26]):

Proposition 2.1. Let f : ℝ^{k} → ℝ be convex. Then f is locally Lipschitz in ℝ^{k}. More precisely, for every x_{0} ∈ ℝ^{k} and r, δ > 0, we have

|f(x) − f(y)| ≤ |x − y| \frac{2}{δ} sup_{B_{r+δ}(x_{0})} f for every x, y ∈ B_{r}(x_{0}).

In particular, ∂f(x) ⊂ (2 sup_{B_{r+1}} f) \overline{B}_{1} for every x ∈ B_{r}.

Given a function f : ℝ^{k} → ℝ, we define its conjugate f^{∗} : ℝ^{k} → (−∞, +∞] as follows:

f^{∗}(x) := sup_{y∈ℝ^{k}}{⟨x, y⟩ − f(y) } for every x ∈ ℝ^{k}.

We record for later use the following well known facts (cf. [26, Theorem 23.5])

Proposition 2.2. Let f : ℝ^{k} → ℝ be superlinear. Then f^{∗} is convex and locally bounded on ℝ^{k}. Moreover

f(x) ≥ f^{∗∗}(x) := sup_{y∈ℝ^{k}}{⟨x, y⟩ − f^{∗}(y) } for every x ∈ ℝ^{k},

with equality holding for every x if and only if f is convex. When f is convex, the following conditions on x, x^{∗} ∈ ℝ^{k} are equivalent to each other:

(i) f(x) + f^{∗}(x^{∗}) ≤ ⟨x, x^{∗}⟩; (ii) f(x) + f^{∗}(x^{∗}) = ⟨x, x^{∗}⟩; (iii) x^{∗} ∈ ∂f(x); (iv) x ∈ ∂f^{∗}(x^{∗}).

We denote by W^{1,1}((a, b), ℝ^{N}) the space of absolutely continuous curves from the interval (a, b) to ℝ^{N}, while W_{loc}^{1,1}(ℝ, ℝ^{N}) denotes the space of locally absolutely continuous curves. We endow W_{loc}^{1,1}(ℝ, ℝ^{N}) with the metrizable topology of uniform convergence on compact subsets of ℝ. We recall that a curve γ : (a, b) → ℝ^{N} is said to be parametrized by arc–length if |γ̇(s)| = 1 for almost every s ∈ (a, b).

Throughout the paper, α, β will always denote two functions from ℝ_{+} to ℝ_{+} that are convex, non–decreasing and superlinear, namely

lim_{h→+∞} \frac{α(h)}{h} = lim_{h→+∞} \frac{β(h)}{h} = +∞.

The following result is a consequence of Dunford–Pettis Theorem (cf. Theorems 2.11 and 2.12 in [7]).

Proposition 2.3. Let I a bounded interval of ℝ and (γ_{n})_{n} a sequence in W^{1,1}(I, ℝ^{N}) such that

sup_{n} ∫_{I} α(|γ̇_{n}|) ds < +∞.  (7)

Then there exists a subsequence (γ_{n_{k}})_{k} and a curve γ ∈ W^{1,1}(I, ℝ^{N}) such that

lim_{k→∞} ∥γ_{n_{k}} − γ∥_{∞} = 0.

Remark 2.4. We point out that condition (7) implies that (γ_{n_{k}})_{k} actually converges to γ weakly in W^{1,1}(I, ℝ^{N}).

Let I(ℝ) be the collection of bounded, open intervals of ℝ. We denote by A = A(α, β) the class of all abstract functionals F : W_{loc}^{1,1}(ℝ; ℝ^{N}) × I(ℝ) → [0, +∞] satisfying the following properties:

F_{a}^{b}(γ) = F_{a}^{b}(ξ)  if γ, ξ ∈ W_{loc}^{1,1}(ℝ; ℝ^{N}) and γ = ξ a.e. on (a, b), (F1)
F_{a+h}^{b+h}( γ(·) ) = F_{a}^{b}( γ(· + h) )  for any a, b, h ∈ ℝ and γ ∈ W_{loc}^{1,1}(ℝ; ℝ^{N}), (F2)
F_{a}^{c}(γ) = F_{a}^{b}(γ) + F_{b}^{c}(γ)  for any a < b < c and γ ∈ W_{loc}^{1,1}(ℝ; ℝ^{N}), (F3)

∫_{a}^{b} α(|γ̇|) dt ≤ F_{a}^{b}(γ) ≤ ∫_{a}^{b} β(|γ̇|) dt for any a, b ∈ ℝ, γ ∈ W_{loc}^{1,1}(ℝ; ℝ^{N}). (F4)

Here the notation F_{a}^{b}(γ) stands for F(γ, (a, b)). When a = 0, F_{a}^{b} will be more simply denoted by F^{b}. Knowing the latter for any b > 0 is sufficient to identify the functional F, in view of property (F2).

We will say that F ∈ A is lower semicontinuous on W_{loc}^{1,1}(ℝ; ℝ^{N}) if F_{a}^{b} is lower semicontinuous on W^{1,1}((a, b), ℝ^{N}) with respect to the uniform convergence, for any (a, b) ∈ I(ℝ).

Given F ∈ A, we will denote by \overline{F} its lower semicontinuous envelope with respect to the local uniform convergence on W_{loc}^{1,1}(ℝ; ℝ^{N}), namely the functional defined as follows:

\overline{F}_{a}^{b}(γ) := inf \{lim inf_{n} F_{a}^{b}(γ_{n}) : (γ_{n})_{n} ⊂ W^{1,1}((a, b), ℝ^{N}), lim_{n} ∥γ_{n} − γ∥_{∞} = 0 \}

for any a < b. As well known, \overline{F} is the greatest among all lower semicontinuous functionals which are less or equal than F on W_{loc}^{1,1}(ℝ; ℝ^{N}) × I(ℝ) (see [14]). Note that \overline{F} still enjoys hypotheses (F1)–(F4).

Choose F in A, and let s, t ∈ ℝ and x, y ∈ ℝ^{N}. We define

d_{F}((s, x), (t, y)) := inf {F_{s}^{t}(γ) : γ ∈ W^{1,1}((s, t), ℝ^{N}), γ(s^{+}) = x, γ(t^{−}) = y} (8)

for s < t, while we agree that d_{F}((s, x), (s, x)) = 0, and d_{F}((s, x), (t, y)) = +∞ when either s > t or s = t and x ≠ y. As a simple consequence of the definitions, we derive the following facts.

Proposition 2.5. Let F ∈ A and d = d_{F}. For any x, y, ζ ∈ ℝ^{N} and s, t, τ ∈ ℝ, the following properties hold:

(d1) d((0, x), (0, x)) = 0 and d((0, x), (t, x)) = +∞ when t < 0;

(d2) d((s, x), (t, y)) = d((0, x), (t − s, y));

(d3) d((s, x), (t, y)) ≤ d((s, x), (τ, ζ)) + d((τ, ζ), (t, y));

(d4) t α (\frac{|y − x|}{t}) ≤ d((0, x), (t, y)) ≤ t β (\frac{|y − x|}{t}) when t > 0.

It is apparent from the definition that

d\overline{F}((0, x), (t, y)) = inf \{lim inf_{n} d_{F}((0, x_{n}), (t, y_{n})) : x_{n} → x, y_{n} → y \} (9)

for every x, y ∈ ℝ^{N} and t > 0. Otherwise stated, for any fixed t > 0 the function d\overline{F}((0, ·), (t, ·)) is the lower semicontinuous envelope of d_{F}((0, ·), (t, ·)). In particular,

d\overline{F} ≤ d_{F} on (ℝ × ℝ^{N}) × (ℝ × ℝ^{N}).

Remark 2.6. It is not clear whether the inequality in the above expression can be actually strict. Indeed, it is not hard to show that d_{F}((0, ·), (t, ·)) is upper semicontinuous on ℝ^{N} × ℝ^{N} for every fixed t > 0, but there is no evidence why it should be lower semicontinuous. We only know that this is true when α and β have the same kind of growth at infinity. To see this, it is enough to argue as in the proof of Theorem 3.2 in [2] by using, in place of Lemma 3.1, the fact that

lim_{η→0^{+}} α (\frac{|γ(s + η) − γ(s)|}{η}) η = 0 for every s ∈ (0, t),

where γ is any curve in W^{1,1}((0, t), ℝ^{N}) such that ∫_{0}^{t} α(|γ̇|) ds < +∞.

To any function d from (ℝ×ℝ^{N})×(ℝ×ℝ^{N}) to [0, +∞] satisfying the statement of Proposition 2.5 we associate a functional L : W_{loc}^{1,1}(ℝ; ℝ^{N}) × I(ℝ) → [0, +∞] defined as follows: for any a < b and γ ∈ W^{1,1}((a, b), ℝ^{N}), we set

L_{a}^{b}(γ) := sup {∑_{i} d(t_{i}, γ(t_{i}), t_{i+1}, γ(t_{i+1})) : a = t_{0} < ⋯ < t_{n} = b, n ∈ ℕ } , (10)

where the supremum is taken over all possible finite partitions of (a, b). The functional L satisfies hypotheses (F1)–(F4), as can be easily checked. According to our previous notation, in the sequel we will write L^{b} in place of L_{0}^{b}.

3. Abstract functionals of autonomous type

3.1. A metric–type analysis. Let d be a function from (ℝ × ℝ^{N}) × (ℝ × ℝ^{N}) to [0, +∞] such to satisfy the statement of Proposition 2.5, and let L be the functional associated to d through (10). The purpose of this section is to show that the functional L admits an integral representation, provided some additional conditions are assumed on d.

We start by recording a result that will be required later in this section (cf. proof of Theorem 3.5). We have stated it at this point to emphasize its independence from any additional hypothesis we will introduce on d. The proof is omitted, for it may be easily recovered from the one provided in [2, Lemma 4.2] for the case β(h) = Λ (1 + h^{p}) with Λ ∈ ℝ_{+} and p > 1.

Lemma 3.1. Let x, v_{1}, v_{2} ∈ ℝ^{N} and δ > 0. For i = 1, 2, set

ℓ_{i}^{−} := lim inf_{h→0^{+}} \frac{d((0, x), (h, x + ξ_{i}(h)))}{h}, ℓ_{i}^{+} := lim sup_{h→0^{+}} \frac{d((0, x), (h, x + ξ_{i}(h)))}{h},

where ξ_{i} : [0, δ] → ℝ^{N} is a function such that

lim_{h→0^{+}} \frac{ξ_{i}(h)}{h} = v_{i}.

For any 0 < c_{1} < c_{2} we have

c_{2}ℓ_{2}^{−} − c_{1}ℓ_{1}^{−} ≤ (c_{2} − c_{1}) β ( \frac{|c_{2}v_{2} − c_{1}v_{1}|}{c_{2} − c_{1}} ) ,
c_{2}ℓ_{2}^{+} − c_{1}ℓ_{1}^{+} ≤ (c_{2} − c_{1}) β ( \frac{|c_{2}v_{2} − c_{1}v_{1}|}{c_{2} − c_{1}} ) .

In particular, ℓ_{1}^{−} = ℓ_{2}^{−} and ℓ_{1}^{+} = ℓ_{2}^{+} whenever v_{1} = v_{2}.

We now add some hypotheses on d. Throughout the section, we assume that the following condition holds:

d((0, ·), (·, ·))  is locally Lipschitz continuous on ℝ^{N} × (0, +∞) × ℝ^{N}. (*)

We first prove that the L^{t}–length of any curve γ ∈ W^{1,1}((0, t), ℝ^{N}) admits an integral representation in terms of its metric derivative.

Definition 3.2 (Metric derivative). Given a curve γ ∈ W^{1,1}((a, b), ℝ^{N}), we define the metric derivative |γ̇|_{d}(s) of γ at the point s ∈ (a, b) as

|γ̇|_{d}(s) := lim sup_{h→0^{+}} \frac{d((s, γ(s)), (s + h, γ(s + h)))}{h}.  (11)

Theorem 3.3. Let condition (*) hold. Then, for any curve γ ∈ W^{1,1}((0, t), ℝ^{N}),

L^{t}(γ) = ∫_{0}^{t} |γ̇|_{d}(s) ds.

Moreover, the lim sup at the right-hand side of (11) is actually a limit for a.e. s ∈ (0, t) whenever L^{t}(γ) < +∞.

Remark 3.4. The definition of L^{t} and the notion of metric derivative, as well as the integral representation formula provided by Theorem 3.3, are well known in classical metric spaces. In some sense, in fact, the function d can be regarded as a degenerate non–symmetric distance on the product space ℝ × ℝ^{N}. The proof below is actually inspired to the one proposed in [4, Theorem 4.1.1].

Proof. Let us set J := (0, t) and let (t_{n})_{n} be a dense sequence in J made up by differentiability points of γ. For each n ∈ ℕ, we define

φ_{n}(s) := \{ \begin{matrix}d((t_{n}, γ(t_{n})), (s, γ(s)) ) & if s ∈ (t_{n}, t) \\ 0 & if s ∈ (0, t_{n}]. \end{matrix}

From (*) we deduce that φ_{n} is absolutely continuous in J \ {t_{n}}, as a composition of a locally Lipschitz function with an absolutely continuous curve, hence its derivative φ̇_{n}(s) exists at almost every point s ∈ J. Let us define

m(s) := sup_{n} φ̇_{n}(s) for a.e. s ∈ J.

We start by proving that

L^{t}(γ) = ∫_{0}^{t} m(ς) dς  (12)

and

m(s) = |γ̇|_{d}(s)  whenever s ∈ J is Lebesgue point for m(·).  (13)

By the definition of φ_{n} and the triangular inequality we infer that

|γ̇|_{d}(s) = lim inf_{h→0^{+}} \frac{d((s, γ(s)), (s + h, γ(s + h)))}{h} ≥ lim inf_{h→0^{+}} \frac{φ_{n}(s + h) − φ_{n}(s)}{h} = φ̇_{n}(s)

for a.e. s ∈ J, hence, taking the sup over n ∈ ℕ,

lim inf_{h→0^{+}} \frac{d((s, γ(s)), (s + h, γ(s + h)))}{h} ≥ m(s)  for a.e. s ∈ J.  (14)

On the other hand, by the properties enjoyed by d we know that

0 ≤ d((t_{n}, γ(t_{n})), (s, γ(s)) ) ≤ (s − t_{n}) β (\frac{|γ(s) − γ(t_{n})|}{|s − t_{n}|}) for every s > t_{n},

hence inf_{n} d((t_{n}, γ(t_{n})), (s, γ(s)) ) = 0 whenever s is a differentiability point of γ. We derive

d((s, γ(s)), (τ, γ(τ))) ≤ sup_{n} (φ_{n}(τ) − φ_{n}(s))
= sup_{n} ∫_{s}^{τ} φ̇_{n}(ς) dς ≤ ∫_{s}^{τ} m(ς) dς

(15)

for almost every s < τ. If s is a Lebesgue point for m(·), we obtain

lim sup_{h→0^{+}} \frac{d((s, γ(s)), (s + h, γ(s + h)))}{h} ≤ lim sup_{h→0^{+}} \frac{1}{h} ∫_{s}^{s+h} m(ς) dς = m(s),

and this inequality, combined with (14), gives (13).

We now prove (12). By (15) it follows

∑_{i=0}^{k−1} d((t_{i}, γ(t_{i})), (t_{i+1}, γ(t_{i+1}))) ≤ ∑_{i=0}^{k−1} ∫_{t_{i}}^{t_{i+1}} m(ς) dς = ∫_{0}^{t} m(ς) dς

for every choiche 0 = t_{0} < t_{1} < ... < t_{k} = t, k ∈ ℕ. Taking the sup over all such partitions we obtain

L^{t}(γ) ≤ ∫_{0}^{t} m(ς) dς.

In order to prove the opposite inequality, choose ε > 0 and let h := t/k, t_{i} := ih, with k ≥ 2 such that h ≤ ε. We observe that

\frac{1}{h} ∫_{0}^{t−ε} d((ς, γ(ς)), (ς + h, γ(ς + h))) dς

≤ \frac{1}{h} ∫_{0}^{h} ∑_{i=0}^{k−2} d((t_{i} + ς, γ(t_{i} + ς)), (t_{i+1} + ς, γ(t_{i+1} + ς))) dς

≤ \frac{1}{h} ∫_{0}^{h} L^{t}(γ) dς = L^{t}(γ).

From Fatou’s Lemma and (14) we derive

∫_{0}^{t−ε} m(ς) dς ≤ ∫_{0}^{t−ε} lim inf_{h→0^{+}} \frac{d((ς, γ(ς)), (ς + h, γ(ς + h)))}{h} dς

≤ lim inf_{h→0^{+}} \frac{1}{h} ∫_{0}^{t−ε} d((ς, γ(ς)), (ς + h, γ(ς + h))) dς ≤ L^{t}(γ),

and we conclude letting ε → 0 by the Monotone Convergence Theorem.

It is now easy to conclude: when L^{t}(γ) = +∞, the statement follows by taking (12) and (14) into account; when L^{t}(γ) < +∞, m(·) is integrable and the statement is a consequence of (12) and (13) since almost every s ∈ J is a Lebesgue point for m(·). □

We are now ready to prove the main result of this section.

Theorem 3.5. Assume condition (*) holds. Then L is lower semicontinuous on W_{loc}^{1,1}(ℝ; ℝ^{N}) with respect to the local uniform convergence, and admits the following integral representation:

L^{t}(γ) = ∫_{0}^{t} L^{−}(γ, γ̇) ds = ∫_{0}^{t} L^{+}(γ, γ̇) ds, γ ∈ W^{1,1}((0, t), ℝ^{N}) and t > 0,

where

L^{−}(x, q) := lim inf_{h→0^{+}} \frac{d((0, x), (h, x + hq))}{h},

L^{+}(x, q) := lim sup_{h→0^{+}} \frac{d((0, x), (h, x + hq))}{h} .

Moreover, the functions L^{±} enjoy the following properties:

(i) L^{±} : ℝ^{N} × ℝ^{N} → ℝ_{+} is Borel–measurable;

(ii) α (|q|) ≤ L^{±}(x, q) ≤ β (|q|) for every (x, q) ∈ ℝ^{N} × ℝ^{N};

(iii) L^{±}(x, ·) is continuous on ℝ^{N} for every x ∈ ℝ^{N}; (iv) L^{±}(x, ·) is convex on ℝ^{N} for a.e. x ∈ ℝ^{N};

(v) for every t_{1} < t_{2} in ℝ and γ ∈ W^{1,∞}((t_{1}, t_{2}), ℝ^{N})

λ ↦ L^{±}(γ(s), λ γ̇(s)) is convex on ℝ for a.e. s ∈ (t_{1}, t_{2}).

Proof. Assumption (*) implies that the functional

γ ↦ ∑_{i=0}^{m−1} d(t_{i}, γ(t_{i}), t_{i+1}, γ(t_{i+1}))

is continuous on W^{1,1}((0, t), ℝ^{N}) with respect to the uniform convergence of curves, for any fixed partition 0 = t_{0} < t_{1} < ... < t_{m} = t, m ∈ ℕ. As a supremum of a family of continuous functionals, the lower semicontinuity of L^{t}(·) follows.

The integral representation formula is a consequence of Theorem 3.3, since Lemma 3.1 assures that

L^{−}(γ(s), γ̇(s)) = L^{+}(γ(s), γ̇(s)) = |γ̇|_{d}(s)

whenever s ∈ (0, t) is a differentiability point of γ, that is almost everywhere.

Items (i) and (ii) are an obvious consequence of the definitions of L^{±}, together with the fact that (x, q) ↦ d((0, x), (h, x + hq))/ h is Borel–measurable (in fact, continuous) on ℝ^{N} × ℝ^{N} for any h > 0.

Items (iii) and (iv) may be proved arguing as in [2] (cf. Theorems 5.1 and 5.2, respectively).

To prove (v), we make use of a ”zig–zag” argument. Let us arbitrarily fix a Lipschitz curve γ : (t_{1}, t_{2}) → ℝ^{N}, and choose λ_{1}, λ_{2} ∈ ℝ and δ ∈ (0, 1) as above. Set

λ := δλ_{1} + (1 − δ)λ_{2}.

We aim to prove that

L^{±}(γ(s), λ γ̇(s)) ≤ δ L^{±}(γ(s), λ_{1} γ̇(s))+(1−δ) L^{±}(γ(s), λ_{2} γ̇(s)) for a.e. s ∈ (t_{1}, t_{2}).

This is enough to conclude. Indeed, the statement follows by letting λ_{1}, λ_{2} and δ vary over two sets, countable and dense in ℝ \ {0} and (0, 1), respectively, and by using the continuity of L^{±}(x, ·) for every x ∈ ℝ^{N}.

To prove the claimed inequality, we will actually show that

∫_{a}^{b} L^{±}(γ, λ γ̇) ds ≤ δ ∫_{a}^{b} L^{±}(γ, λ_{1} γ̇) ds + (1 − δ) ∫_{a}^{b} L^{±}(γ, λ_{2} γ̇) ds (16)

for any [a, b] ⊂ (t_{1}, t_{2}), which is equivalent. Let us fix such an interval. Up to replacing the curve γ(·) with γ(a + ·), we can assume a = 0. By reversing the orientation of γ if necessary, we can additionally assume λ > 0. For every fixed n ∈ ℕ, we define a picewise affine map ψ_{n} on [0, b/(λ n) ] as

ψ_{n}(s) := \{ \begin{matrix}λ_{1} s & if s ∈ [0, δ \frac{b}{λ n}] \\ λ_{1} δ \frac{b}{λ n} + λ_{2} (s − δ \frac{b}{λ n}) & if s ∈ [δ \frac{b}{λ n}, \frac{b}{λ n}]. \end{matrix}

Consider the partition 0 = t_{0}^{n} < t_{1}^{n} < ⋯ < t_{n}^{n} = b/λ of the interval [0, b/λ] obtained by choosing

t_{i}^{n} := i\frac{b}{λ n} for each 0 ≤ i ≤ n,

and define φ_{n} : [0, b/λ] → ℝ by setting

φ_{n}(s) = λ t_{i}^{n} + ψ(s − t_{i}^{n}) if s ∈ [t_{i}^{n}, t_{i+1}^{n}], for each i = 0, …, n − 1.

The sequence (φ_{n})_{n} uniformly converges to the function φ(s) := λ s on [0, b/λ], hence, for n sufficiently large, the curves ξ_{n} := γ∘φ_{n} are well defined and uniformly converge to ξ := γ∘φ on [0, b/λ]. By the lower semicontinuity of L we infer

L^{b/λ}(ξ) ≤ lim inf_{n→+∞} L^{b/λ}(ξ_{n}).

A simple computation shows that this is equivalent to saying that

∫_{0}^{b} L^{±}(γ, λ γ̇) ds

≤ lim inf_{n→+∞} ∑_{i=0}^{n−1} \frac{b}{n} (δ∫ ; −_{λt_{i}^{n}}^{λt_{i}^{n}+ \frac{δλ_{1}}{λ} \frac{b}{n}} L^{±}(γ, λ_{1} γ̇) ds + (1 − δ)∫ ; −_{λt_{i}^{n}+ \frac{δλ_{1}}{λ} \frac{b}{n}}^{λ t_{i+1}^{n}} L^{±}(γ, λ_{2} γ̇) ds) ,

and (16) follows thanks to Lemma 3.6 below. □

Lemma 3.6. Let (a, b) be a bounded interval of ℝ and μ ∈ (0, 1). For each n ∈ ℕ, let {I_{i}^{n} : 1 ≤ i ≤ n } and {J_{i}^{n} : 1 ≤ i ≤ n } be two collections of pairwise disjoint intervals such that:

(a) ⋃_{i=1}^{n} I_{i}^{n} = (a, b)  and  max_{1≤i≤n} |I_{i}^{n}| → 0 as n → +∞;
(b) J_{i}^{n} ⊆ I_{i}^{n}  and  |J_{i}^{n}| ≥ μ |I_{i}^{n}| > 0 for every 1 ≤ i ≤ n.

Then

lim_{n→+∞} ∑_{i=1}^{n} |I_{i}^{n}| ∫ ; −_{J_{i}^{n}}g(s) ds = ∫_{a}^{b} g(s) ds  for any g ∈ L^{1}((a, b)). (17)

Proof. For each n ∈ ℕ and 1 ≤ i ≤ n, set μ_{i}^{n} := L^{1} (I_{i}^{n}) /L^{1} (J_{i}^{n}) and

u_{n}(x) := ∑_{i=1}^{n} μ_{i}^{n} χ_{J_{i}^{n}}(x), x ∈ (a, b).

Claim (17) amounts to saying that u_{n} ⇀^{∗} χ_{(a,b)} in L^{∞}((a, b)). It is easy to see that (17) holds whenever g is continuous. As sup_{n} ∥u_{n}∥_{∞} ≤ 1/μ, the statement follows by density of the continuous functions into L^{1}((a, b)). □

Remark 3.7. We record for later use that the results of this section can be easily generalized to functions d which satisfy, in place of property (d4) of Proposition 2.5, the following condition

(d4′)  t α (\frac{|y − x|}{t}) ≤ d((0, x), (t, y)) ≤ t β_{n} (\frac{|y − x|}{t}) ∀ x, y ∈ B_{n}, ∀ n ∈ ℕ,

where (β_{n})_{n∈ℕ} is a family of convex, non–decreasing and superlinear functions from ℝ_{+} to ℝ_{+}. In particular, the results of Theorem 3.5 still hold, provided claim (ii) is modified as follows:

(ii)′  α (|q|) ≤ L^{±}(x, q) ≤ β_{n} (|q|)  for any (x, q) ∈ B_{n} × ℝ^{N} and n ∈ ℕ.

3.2. Integral representation of abstract functionals. We now proceed to show that condition (*) holding with d = d_{F} characterizes all lower semicontinuous functionals F ∈ A that admit an integral representation; i.e., such that

F_{t_{1}}^{t_{2}}(γ) = ∫_{t_{1}}^{t_{2}} F (γ, γ̇) ds,  γ ∈ W^{1,1}((t_{1}, t_{2}), ℝ^{N})  (18)

for any t_{1} < t_{2}, with F : ℝ^{N} × ℝ^{N} → ℝ_{+} Borel–measurable. We note that, up to suitably modifying the integrand F on a subset of ℝ^{N} × ℝ^{N} which is transversal to s ↦ (γ(s), γ̇(s)) for any γ ∈ W_{loc}^{1,1}(ℝ; ℝ^{N}), we can always assume that

α(|q|) ≤ F (x, q) ≤ β(|q|) for any (x, q) ∈ ℝ^{N} × ℝ^{N}.  (19)

Since F enjoys (F2), it will be enough to prove (18) for any t_{2} > 0 and t_{1} = 0.

Theorem 3.8. Let F be a lower semicontinuous abstract functional belonging to A. Then F admits an integral representation if and only if the function d = d_{F} associated to F through (8) satisfies condition (*). In this instance we have:

F^{t}(γ) = ∫_{0}^{t} F^{−}(γ, γ̇) ds = ∫_{0}^{t} F^{+}(γ, γ̇) ds, γ ∈ W^{1,1}((0, t), ℝ^{N}) and t > 0,

where

F^{−}(x, q) := lim inf_{h→0^{+}} \frac{d_{F}((0, x), (h, x + hq))}{h},

F^{+}(x, q) := lim sup_{h→0^{+}} \frac{d_{F}((0, x), (h, x + hq))}{h} .

Moreover, the functions F^{±} enjoy the following properties:

(i) F^{±} : ℝ^{N} × ℝ^{N} → ℝ_{+} is Borel–measurable;

(ii) α (|q|) ≤ F^{±}(x, q) ≤ β (|q|) for every (x, q) ∈ ℝ^{N} × ℝ^{N};

(iii) F^{±}(x, ·) is continuous on ℝ^{N} for every x ∈ ℝ^{N};

(iv) F^{±}(x, ·) is convex on ℝ^{N} for a.e. x ∈ ℝ^{N};

(v) for every t_{1} < t_{2} in ℝ and γ ∈ W^{1,∞}((t_{1}, t_{2}), ℝ^{N})

λ ↦ F^{±}(γ(s), λ γ̇(s)) is convex on ℝ for a.e. s ∈ (t_{1}, t_{2}).

Proof. Let us assume that F admits an integral representation of the form (18) for some Borel–measurable F : ℝ^{N} × ℝ^{N} → ℝ_{+} satisfying (19). We want to show that condition (*) holds with d = d_{F}. This basically follows by what proved in [15]. We provide a proof for the reader’s convenience.

Fix x, y ∈ ℝ^{N} and t > 0. For 0 < r < t/2, let us denote by U_{r} the open set B_{r}(y) × (t − r, t + r) × B_{r}(x). We claim that there exists a constant K := K(t, α, β) such that

d((0, ·), (·, ·)) is K–Lipschitz continuous in U_{r}.

Choose (y_{1}, t_{1}, x_{1}) and (y_{2}, t_{2}, x_{2}) in U_{r}, and set

h := |t_{1} − t_{2}| + |x_{1} − x_{2}| + |y_{1} − y_{2}|, s_{0} := \frac{t_{1} − t_{2}}{2} + h.

Since U_{r} is convex, it suffices to prove the statement locally, namely for small values of h. Choose h < t_{2}/2 so that s_{0} < t_{1}/2. Let γ_{1} be a curve in W^{1,1}((0, t_{1}), ℝ^{N}) connecting y_{1} to x_{1} such that

d((0, y_{1}), (t_{1}, x_{1})) = ∫_{0}^{t_{1}} F (γ_{1}, γ̇_{1}) ds,

which does exist in force of Proposition 2.3 and of the lower semicontinuity of \overline{F}. Theorem 2.2 in [15] asserts that γ_{1} is Lipschitz continuous and ∥γ̇_{1}∥_{∞} ≤ κ for some constant κ depending on t, α and β only. Choose u_{1}, v_{1} ∈ ℝ^{N} so that

γ_{1}(s_{0}) = y_{2} + hu_{1}, γ_{1}(t_{1} − s_{0}) = x_{2} + hv_{1}.

Note that |u_{1}|, |v_{1}| < 1 + 2κ. We now define a curve γ_{2} : [0, t_{2}] → ℝ^{N} connecting y_{2} to x_{2} as follows:

γ_{2}(s) := \{ \begin{matrix}y_{2} + su_{1} & if s ∈ [0, h], \\ γ_{1}(s_{0} + s − h) & if s ∈ [h, t_{2} − h] \\ x_{2} + (t_{2} − s)v_{1} & if s ∈ [t_{2} − h, t_{2}] \end{matrix}

Recalling that F is positive, we get

d((0, y_{2}), (t_{2}, x_{2})) − d((0, y_{1}), (t_{1}, x_{1})) ≤ ∫_{0}^{t_{2}} F (γ_{2}, γ̇_{2}) ds − ∫_{0}^{t_{1}} F (γ_{1}, γ̇_{1}) ds
≤ ∫_{0}^{h} F (γ_{2}, u_{1}) ds + ∫_{t_{2}−h}^{t_{2}} F (γ_{2}, u_{2}) ds ≤ 2 β(1 + 2κ) h,

so, setting K̃ := 2 β(1 + 2κ_{A}), we obtain

d((0, y_{2}), (t_{2}, x_{2})) − d((0, y_{1}), (t_{1}, x_{1})) ≤ K̃ (|t_{1} − t_{2}| + |x_{1} − x_{2}| + |y_{1} − y_{2}|) .

The claim follows by interchanging the roles of (y_{1}, t_{1}, x_{1}) and (y_{2}, t_{2}, x_{2}) and by setting K := \sqrt{2N + 1} K̃.

Conversely, let us assume that d_{F} satisfies condition (*). Since F coincides with its lower semicontinuous envelope \overline{F}, the assertion follows in view of Theorem 3.5 and of Proposition 3.9 below. □

Proposition 3.9. Let L be defined via (10) with d = d_{F}, and assume d_{F} satisfies condition (*). Then L is the lower semicontinuous envelope of F, namely

L^{t}(·) = \overline{F}^{t}(·)  on W^{1,1}((0, t), ℝ^{N})  (20)

for every t > 0.

Proof. Fix t > 0. By Theorem 3.5, we already know that L^{t}(·) is lower semicontinuous on W^{1,1}((0, t), ℝ^{N}) with respect to the uniform convergence. This and the inequality L^{t}(·) ≤ F^{t}(·), which is apparent from the definition, implies in particular that

L^{t}(·) ≤ \overline{F}^{t}(·) on W^{1,1}((0, t), ℝ^{N}).

To prove the opposite inequality, we will show that, for any fixed γ ∈ W^{1,1}((0, t), ℝ^{N}), there exists a sequence of curves (γ_{k})_{k} ⊂ W^{1,1}((0, t), ℝ^{N}) with γ_{k}(0) = γ(0), γ_{k}(t) = γ(t) such that

lim sup_{k→+∞} F^{t}(γ_{k}) ≤ L^{t}(γ)  and  lim_{k→+∞} ∥γ_{k} − γ∥_{∞} = 0.

Of course, we may assume that L^{t}(γ) < +∞. Fix k ∈ ℕ and choose a finite partition 0 = t_{0} < t_{1} < ⋯ < t_{m} = t with |t_{i+1} − t_{i}| < 1/k for each i, such that

∑_{i=0}^{m−1} d(t_{i}, γ(t_{i}), t_{i+1}, γ(t_{i+1})) < L^{t}(γ) + \frac{1}{k} .

By the definition of d, it is easy to see that there exists a curve γ_{k} ∈ W^{1,1}((0, t), ℝ^{N}) with γ_{k}(t_{i}) = γ(t_{i}) for each i such that

F^{t}(γ_{k}) < L^{t}(γ) + \frac{1}{k} .

Let us show that ∥γ_{k} − γ∥_{∞} → 0 as k → ∞. Let α_{1} > 0 such that α(|q|) ≥ |q| − α_{1} for any q ∈ ℝ^{N}. For each i = 0, …, m − 1, we have

∫_{t_{i}}^{t_{i+1}} |γ̇_{k}(s)| ds ≤ L_{t_{i}}^{t_{i+1}}(γ) + \frac{1 + α_{1}}{k} =: ρ_{k},

so γ_{i} ([t_{i}, t_{i+1}]) ⊂ γ ([t_{i}, t_{i+1}])+B_{ρ_{k}}. The conclusion follows since ρ_{k} → 0 for k → +∞ by the absolute continuity of the map s ↦ L^{s}(γ). □

In view of what seen so far, we also derive the following

Theorem 3.10. Let F ∈ A and assume the associated function d_{F} satisfies condition (*). Then its lower semicontinuous functional \overline{F} admits the following integral representation:

\overline{F}^{t}(γ) = ∫_{0}^{t} F^{−}(γ, γ̇) ds = ∫_{0}^{t} F^{+}(γ, γ̇) ds, γ ∈ W^{1,1}((0, t), ℝ^{N}) and t > 0,

where

F^{−}(x, q) := lim inf_{h→0^{+}} \frac{d_{F}((0, x), (h, x + hq))}{h},

F^{+}(x, q) := lim sup_{h→0^{+}} \frac{d_{F}((0, x), (h, x + hq))}{h} .

Moreover, the functions F^{±} enjoy properties (i)–(v) in the statement of Theorem 3.8.

Proof. By assumption, d((0, ·), (·, ·)) is locally Lipschitz in ℝ^{N} × (0, +∞) × ℝ^{N}, in particular it is continuous. It follows from the definitions (cf. (9)) that d_{F} = d\overline{F} on (ℝ × ℝ^{N}) × (ℝ × ℝ^{N}), and the statement follows by Theorem 3.8. □

Remark 3.11. When α and β have the same kind of growth at infinity, we know by Remark 2.6 that d\overline{F} = d_{F} on (ℝ × ℝ^{N}) × (ℝ × ℝ^{N}), hence from Theorem 3.8 we derive that condition (*) holding for d = d_{F} is necessary as well in order to have an integral representation result for \overline{F}. This might be no longer true in general. In other words, there might exist functionals F ∈ A whose associated functions d_{F} are not locally Lipschitz, while d\overline{F} are.

Remark 3.12. All results of this section can be generalized to functionals F which satisfy, in place of (F4), the following assumption:

for every n ∈ ℕ and a, b ∈ ℝ

∫_{a}^{b} α(|γ̇|) dt ≤ F_{a}^{b}(γ) ≤ ∫_{a}^{b} β_{n}(|γ̇|) dt for any γ ∈ W_{loc}^{1,1}(ℝ, B_{n}).

(F4′)

where (β_{n})_{n∈ℕ} is a family of convex, non–decreasing and superlinear functions from ℝ_{+} to ℝ_{+}. To prove Theorems 3.8 and 3.10, and Proposition 3.9 in this more general setting, we notice that the function d_{F} associated to any such F enjoys assertions (d1)–(d3) of Proposition 2.5, and assumption (d4′) of Remark 3.7, which is immediately obtained by choosing a = 0, b = t and γ(s) = x + s(y − x)/t in (F4′) for every x, y ∈ B_{n} and n ∈ ℕ. By taking into account Remark 3.7, we can easily conclude via the same arguments. Of course, claim (ii) in Theorem 3.10 must be modified as follows:

(ii)′  α (|q|) ≤ L^{±}(x, q) ≤ β_{n} (|q|)  for any (x, q) ∈ B_{n} × ℝ^{N} and n ∈ ℕ.

4. A class of integral functionals

In this Section we will show that condition (*) is enjoyed by a wide class of (non lower semicontinuous) integral functionals, including in particular the ones associated to integrands satisfying conditions (i)–(v) in the statement of Theorem 3.8. By this mean, we will in particular single out a family of integral functionals which is closed with respect to the Γ–convergence. More precisely, we consider an autonomous Lagrangian L : ℝ^{N} × ℝ^{N} → ℝ_{+} which satisfies the following assumptions:

(L1) L is Borel–measurable on ℝ^{N} × ℝ^{N},

(L2) α (|q|) ≤ L(x, q) ≤ β (|q|) for all (x, q) ∈ ℝ^{N} × ℝ^{N},

(L3) for every for every t_{1} < t_{2} in ℝ and γ ∈ W^{1,∞}((t_{1}, t_{2}), ℝ^{N})

λ ↦ L(γ(s), λ γ̇(s)) is convex on ℝ for a.e. s ∈ (t_{1}, t_{2}).

Remark 4.1. The condition L(x, q) ≤ β (|q|) for every (x, q) ∈ ℝ^{N} × ℝ^{N} is not so restrictive as it might appear. Indeed, it amounts to saying that (cf. Lemma 2.3 in [17])

sup {L(x, q) : (x, q) ∈ ℝ^{N} × B_{R} } < +∞ for any R > 0.

Condition (L3) is in particular satisfied when L(x, ·) is convex for every x ∈ ℝ^{N}.

We define the associated action functional 𝕃 : W_{loc}^{1,1}(ℝ; ℝ^{N}) × I(ℝ) → [0, +∞] by setting

𝕃(γ, (a, b)) := ∫_{a}^{b} L(γ, γ̇) ds,  γ ∈ W^{1,1}((a, b), ℝ^{N}),  (21)

for any (a, b) ∈ I(ℝ). Clearly, 𝕃 is a functional belonging to A.

The main goal of this section will be proving the following result.

Theorem 4.2. Let L : ℝ^{N} × ℝ^{N} → ℝ_{+} be an autonomous Lagrangian satisfying conditions (L1)–(L3), and 𝕃 the integral functional defined via (21). Then the associated function d = d_{𝕃} defined through (8) satisfies condition (*). More precisely, for every M > 0 there exists K = K(M, α, β) such that

d((0, ·), (·, ·)) is K–Lipschitz continuous in \overline{C_{M}},

where C_{M} := {(y, t, x) ∈ ℝ^{N} × (0, +∞) × ℝ^{N} : |x − y| < M t }.

Looking back at the proof of Theorem 3.8, we see that Theorem 4.2 will easily follow via a similar argument as soon as we get some a priori estimates on the Lipschitz constant of quasi–optimal curves parametrized in (0, t) and connecting y to x, for every (y, t, x) ∈ C_{M}. That is precisely the content of Lemma 4.16. Its proof relies on a careful analysis on the role played by reparametrizations, which will be carried out in the next two subsections.

4.1. Preliminary tools. We start by introducing a piece of notation. Set

Ω := {(x, q) ∈ ℝ^{N} × ℝ^{N} : λ ↦ L(x, λq) is convex on ℝ }

and

C := Ω ∩ (ℝ^{N} × 𝕊^{N−1}).

For every (x, q) ∈ Ω, let f(x, q, ·) be the conjugate of the map λ → L(x, λq), namely

f(x, q, u) := max_{λ∈ℝ}{uλ − L(x, λ q) } for every u ∈ ℝ,

which is convex and superlinear, due to (L2). Proposition 2.2 implies that

L(x, λ q) = f^{∗}(x, q, λ) := max_{u∈ℝ}{λ u − f(x, q, u) }

for every λ ∈ ℝ and (x, q) ∈ Ω, in particular

min_{ℝ}f(x, q, ·) = −L(x, 0) for any (x, q) ∈ Ω.

For any a ∈ ℝ, let us define

σ_{a}(x, q) := max { u : f(x, q, u) ≤ a } for any (x, q) ∈ ℝ^{N} × ℝ^{N}.

The set appearing above is void whenever a < −L(x, 0). In this case, we agree that σ_{a}(x, q) = −∞. The definition of σ_{a} can be extended to the whole ℝ^{N} × ℝ^{N} by setting σ_{a}(x, q) = −∞ for every (x, q) ∉ Ω.

Remark 4.3. Under the additional assumption that L(x, ·) is convex for any fixed x ∈ ℝ^{N}, the definition of σ_{a} given above reduces to

σ_{a}(x, q) := max{⟨q, p⟩ : H(x, p) ≤ a} for every q ∈ ℝ^{N}, a ∈ ℝ,

where H : ℝ^{N} × ℝ^{N} → ℝ is the Hamiltonian associated to L through the Fenchel transform, namely

H(x, p) := max_{q∈ℝ^{N}} {⟨p, q⟩ − L(x, q)} .

In this case, c(x, q) is actually independent of q and coincides with min_{ℝ^{N}} H(x, ·). In the above definition it is understood that σ_{a}(x, q) = −∞ whenever a < min_{ℝ^{N}} H(x, ·).

Proposition 4.4. For any a ∈ ℝ, the following properties hold:

(i) σ_{a}(x, λ q) = λ σ_{a}(x, q) for every (x, q) ∈ ℝ^{N} × ℝ^{N} and λ > 0;

(ii) L(x, q) ≥ σ_{a}(x, q) − a for every (x, q) ∈ ℝ^{N} × ℝ^{N}.

Proof. To prove (i), we can assume (x, q) ∈ Ω and a ≥ −L(x, 0), being the statement otherwise trivial by definition of σ_{a}. Then the equality f(x, λ q, u) = f(x, q, u/λ), which holds true for every u ∈ ℝ, implies

σ_{a}(x, λ q) = max { u : f(x, q, u/λ) ≤ a } = max { λ u : f(x, q, u) ≤ a } = λ σ_{a}(x, q).

Let us prove (ii). Up to trivial cases, we can assume (x, q) ∈ Ω and a ≥ −L(x, 0). By Proposition 2.2, we get

L(x, q) = max_{u∈ℝ}{λ u − f(x, q, u) } ≥ max_{f(x,q,u)≤a} {λ u − a } = σ_{a}(x, q) − a,  (22)

as claimed. □

For any (x, q) ∈ Ω and a ∈ ℝ, we set

Λ_{a}(x, q) := { λ ∈ [0, +∞) : L(x, λ q) = σ_{a}(x, λ q) − a },  (23)

and

\underline{λ}_{a}(x, q) := inf Λ_{a}(x, q), \overline{λ}_{a}(x, q) := sup Λ_{a}(x, q).

We agree that \underline{λ}_{a}(x, q) = \overline{λ}_{a}(x, q) = 0 whenever Λ_{a}(x, q) = ∅, that is, when either q ≠ 0 and a < −L(x, 0), or q = 0 and a ≠ −L(x, 0). Last, we extend the functions \underline{λ}_{a}, \overline{λ}_{a} to the whole ℝ^{N} × ℝ^{N} by setting

\underline{λ}_{a}(x, q) = \overline{λ}_{a}(x, q) = −∞ whenever (x, q) ∉ Ω.

We define the following functions:

α_{∗}(u) := max_{λ∈ℝ}{uλ − α(|λ|) } , β_{∗}(u) := max_{λ∈ℝ}{uλ − β(|λ|) } for every u ∈ ℝ,

and we remark that they are convex and superlinear as α(| · |) and β(| · |) are so. For every a ∈ ℝ, set

R_{a} := max {|u| : β_{∗}(u) ≤ a }  (24)

and

κ_{a} := 2 max {α_{∗}(u) : |u| ≤ R_{a} + 1 } .  (25)

The following compactness result holds.

Lemma 4.5. Let a ∈ ℝ. Then

{u ∈ ℝ : f(x, q̂,u) ≤ a } ⊆ B_{R_{a}},  Λ_{a}(x, q̂) ⊆ B_{κ_{a}}  for every (x, q̂) ∈ C.

Proof. The first assertion follows at once from the fact that β^{∗}(·) ≤ f(x, q̂,·) for every (x, q̂) ∈ Ω. To prove the second one, pick up (x, q̂) ∈ C and set f(·) := f(x, q̂, ·). From Proposition 2.2 we infer that λ ∈ Λ_{a}(x, q̂) if and only if λ ∈ ∂f(u) for some f(u) ≤ a, so

Λ_{a}(x, q̂) ⊆ {∂f(u) : u ∈ B_{R_{a}} } .

As f(·) ≤ α^{∗}(·), we conclude thanks to Proposition 2.1. □

Now we fix (x, q̂) ∈ C and we examine the properties of the multifunction a ↦ Λ_{a}(x, q̂). To ease notation, we will write g(λ) and f(u) in place of L(x, λ q̂) and f(x, q̂, u), respectively. The duality between f and g implies, by Proposition 2.2, that

g(λ) = λ u − f(u)  for every u ∈ ∂g(λ),  (26)

for any given λ ∈ ℝ. In view of (22), we infer that λ ∈ Λ_{a}(x, q̂) if and only if a ∈ f (∂g(λ)). We start by considering the set–valued map A(λ) := f (∂g(λ)) on [0, +∞), which is the inverse of a ↦ Λ_{a}(x, q̂), in the sense of set–valued analysis (see [27, Chapter 5]). Indeed, note that

Λ_{a}(x, q̂) = { λ ∈ [0, +∞) : a ∈ A(λ) }.  (27)

We also remark for further use that, by classical results of non–smooth analysis (cf. [12, Theorem 2.3.10]),

∂g(λ) = ⟨∂_{q}L(x, λ q̂), q̂⟩  for every λ ∈ ℝ.  (28)

Proposition 4.6. Let A(·) as above. The following facts hold.

(i) For any λ ∈ ℝ

A(λ) = [\underline{a}(λ), \overline{a}(λ)] for some −L(x, 0) ≤ \underline{a}(λ) ≤ \overline{a}(λ) < +∞.

(ii) The set–valued map A(·) is upper semicontinuous on [0, +∞). In particular, \underline{a}(·) is lower semicontinuous and \overline{a}(·) is upper semicontinuous on [0, +∞). Moreover

A(0) = \{ −L(x, 0) \},  lim_{λ→+∞} \underline{a}(λ) = +∞.

(iii) The set–valued map λ ↦ A(λ) is non–decreasing on [0, +∞).

(iv) ⋃_{λ≥0} A(λ) = [−L(x, 0), +∞).

Proof. As g is convex, its subgradient ∂g(λ) is a compact interval of ℝ, so the same is true for A(λ). That proves (i). The upper semicontinuity of A(·) comes from the fact that the multifunction λ ↦ ∂g(λ) is upper semicontinuous and f is continuous. The equality A(0) = {−L(x, 0)} is an immediate consequence of (26) and of the fact that g(0) = f^{∗}(0) = L(x, 0). The remainder of (ii) follows by definition of \underline{a}(·), \overline{a}(·) and by superlinearity of f and g.

Let us prove (iii). Since f and g are convex, the multimappings u ↦ ∂f(u) and λ ↦ ∂g(λ) are non–decreasing on ℝ. By superlinearity, we get in particular

⋃_{λ≥0} ∂g(λ) = [\underline{u}(0), +∞) with \underline{u}(0) ∈ ∂g(0).

By duality (cf. Proposition 2.2), 0 ∈ ∂f(\underline{u}(0)), so the monotonicity of ∂f(·) yields that f is non–decreasing on [\underline{u}(0), +∞).

Item (iv) comes from (ii) and (iii). □

We use this information to prove a result that will be crucial for our future analysis.

Proposition 4.7. Let (x, q̂) ∈ C. The following facts hold.

(i) For any a ≥ −L(x, 0), we have

Λ_{a}(x, q) = [\underline{λ}_{a}(x, q), \overline{λ}_{a}(x, q)] for some 0 ≤ \underline{λ}_{a}(x, q) ≤ \overline{λ}_{a}(x, q) < +∞.

Moreover,

\underline{λ}_{−L(x,0)}(x, q) = 0,  lim_{a→+∞} \underline{λ}_{a}(x, q) = +∞.

(ii) The set–valued map a ↦ Λ_{a}(x, q) is upper semicontinuous and non–decreasing on [−L(x, 0), +∞).

(iii)  \underline{λ}_{a}(x, q) = sup_{b<a} \overline{λ}_{b}(x, q)  for any a > −L(x, 0)

and

\overline{λ}_{a}(x, q) = inf_{b>a} \underline{λ}_{b}(x, q) for any a ≥ −L(x, 0).

(iv) \underline{λ}_{a}(x, q) ≥ \frac{a + L(x, 0)}{2R_{a}} for any a > −L(x, 0), with R_{a} defined by (24).

Proof. We recall that Λ_{a}(x, q̂) = {λ ≥ 0 : a ∈ A(λ) } . The monotonicity property of the set–valued map a ↦ Λ_{a}(x, q̂) is a consequence of Proposition 4.6-(iii). In particular, Λ_{a}(x, q̂) is a bounded interval for any a ≥ −L(x, 0).

To prove the upper semicontinuity of a ↦ Λ_{a}(x, q̂), we need to show that, for each pair of sequences (a_{n})_{n} and (λ_{n})_{n} such that a_{n} → a ∈ ℝ, λ_{n} → λ ∈ ℝ and λ_{n} ∈ Λ_{a_{n}}(x, q̂) for every n ∈ ℕ, we have λ ∈ Λ_{a}(x, q̂). That easily follows by the upper semicontinuity of A(·). In particular, this implies that Λ_{a}(x, q̂) is closed for any a ≥ −L(x, 0).

The equality \underline{λ}_{−L(x,0)}(x, q̂) = 0 is a trivial consequence of definition (23). The coercivity of a ↦ \underline{λ}_{a}(x, q̂) comes from Proposition 4.6-(ii). Item (iii) immediately follows from the monotone and semicontinuous character of the map a ↦ Λ_{a}(x, q̂).

Let us prove (iv). Choose a > −L(x, 0) and set λ := \underline{λ}_{a}(x, q̂). By Proposition 4.4-(ii) we get

σ_{a}(x, λq̂) = L(x, λq̂) + a ≥ σ_{−L(x,0)}(x, λq̂) + a + L(x, 0),

hence, by Lemma 4.5,

a + L(x, 0) ≤ λ (σ_{a}(x, q̂) − σ_{−L(x,0)}(x, q̂)) ≤ λ (R_{a} + R_{−L(x,0)}) |q̂|,

and the statement follows as R_{−L(x,0)} < R_{a} by definition. □

4.2. Optimal reparametrizations. Let us now consider a Lipschitz curve γ defined on a bounded interval J := (0, ℓ).

Definition 4.8. A curve ξ defined on a bounded interval (0, t) is said to be a reparametrization of γ if there exists an absolutely continuous map φ : [0, t] → [0, ℓ], surjective and non–decreasing, such that

ξ = γ∘φ on (0, t).

We furthermore say that ξ is a (bi–)Lipschitz reparametrization of γ if φ is a (bi–) Lipschitz homeomorphism.

Remark 4.9. For reasons that will be clear soon, we want to allow a reparametrization to stop at a point for some time. This accounts for the choice of the unusual definition given above.

We introduce the following notation:

[γ]_{t} := {ξ ∈ W^{1,1}((0, t), ℝ^{N}) : ξ is a reparametrization of γ }
[γ]_{t}^{♭} := {ξ ∈ W^{1,1}((0, t), ℝ^{N}) : ξ is a bi–Lipschitz reparametrization of γ }.

The following lemma comes from classical results of analysis in metric spaces (see e.g. Section VII.2 in [21]).

Lemma 4.10. Let ξ ∈ W^{1,1}((0, t), ℝ^{N}). Then there exists a Lipschitz curve γ, defined on a bounded interval (0, ℓ), such that ξ ∈ [γ]_{t}. We can furthermore assume that γ is parametrized by arc–length.

A further step in the analysis is carried out by picking up some special reparametrizations of the curve γ.

Definition 4.11. A curve ξ defined on a bounded interval (0, t) is said to have an a–Lagrangian parametrization if

L(ξ(s), ξ̇(s)) = σ_{a}(ξ(s), ξ̇(s)) − a for a.e. s ∈ (0, t), a ∈ ℝ.

For any a ∈ ℝ and t > 0, we define

[γ](a, t) := {ξ ∈ [γ]_{t} : ξ has an a–Lagrangian parametrization },
[γ]^{♭}(a, t) := {ξ ∈ [γ]_{t}^{♭} : ξ has an a–Lagrangian parametrization }.

Now assume γ parametrized by arc–length, and let

c_{γ} := ess sup_{s∈J} − L (γ(s), 0)

We define a multifunction T_{γ} : (c_{γ}, +∞) → P(ℝ_{+}) by setting

T_{γ}(a) := {t > 0 : [γ]^{♭}(a, t) is non–empty }.

The properties of the multifunction T_{γ}(·) are stated below.

Proposition 4.12. Let γ and T (·) := T_{γ}(·) as above. The following facts hold. (i) For any a > c_{γ}, T (a) is a compact interval in (0, +∞), namely

T (a) := [\underline{T} (a), \overline{T} (a)] for some \overline{T} (a) ≥ \underline{T} (a) > 0.

(ii) The multifunction T (·) is non–increasing and upper semicontinuous on (c_{γ}, +∞). Moreover inf_{a>c_{γ}} \overline{T} (a) = 0.

(iii) Let \underline{T} (c_{γ}) := sup_{a>c_{γ}} \overline{T} (a). If \underline{T} (c_{γ}) is finite, then [γ](c_{γ}, \underline{T} (c_{γ})) ≠ ∅.

In particular, for any 0 < t ≤ \underline{T} (c_{γ}) with t < +∞, there exists a ≥ c_{γ} such that γ admits an a–Lagrangian Lipschitz reparametrization on (0, t).

We first prove an auxiliary lemma.

Lemma 4.13. Let γ : (0, ℓ) → ℝ^{N} be a Lipschitz curve parametrized by arc–length and a ∈ ℝ. The following facts hold true.

(i) For every t > 0 and ξ ∈ [γ]_{t}, the map σ_{a}(ξ(·), ξ̇(·)) is Lebesgue–measurable on (0, t), and

∫_{0}^{t} σ_{a}(ξ(s), ξ̇(s)) ds = ∫_{0}^{ℓ} σ_{a}(γ(s), γ̇(s)) ds.  (29)

(ii) The maps \underline{λ}_{a}(γ(·), γ̇(·)), \overline{λ}_{a}(γ(·), γ̇(·)) are Lebesgue–measurable on (0, ℓ).

Proof. Let Ω := {(x, λ q̂) : (x, q̂) ∈ C, λ ∈ ℝ }. Take t > 0 and ξ ∈ [γ]_{t}. In order to prove (i), it suffices to define a function σ̃_{a} such to be Borel–measurable on ℝ^{N} × ℝ^{N} and coinciding with σ_{a} on Ω. Indeed the map s ↦ (ξ(s), ξ̇(s)) is Lebesgue measurable and takes values in Ω for a.e. s ∈ (0, t) by assumption (L3), hence the function σ_{a}(ξ(·), ξ̇(·)) coincides, almost everywhere on (0, ℓ), with σ̃_{a}(γ(·), γ̇(·)), which is Lebesgue–measurable as a composition of a Borel–measurable map with a Lebesgue–measurable one.

To this aim, let us denote by (λ_{n})_{n} and (u_{n})_{n} two dense sequences in ℝ with 0 ∈ (u_{n})_{n}, and define, for every (x, q) ∈ ℝ^{N} × ℝ^{N} and a ∈ ℝ,

f̃(x, q, u) := max_{n∈ℕ}{uλ_{n} − L(x, λ_{n} q) } for every u ∈ ℝ,

and

σ̃_{a}(x, q) := inf_{k} (sup_{n} {u_{n} ϑ_{E_{n}^{k}}(x, q)}) ,

where E_{n}^{k} := {(x, q) ∈ ℝ^{N} × ℝ^{N} : f̃(x, q, u_{n}) ≤ a + 1/k } and ϑ_{E_{n}^{k}} denotes the function identically 1 on E_{n}^{k} and −∞ elsewhere. The function σ̃_{a} is Borel–measurable on ℝ^{N} × ℝ^{N} for any a ∈ ℝ by construction. Moreover

σ_{a}(x, q) = σ̃_{a}(x, q) for every (x, q) ∈ Ω,

which holds true since f̃(x, q, ·) = f(x, q, ·) by the continuity of the map λ ↦ L(x, λ q). The equality (29) is a consequence of the fact that σ_{a}(x, ·) is positively 1–homogeneous.

To prove (ii), we notice that the map s ↦ (γ(s), γ̇(s)) takes values in C for a.e. s ∈ (0, ℓ), due to assumption (L3) and to the fact that it is parameterized by arc– length, hence it suffices to define two functions \underline{λ̃}_{a}, \overline{λ̃}_{a} such to be Borel–measurable on ℝ^{N} × 𝕊^{N−1} and coinciding on C with \underline{λ}_{a}, \overline{λ}_{a}, respectively.

For each n ∈ ℕ, let

F_{n} := {(x, q) ∈ ℝ^{N} × ℝ^{N} : f̃(x, q, ⟨∂_{q}L(x, λ_{n} q), q⟩) ∩ (−∞, a) ≠ ∅},

which is Borel measurable for the multifunction (x, q) ↦ f̃(x, q, ⟨∂_{q}L(x, λ_{n} q), q⟩) is so. Set

\underline{λ̃}_{a}(x, q) := sup_{n} λ_{n} χ_{F_{n}} (x, q) for every (x, q) ∈ ℝ^{N} × 𝕊^{N−1}.

Since f̃(x, q, ·) = f(x, q, ·) for every (x, q) ∈ C, we conclude that \underline{λ̃}_{a}(x, q) = \underline{λ}_{a}(x, q) on C in view of (27), (28) and of Proposition 4.7. The analogous statement for \overline{λ}_{a} can be proved in a similar way. □

Proof of Proposition 4.12. (i) Fix a > c_{γ}, and set

\underline{λ}_{a}(ς) := \underline{λ}_{a}(γ(ς), γ̇(ς)), \overline{λ}_{a}(ς) := \overline{λ}_{a}(γ(ς), γ̇(ς)) for a.e. ς ∈ (0, ℓ).

Let

\underline{T} (a) := ∫_{0}^{ℓ} \frac{1}{\overline{λ}_{a}(ς)} dς, \overline{T} (a) := ∫_{0}^{ℓ} \frac{1}{\underline{λ}_{a}(ς)} dς.

Such quantities are well defined, positive real values, thanks to Proposition 4.7-(iv) and to the measurable character of \underline{λ}_{a}(·), \overline{λ}_{a}(·). To show that they belong to T (a), we will prove the existence of two curves \underline{γ}_{a}, \overline{γ}_{a}, defined on (0, \underline{T} (a)) and (0, \overline{T} (a)),

23

respectively, which are a–Lagrangian bi–Lipschitz reparametrizations of γ. To this aim, let us define

\underline{f} _{a}(s) := ∫_{0}^{s} \frac{1}{\overline{λ}_{a}(ς)} dς, \overline{f}_{a}(s) := ∫_{0}^{s} \frac{1}{\underline{λ}_{a}(ς)} dς for any s ∈ (0, ℓ),

and set

\underline{φ}_{a} := (\underline{f} _{a})^{−1}, \overline{φ}_{a} := (\overline{f}_{a})^{−1} ,

defined on (0, \underline{T} (a)) and (0, \overline{T} (a)), respectively. As

˙\underline{φ}_{a}(τ) = \overline{λ}_{a}(\underline{φ}_{a}(τ)),  ˙  \overline{φ}_{a}(τ) = \underline{λ}_{a}(\overline{φ}_{a}(τ))  for a.e. τ,

we immediately derive that \underline{φ}_{a} and \overline{φ}_{a} are order–preserving bi–Lipschitz diffeomorphisms. Let us set

\underline{γ}_{a} := γ ∘ \underline{φ}_{a} on (0, \underline{T} (a)), \overline{γ}_{a} := γ ∘ \overline{φ}_{a} on (0, \overline{T} (a)).

Since

˙\underline{γ}_{a}(·) := \overline{λ}_{a}(\underline{φ}_{a}(·)) γ̇(\underline{φ}_{a}(·)) a.e. on (0, \underline{T} (a))

and

˙\overline{γ}_{a}(·) := \underline{λ}_{a}(\overline{φ}_{a}(·)) γ̇(\overline{φ}_{a}(·)) a.e. on (0, \overline{T} (a)),

we conclude that the curves \underline{γ}_{a} and \overline{γ}_{a} has an a–Lagrangian parametrization by the very definition of \overline{λ}_{a} and \underline{λ}_{a}.

In order to prove that [\underline{T} (a), \overline{T} (a)] ⊆ T (a), we will show that

δ \underline{T} (a) + (1 − δ) \overline{T} (a) ∈ T (a)  for any δ ∈ (0, 1).  (30)

Fix δ ∈ (0, 1), and set

δ(ς) := \frac{δ \overline{λ}_{a}(ς)}{δ \overline{λ}_{a}(ς) + (1 − δ) \underline{λ}_{a}(ς)}, λ(ς) := δ(ς) \underline{λ}_{a}(ς) + (1 − δ(ς)) \overline{λ}_{a}(ς)

for almost every ς ∈ (0, ℓ), and

f(s) := ∫_{0}^{s} \frac{1}{λ(ς)} dς for s ∈ (0, ℓ), φ := f^{−1} on [0, f(ℓ)].

Since δ(ς) ∈ [0, 1] for almost every ς ∈ (0, ℓ), we get that λ_{a}(ς) ∈ Λ_{a}(γ(ς), γ̇(ς)) for almost every ς ∈ (0, ℓ), in particular φ is an order–preserving bi–Lipschitz diffeomorphism. Arguing as above, we see that the curve γ_{a} := γ ∘ φ is an a–Lagrangian bi–Lipschitz reparametrization of γ on (0, f(ℓ)), so f(ℓ) ∈ T (a). Now it is easy to check, by definition of δ(·), that f(ℓ) = δ \overline{T} (a) + (1 − δ) \underline{T} (a). That proves (30) as δ was arbitrarily chosen in (0, 1).

Let us now prove that T (a) ⊆ [\underline{T} (a), \overline{T} (a)]. Let T ∈ T (a) and γ̃ := γ ∘ φ be an a–Lagrangian reparametrization of γ for some order–preserving bi–Lipschitz diffeomorphism φ : (0, T) → (0, ℓ). Then

φ̇(τ) ∈ Λ_{a} (γ(φ(τ)), γ̇(φ(τ))) for a.e. τ ∈ (0, T).

Let f := φ^{−1}. We have

T = f(ℓ) = ∫_{0}^{ℓ} ḟ(ς) dς = ∫_{0}^{ℓ} \frac{1}{φ̇(f(ς))} dς,

and since φ̇(f(ς)) ∈ Λ_{a}(γ(ς), γ̇(ς)) = [\underline{λ}_{a}(ς), \overline{λ}_{a}(ς)] for a.e. ς ∈ (0, ℓ), we clearly get T ∈ [\underline{T} (a), \overline{T} (a)].

(ii) Let b > a > c_{γ}. Then \underline{λ}_{b}(ς) ≥ \overline{λ}_{a}(ς) for almost every ς ∈ (0, ℓ), hence \overline{T} (b) ≤ \underline{T} (a). That proves that T (·) is a non–increasing multifunction. To prove that T (·) is u.s.c. on (c_{γ}, +∞), it will be enough to show that

\underline{T} (a) = sup_{b>a} \overline{T} (b),  \overline{T} (a) = inf_{b<a} \underline{T} (b)  for any a > c_{γ}.

This actually follows as a simple application of the Monotone Convergence Theorem and by the monotonicity poperties of \underline{λ}_{a}, \overline{λ}_{a} (cf. Proposition 4.7-(iii)). The last assertion holds by definition of \underline{T} (a) since sup_{a>c_{γ}} \overline{λ}_{a}(ς) = +∞ for almost every ς ∈ (0, ℓ).

(iii) Let \underline{T} (c_{γ}) be finite. Arguing as in (i), we may find a non–increasing sequence of Borel–measurable maps λ_{n} : (0, ℓ) → [0, +∞) such that, for each n ∈ ℕ,

T_{n} = ∫_{0}^{ℓ} \frac{1}{λ_{n}(ς)} dς  and  λ_{n}(ς) ∈ Λ_{c_{γ}+1/n}(γ(ς), γ̇(ς))  for a.e. ς ∈ (0, ℓ),

with sup_{n} T_{n} = \underline{T} (c_{γ}). Set

λ(ς) = inf_{n} λ_{n}(ς) for every ς ∈ (0, ℓ).

Then λ(·) is measurable and λ(ς) ∈ Λ_{c_{γ}}(γ(ς), γ̇(ς)) for almost every ς ∈ (0, ℓ). Moreover the Monotone Convergence Theorem yields

\underline{T} (c_{γ}) = sup_{n∈ℕ} T_{n} = sup_{n∈ℕ}∫_{0}^{ℓ} \frac{1}{λ_{n}(ς)} dς = ∫_{0}^{ℓ} \frac{1}{λ(ς)} dς,

in particular the map

f(s) := ∫_{0}^{s} \frac{1}{λ(ς)} dς

is increasing and absolutely continuous on (0, ℓ). A c_{γ}–Lagrangian Lipschitz reparametrization of γ defined on (0, \underline{T} (c_{γ})) can be now obtained by setting γ̃ := γ ∘ φ with φ := (f)^{−1} on (0, \underline{T} (c_{γ})).

Last, the fact that the multifunction is upper semicontinuous, monotone and convex–set–valued implies that

⋃_{a>c_{γ}} T (c_{γ}) = (0, \underline{T} (c_{γ}))

and this is enough to obtain the remainder of the statement. □

We now seek for an optimal reparametrization of γ on the interval (0, t), for any given t ∈ (0, +∞). Such a reparametrization does not exist in general, as we will see. In any case, however, we are able to derive an estimate on the Lipschitz constants of quasi–optimal reparametrizations. This is a crucial step for our study.

Theorem 4.14. Let γ : (0, ℓ) → ℝ^{N} be a Lipschitz curve parametrized by arc–length. Then, for every t ∈ (0, +∞), there exists a ≥ c_{γ} such that

inf_{ξ∈[γ]_{t}} ∫_{0}^{t} L(ξ, ξ̇) ds = inf_{ξ∈[γ]_{t}} {∫_{0}^{t} L(ξ, ξ̇) ds : ∥ξ̇∥_{∞} ≤ κ_{a}} = ∫_{0}^{ℓ} σ_{a}(γ, γ̇) ds − a t,

with κ_{a} given by (25). The above infimum is a minimum whenever t ≤ \underline{T}_{γ}(c_{γ}), and is in particular attained by some curve belonging to [γ]^{♭}(a, t) with a > c_{γ} when t < \underline{T}_{γ}(c_{γ}).

Proof. By Proposition 4.4 and Lemma 4.13, we get

∫_{0}^{t} L(ξ, ξ̇) ds ≥ ∫_{0}^{t} (σ_{a}(ξ, ξ̇) − a) ds = ∫_{0}^{ℓ} σ_{a}(γ, γ̇) ds − a t  (31)

for any a ≥ c_{γ} and ξ ∈ [γ]_{t}, and (31) is an equality whenever ξ ∈ [γ](a, t). The assertion for t ≤ \underline{T}_{γ}(c_{γ}) hence follows in force of Proposition 4.12 and Lemma 4.5.

Let us now assume t > \underline{T}_{γ}(c_{γ}) and set h := t − \underline{T}_{γ}(c_{γ}). Let ξ ∈ [γ](c_{γ}, \underline{T}_{γ}(c_{γ})). By definition of c_{γ}, there exists, for each n ∈ ℕ, s_{n} ∈ (0, \underline{T}_{γ}(c_{γ})) such that

c_{γ} + L(γ(s_{n}), 0) < \frac{1}{n}.

To ease notation, we will write c_{n} in place of −L(γ(s_{n}), 0). We define

ξ_{n}(s) := \{ \begin{matrix}ξ(s) & if s ∈ (0, s_{n}] \\ ξ(s_{n}) & if s ∈ [s_{n}, s_{n} + h] \\ ξ(s − h) & if s ∈ [s_{n} + h, t). \end{matrix}

We have

∫_{0}^{t} L(ξ_{n}, ξ̇_{n}) ds = ∫_{0}^{\underline{T}_{γ}(c_{γ})} L(ξ, ξ̇) ds − h c_{n} = ∫_{0}^{\underline{T}_{γ}(c_{γ})} σ_{c_{γ}}(ξ, ξ̇) ds − \underline{T}_{γ}(c_{γ}) c_{γ}

−h c_{n} = ∫_{0}^{ℓ} σ_{c_{γ}}(γ, γ̇) ds − c_{γ} t + h (c_{γ} − c_{n}) < ∫_{0}^{ℓ} σ_{c_{γ}}(γ, γ̇) ds − c_{γ} t + \frac{h}{n}.

Taking (31) into account, we derive

∫_{0}^{ℓ} σ_{c_{γ}}(γ, γ̇) ds − c_{γ} t ≤ ∫_{0}^{t} L(ξ_{n}, ξ̇_{n}) ds < ∫_{0}^{ℓ} σ_{c_{γ}}(γ, γ̇) ds − c_{γ} t + \frac{h}{n},

and we conclude letting n → +∞. □

Remark 4.15. If in Theorem 4.14 the Lagrangian is assumed lower semicontinuous in x, we can furthermore say that, for every t > 0,

inf_{ξ∈[γ]_{t}} ∫_{0}^{t} L(ξ, ξ̇) ds = min {∫_{0}^{t} L(ξ, ξ̇) ds : ξ ∈ [γ](a, t) }

for some constant a ≥ c_{γ}. This can be proved by considering, in place of T_{γ}(·), the set–valued map defined as

T_{γ}^{∗}(a) := {t > 0 : [γ](a, t) is non–empty }

for every a ≥ c_{γ}^{∗} := sup_{s∈J} −L (γ(s), 0). The multifunction T_{γ}^{∗}(·) agrees with T_{γ}(·) on (c_{γ}^{∗}, +∞). Indeed, the inequality

\underline{λ}_{a}(γ(s), γ̇(s)) ≥ \frac{a − c_{γ}^{∗}}{2R_{a}} for a.e. s ∈ (0, ℓ),

which holds true by Proposition 4.7, implies that [γ](a, t) = [γ]^{♭}(a, t) for every a > c_{γ}^{∗} and t > 0 (cf. the argument showing that T (a) ⊆ [\underline{T} (a), \overline{T} (a)] in the proof of Proposition 4.12). On the other hand, we always have

T_{γ}^{∗}(c_{γ}^{∗}) = [\underline{T}_{γ}(c_{γ}^{∗}), +∞)  (32)

when \underline{T}_{γ}(c_{γ}^{∗}) is finite, and that is enough to get the statement in view of (31). To prove (32), let ξ be a curve belonging to [γ](c_{γ}^{∗}, \underline{T} (c_{γ}^{∗})) (which does exist in force of Proposition 4.12) and take s_{0} ∈ [0, \underline{T}_{γ}(c_{γ}^{∗})] such that L(ξ(s_{0}), 0) = −c_{γ}^{∗}. Such an s_{0} always exists by the upper semicontinuity of −L(γ(·), 0) on [0, ℓ]. For every h > 0, define ξ_{h} : [0, \underline{T} (c_{γ}^{∗}) + h] → ℝ^{N} as

ξ_{h}(s) := \{ \begin{matrix}ξ(s) & if s ∈ [0, s_{0}] \\ ξ(s_{0}) & if s ∈ [s_{0}, s_{0} + h] \\ ξ(s − h) & if s ∈ [s_{0} + h, \underline{T}_{γ}(c_{γ}^{∗}) + h]. \end{matrix}

It is easily seen that ξ_{h} is a c_{γ}^{∗}–Lagrangian reparametrization of γ. This shows that

\underline{T}_{γ}(c_{γ}^{∗}) + h ∈ T_{γ}^{∗}(c_{γ}^{∗}) for every h > 0,

as claimed.

4.3. Proof of Theorem 4.2 and further extensions. With the aid of the results obtained so far, we can now establish the following key lemma.

Lemma 4.16. Let x, y ∈ ℝ^{N} and t > 0 such that d((0, y), (t, x)) < M t. Then there exists a constant κ = κ(M, α, β) such that

d((0, y), (t, x)) = inf {∫_{0}^{t} L(ξ, ξ̇) ds : ξ(0) = y, ξ(t) = x, ∥ξ̇∥_{∞} ≤ κ } .

Proof. Choose \overline{n} ∈ ℕ such that M/α(\overline{n}) < 1/2 and set

A = A(\overline{n}) := max\{α_{∗}(u) : |u| ≤ 2 β(\overline{n} + 1) \},

where we recall that α_{∗}(u) := max_{λ∈ℝ}{λu − α(|λ|) }. We claim that the statement holds true with κ := κ_{A} defined according to (25).

Indeed, pick up a curve ξ ∈ W^{1,1}((0, t), ℝ^{N}) such that

∫_{0}^{t} L(ξ, ξ̇) ds < M t,

and let γ : (0, ℓ) → ℝ^{N} be a Lipschitz curve, parametrized by arc–length, such that ξ ∈ [γ]_{t}, according to Lemma 4.10. In view of Theorem 4.14, up to choosing a different ξ in [γ]_{t} without increasing the action, we can always assume that either ∥ξ̇∥_{∞} ≤ κ_{c_{γ}}, or ξ ∈ [γ]^{♭}(a, t) for some a > c_{γ}. In the first case, we note that

c_{γ} ≤ α_{∗}(0),

for −L(x, 0) ≤ −α(0) ≤ α_{∗}(0) for every x ∈ ℝ^{N}. The claim follows by definition of κ_{A} as α_{∗}(0) ≤ A.

Let us instead assume that ξ belongs to [γ]^{♭}(a, t) for some a > c_{γ}. In particular, |ξ̇(s)| ≠ 0 almost everywhere on (0, t). Set J := \{s ∈ (0, t) : 0 < |ξ̇(s)| < \overline{n} \}. We have

M t > ∫_{0}^{t} L(ξ, ξ̇) ds ≥ ∫_{0}^{t} α(|ξ̇|) ds ≥ α(\overline{n}) |(0, t) \ J |,

hence |J| > t/2. Pick up a differentiability point \overline{s} ∈ J for ξ, and denote by f and g the functions u ↦ f(ξ(\overline{s}), ξ̇(\overline{s})/|ξ̇(\overline{s})|, u), λ ↦ L(x, λ ξ̇(\overline{s})/|ξ̇(\overline{s})|), respectively. By the fact that ξ has an a–Lagrangian parametrization we derive that

a ∈ f (∂g( |ξ̇(\overline{s})|)) ,

in particular a ≤ A by Proposition 2.1. As |ξ̇(s)| ∈ Λ_{a}(ξ(s), ξ̇(s)/|ξ̇(s)|) for a.e. s ∈ (0, t), the claim follows in force of Lemma 4.5 since κ_{a} ≤ κ_{A} by (25). □

Proof of Theorem 4.2. For a fixed M > 0, choose (y_{1}, t_{1}, x_{1}) and (y_{2}, t_{2}, x_{2}) in C_{M}, and set

h := |t_{1} − t_{2}| + |x_{1} − x_{2}| + |y_{1} − y_{2}|, s_{0} := \frac{t_{1} − t_{2}}{2} + h.

Since C_{M} is convex, it suffices to prove the statement locally, namely for h ≪ 1. Choose h < t_{2}/2 so that s_{0} < t_{1}/2. Fix ε > 0 and let γ_{1} ∈ W^{1,1}((0, t_{1}), ℝ^{N}) be an ε–minimizer connecting y_{1} to x_{1}. Thanks to Lemma 4.16, we can assume ∥γ̇∥_{∞} ≤ κ_{A} for some constant A = A(M, α, β). Choose u_{1}, v_{1} ∈ ℝ^{N} so that

γ_{1}(s_{0}) = y_{2} + hu_{1}, γ_{1}(t_{1} − s_{0}) = x_{2} + hv_{1},

and note that |u_{1}|, |v_{1}| < 1 + 2κ_{A}. Define a curve γ_{2} : [0, t_{2}] → ℝ^{N} connecting y_{2} to x_{2} as in the proof of Theorem 3.8. By arguing similarly, we get

d((0, y_{2}), (t_{2}, x_{2})) − d((0, y_{1}), (t_{1}, x_{1})) ≤ 2 β(1 + 2κ_{A}) h + ε,

so the statement follows by interchanging the roles of (y_{1}, t_{1}, x_{1}) and (y_{2}, t_{2}, x_{2}) and since ε is arbitrary. □

Let us now consider a Lagrangian L : ℝ^{N} × ℝ^{N} → ℝ_{+} which satisfies, in place of (L2), the following condition:

(L2′) α (|q|) ≤ L(x, q) ≤ β_{n} (|q|) for all (x, q) ∈ B_{n} × ℝ^{N} and n ∈ ℕ,

where (β_{n})_{n∈ℕ} is a family of convex, non–decreasing and superlinear functions from ℝ_{+} to ℝ_{+}.

Remark 4.17. We point out that condition (L2′) amounts to requiring that L is uniformly superlinear in q, and locally bounded in ℝ^{N} ×ℝ^{N} (cf. Lemma 2.3 in [17]).

In view of the results proved so far, it is now easy to generalize Theorem 4.2 as follows.

Theorem 4.18. Let L : ℝ^{N} × ℝ^{N} → ℝ_{+} be an autonomous Lagrangian satisfying conditions (L1), (L2′),(L3), and let 𝕃 be the integral functional defined via (21). Then the associated function d = d_{𝕃} defined through (8) satisfies condition (*). More precisely, for every M, r > 0 there exist a constant K = K(M, r, α, (β_{n})_{n}) such that

d((0, ·), (·, ·)) is K–Lipschitz continuous in \overline{C_{M}(r)},

where C_{M}(r) := {(y, t, x) ∈ B_{r} × (0, r) × B_{r} : |x − y| < M t }.

Proof. For every n ∈ ℕ, let us denote by 𝕃_{n} the integral functional associated through (21) to the Lagrangian L_{n}(x, q) := L(x, q) χ_{B_{n}}(x) + β_{n}(q) χ_{ℝ^{N} \B_{n}}(x). We claim that, for every M, r > 0, there exists an index k = k(M, r, α, (β_{n})_{n}) such that

d = d_{𝕃_{k}} on C_{M}(r).  (33)

Clearly, that is enough to conclude in force of Theorem 4.2.

Let us fix M, r > 0. We first notice that d enjoys assertions (d1)–(d3) of Proposition 2.5, and assumption (d4′) of Remark 3.7; in particular,

d((0, y), (t, x)) < r β_{m} (M)  for any (y, t, x) ∈ C_{M}(r),  (34)

where m := [r] + 1. Let γ be a curve in W^{1,1}((0, t), ℝ^{N}) connecting y to x such to be quasi–optimal for d((0, y), (t, x)). By (34), it is not restrictive to assume that

∫_{0}^{t} L(γ, γ̇) ds < r β_{m} (M) ,

in particular

∫_{0}^{t} |γ̇| ds < r(α_{1} + β_{m}(M)),

with α_{1} > 0 such that α(|q|) ≥ |q| − α_{1} for any q ∈ ℝ^{N}. As γ has end–points lying in B_{r}, we deduce that γ is entirely contained in the open ball B_{k} with

k := [r(1 + α_{1} + β_{m}(M))] + 1.

Thus

d((0, y), (t, x)) = inf {∫_{0}^{t} L(γ, γ̇) ds : γ(0) = y, γ(t) = x, γ([0, t]) ⊂ B_{k} }

for every (y, t, x) ∈ C_{M}(r), and claim (33) follows at once as L coincide with L_{k} on B_{k} × ℝ^{N}. □

4.4. Compactness with respect to Γ–convergence. As a simple consequence of the analysis carried out in the preceding sections, we derive a compactness result for a class of locally bounded, discontinuous autonomous Lagrangians.

We will say that a sequence of functionals F_{k} : W_{loc}^{1,1}(ℝ; ℝ^{N}) × I(ℝ) → [0, +∞] Γ–converges to a functional \overline{F} : W_{loc}^{1,1}(ℝ; ℝ^{N}) × I(ℝ) → [0, +∞] if, for every γ ∈ W_{loc}^{1,1}(ℝ; ℝ^{N}) and (a, b) ∈ I(ℝ), the following two conditions hold:

1) for every (γ_{k})_{k} ⊂ W^{1,1}((a, b), ℝ^{N}) such that lim_{k} ∥γ_{k} − γ∥_{∞} = 0 we have

lim inf_{k} F_{k}(γ_{k}, (a, b)) ≥ \overline{F}(γ, (a, b));

2) there exists (γ_{k})_{k} ⊂ W^{1,1}((a, b), ℝ^{N}) with lim_{k} ∥γ_{k} − γ∥_{∞} = 0 such that

lim sup_{k} F_{k}(γ_{k}, (a, b)) ≤ \overline{F}(γ, (a, b)).

The Γ–limit of a sequence of functionals is lower semicontinuous with respect to the convergence at issue. It is furthermore known (see [14, Theorem 8.5]) that the Γ–convergence is sequentially compact, that is, any sequence (F_{k})_{k} admits a Γ–convergent subsequence. For a general survey on the theory of Γ–convergence, we refer to [5, 14].

Let now (β_{n})_{n∈ℕ} a sequence of convex, non–decreasing and superlinear functions from ℝ_{+} to ℝ_{+}, and denote by N := N (α, (β_{n})_{n}) the family of Lagrangians L satisfying conditions (L1), (L2)′, (L3). With a slight abuse of notation, we will say that a sequence (L_{k})_{k} Γ–converges to L in N if the associated action functionals Γ–converge to the integral functional associated to L.

Theorem 4.19. Let N := N (α, (β_{n})_{n}) as above. Then N is sequentially compact with respect to Γ–convergence; i.e., any sequence in N admits a Γ–converging subsequence.

Proof. Let (L_{k})_{k} be a sequence in N, and denote by 𝕃_{k} the integral functional associated to each L_{k} through (21). By what remarked above, we already know that, up to subsequences, (𝕃_{k})_{k} Γ–converges to some lower semicontinuous functional \overline{𝕃} satisfying assumptions (F1)–(F3) and (F4′). In order to conclude, we need to show that \overline{𝕃} admits an integral representation for some L ∈ N.

For every x, y ∈ ℝ^{N} and t > 0, set

X^{t}(y, x) := {γ ∈ W^{1,1}((0, t), ℝ^{N}) : γ(0) = y, γ(t) = x } ,
Y^{t}(y, x) := \{γ ∈ X^{t}(y, x) : ∫_{0}^{t} α(|γ̇|) ds ≤ β_{n} (\frac{|x − y|}{t}) t \} ,

where n = n(x, y) is a sufficiently large integer number such that x, y ∈ B_{n}. The growth assumptions (L2) implies that

d_{𝕃_{k}}((0, y), (t, x)) = inf {𝕃_{k}(γ) : γ ∈ Y^{t}(y, x) }

for every x, y ∈ ℝ^{N}, t > 0 and k ∈ ℕ. By Proposition 2.3 we know that Y^{t}(y, x) is compact in X^{t}(y, x) with respect to the uniform convergence, hence by the crucial result of Γ–convergence (cf. [14, Theorem 7.4]) we infer that

d_{\overline{𝕃}}((0, y), (t, x)) = lim_{k→+∞} d_{𝕃_{k}}((0, y), (t, x)) for every x, y ∈ ℝ^{N} and t > 0.

Moreover, for any M, r > 0 the functions d_{𝕃_{k}}((0, ·), (·, ·)) are equi–Lipschitz continuous on \overline{C_{M}(r)} by Theorem 4.18, so the same is true for d_{\overline{𝕃}}((0, ·), (·, ·)). The conclusion now follows in force of Theorem 3.8 and Remark 3.12. □

Remark 4.20. We note that the convergence of d_{𝕃_{k}}((0, ·), (·, ·)) to d_{\overline{𝕃}}((0, ·), (·, ·)) is actually uniform in \overline{C_{M}(r)}, for any M, r > 0.

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[29] L. Tonelli, Fondamenti di Calcolo delle Variazioni. Vol. 1 (1921), Vol. 2 (1923), Zanichelli, Bologna.

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