Semicontinuity and supremal representation in the Calculus of Variations

Francesca Prinari

Dipartimento di Matematica ”Ennio De Giorgi” Universitá del Salento Via Prov.le Lecce-Arnesano, 73100 Lecce, Italy

Abstract

We study the weak* lower semicontinuity properties of functionals of the form

F (u) = ess sup_{x∈Ω} f(x, Du(x))

where Ω is a bounded open set of R^{N} and u ∈ W^{1,∞}(Ω). Without a continuity assumption on f(·, ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub levels). In particular if F is weakly* lower semicontinuous, than it can be represented through a level convex function. Finally a counterexample shows that it is not possible to represent F through the level convex envelope of f.

Mathematics Subject Classification (2000): 47J20, 58B20, 49J45. Keywords: supremal functionals, Calculus of Variations in L^{∞}, level convex function, absolute minimizers.

Contents

1 Introduction 2 2 Necessary and sufficient conditions for the w* lower semicontinuity 4 3 The class of the difference quotients 7 4 Approximation for supremal functionals 10

4.1 Moreau-Yosida Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2 An approximation through coercive functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Some representation results 12 6 The proofs 19 7 An existence theorem of AMLs and a principle of comparison 23 8 A counterexample 25

1 Introduction

In the last years a new class of functionals has been considered with growing interest in the mathematical literature: these functionals are represented in the so called supremal form

F (u) = ess sup_{x∈Ω} f(x, Du(x))  (1.1)

where Ω is a bounded open set of R^{N} and u ∈ W^{1,∞}(Ω). According to part of the already existing literature we will refer to a functional of the type (1.1) as supremal functional or L^{∞}-functional while we refer to the function f which represents F as an admissible supremand. The definition of this class is very meaningful because in many situations one would like to minimize a quantity which cannot be expressed as an integral: for example, a quantity which does not express a mean property of a body or whose values can be relevant on sets of arbitrarily small measure. In these cases the problem could be formulated as the minimization of a supremal functional: see for example the classical problem of finding optimal Lipschitz extensions, first considered by McShane in [18] or the recent formulation of the first dielectric breakdown for composite conductors given in [17].

In order to apply the direct methods of Calculus of Variations to this class of functionals, the main issue to be solved is the identification of the qualitative conditions on the supremand f which imply the lower semicontinuity of F with respect to the weak* W^{1,∞} topology. In fact under reasonable growth conditions for f this is the right topology in order to have the compactness of minimizing sequences. The characterization of lower semicontinuity for a functional expressed by a supremum requires a new notion of convexity: the level convexity. A function f = f(ξ) is said to be level convex (or quasi-convex) if it has convex sub levels. Namely f is level convex if the set {ξ : f(ξ) ≤ λ} is convex for every λ ∈ R; equivalently if

f(θξ + (1 − θ)η) ≤ f(ξ) ∨ f(η)

for every ξ, η ∈ R^{N} and θ ∈ [0, 1]. In [6] Barron, Jensen, Wang show the following sufficient condition:

Theorem 1.1 (Sufficient condition, Theorem 3.4 in [6]) Let f : Ω × R^{N} → R be a Borel function such that f(x, ·) is lower semicontinuous and level convex. Then for any open subset A ⊂ Ω the functional F (u, A) = ess sup_{x∈A} f(x, Du(x)) is sequentially weakly* lower semicontinuous on W^{1,∞}(Ω).

In the same paper, they show that this condition is also necessary.

Theorem 1.2 (Necessary condition, Theorem 2.7 in [6]) Let f : Ω × R^{N} → R be a Borel function such that there exists a function w : R × R → R which is continuous in its first variable with w(0, ξ) = 0 for every ξ ∈ R^{N} and non-decreasing in its second variable, such that

|f(x_{1}, ξ) − f(x_{2}, ξ)| ≤ w(|x_{1} − x_{2}|, |ξ|)

for any x_{1}, x_{2} ∈ Ω and ξ ∈ R^{N}. Let F (u, A) = ess sup_{x∈A} f(x, Du(x)) for every open subset A ⊂ Ω and assume that F (·, A) is sequentially weakly* lower semicontinuous on W^{1,∞}(A). Then for every x ∈ Ω f(x, ·) is a level convex function.

Note that the last theorem requires that the localized functional F (·, A) is weakly* lower semicontinuous for every open subset A ⊂ Ω. Moreover in the case in which only F (·, Ω) is weakly* lower semicontinuous we cannot apply the previous result in order to deduce some convexity property for f. The proof of Theorem 1.2 heavily relies on the continuity assumption on f(·, ξ). If one drops this assumption, then the statement of Theorem 1.2 can be false as it is shown in Remark 3.1 of [16]. This counterexample is based on the fact that in general a supremal functional does not admit a unique representation. More precisely the Authors construct a dense open set A ⊂ Ω with |A| > 0, |Ω \ A| > 0, two admissible level convex supremands φ_{1}, φ_{2} with φ_{1} ≤ φ_{2} on Ω and φ_{1} < φ_{2} on (Ω \ A) × R^{N} such that ess sup_{Ω} φ_{1}(x, Du(x)) = ess sup_{Ω} φ_{2}(x, Du(x)) := F (u) for every u ∈ W^{1,∞}(Ω). Moreover F turns out to be weakly* lower semicontinuous since it admits a level convex supremand, but it can be also represented by any function f, possibly non level convex, such that φ_{1} ≤ f ≤ φ_{2}. This means that without having a continuity property on f(·, ξ), one cannot expect that any admissible supremand for a weakly* l.s.c. L^{∞} functional is a level convex function.

Despite to these facts, most of the results concerning the class of supremal functionals suppose a priori that the weakly* lower semicontinuous functional F is represented by a level convex function. For example the existence of absolute minimizers (the so called AML) shown in [7], the Γ-convergence result given in [12], the homogenization theorem in [9], the principles of comparison with distance functions for AML stated in [13] all assume such a representation.Thus the question whether a weakly* lower semicontinuous functional always admits a level convex supremand turns out to be interesting and useful for applications.

The first positive answer to this problem is given in the 1-dimensional case in [19] where in Corollary 3.1 it is shown that if F (u) = ess sup_{Ω} f(x, u′(x)) is weakly* lower semicontinuous on W^{1,∞}(Ω) then there exists a level convex supremand f̃ which represents F.

The main contribution of this paper is the extension of this result to the N-dimensional case (see Section 2, Theorems 2.5, 2.6 and 2.7). With a completely different technique, under mild assumptions on f(x, ·) and without requiring a continuity property on f(·, ξ), we prove that all weakly* lower semicontinuous supremal functionals

• F : W^{1,∞}(Ω) → R of the form (1.1)

• F : W^{1,∞}(Ω) × A → R of the form

F (u, A) = ess sup_{A} f(x, Du(x))  (1.2)

(where A is the class of the open subsets of Ω) can be represented by a level convex supremand. The proofs of these results are given in Section 6 and are achieved through two steps. First we show that if a supremal functional of the form (1.1) is weakly* lower semicontinuous on W^{1,∞}(Ω) then F is a level convex functional on W^{1,∞}(Ω), i.e. the sub level sets

E_{λ} := {u ∈ W^{1,∞}(Ω) : F (u) ≤ λ}

are convex. The strategy used to prove this property follows the metric approach used in [16] where among other results it is shown that a 1-homogeneous supremal functional can be written in terms of intrinsic distances associated with the functionals (see Section 3). The second step concerns the representation in terms of a level convex supremand. Since we know that the representation is not unique, the main issue is to identify a good candidate. As shows the example constructed in Section 8, given a weakly* lower semicontinuous F of the form (1.1), in general it is not possible to choose as an admissible level convex supremand of F the level convex envelope f^{lc} of f given by

f^{lc}(x, ·) = sup\{h : R^{N} → \overline{R} : h lower semicontinuous and level convex, h(·) ≤ f(x, ·)\}.

In the choice of a suitable supremand we have been inspired by the one constructed in [11] (see Theorem 2.2 and Lemma 3.4 therein). In this paper given an abstract functional F : W^{1,∞}(Ω) × A → R such that F (·, A) is weakly* lower semicontinuous for every A ∈ A, the Authors construct a function f̃ in the following way

f̃(x, ξ) := inf {F (u, B_{r}(x)) | r > 0, u ∈ W^{1,∞}(Ω) s.t. x ∈ û, with Du(x) = ξ }  (1.3)

where û := {x ∈ Ω : x is a differentiability point of u and a Lebesgue point of Du}

and under some suitable assumptions on F, they represent the functional in the supremal form

F (u, A) = ess sup_{A} f̃(x, Du(x))  (1.4)

but they cannot deduce that f̃ is a level convex function. Inspired by the above result, we devote Section 5 to show that if F is a coercive supremal functional of the form (1.2) (eventually not weak* lower semicontinuous), then the function f̃ defined by (1.3) is an admissible supremand of F (see Theorem 5.4). In the case in which F is weakly* l.s.c. on W^{1,∞}(Ω), we show that f̃ is a level convex supremand of F. As easy consequence we obtain also that the function

φ(x, ξ) := inf {F (u) | u ∈ W^{1,∞}(Ω) s.t. x ∈ û, with Du(x) = ξ }  (1.5)

is an admissible level convex supremand of a weakly* l.s.c functional F of the form (1.1). Finally, as special case we deal with to the class of the 1-homogeneous supremal functionals already considered by Garroni, Ponsiglione, Prinari in [16] (see Theorem 2.6).

As consequence of these results in Section 7 we show the existence of Absolute minimizers for a weakly* l.s.c. supremal functional. An absolute minimizer (or AML) of the functional (1.2) is a function u ∈ W^{1,∞}(Ω) such that for all subdomain V ⊂ Ω one has

F (u, V) ≤ F (v, V)

for all v in W^{1,∞}(V) such that v = u on ∂V. In [7] and in [12] it is shown that if the functional (1.2) is coercive and represented by a level convex function f then there exists at least an absolute minimizer of F. Thanks to Theorem 2.7 we can give a result of existence of AML under the natural assumptions that F is weakly* l.s.c. and coercive (see Theorem 7.2). Moreover we discuss the problem of characterizing the AMLs by extending the principle of comparison with cones introduced by Crandall, Evans and Gariepy in [1] for the minimizing Lipschitz Extension Problem. In [13] Champion and De Pascale give a comparison principle with distances but they restrict to the case when f is globally l.s.c.. Now if f is not globally l.s.c. it is an open problem if it is possible to give an analogous principle. We obtain a partial result by showing that if u satisfies a comparison principle with the distances functions introduced in Section 7 and associated to the supremal functional F then u is an AML of F.

Finally, a paper in preparation (see [20]), based on the above results, is devoted to study the weakly* l.s.c. envelope of a supremal functional and to show that the lower semicontinuous envelope of a supremal functional is a level convex functional.

We shall fix some notations useful in the sequel.

Notations

We denote by Ω a generic open bounded domain of R^{N} and by A the family of open subsets of Ω. For every x ∈ R^{N} and r > 0 we denote by B_{r}(x) the open ball {y ∈ R^{N} : |x − y| < r} where | · | is

the euclidean norm on R^{N}.

For any set B ⊂ R^{N} we denote by H^{1}(B) its one dimensional Hausdorff measure. Moreover if B ⊂ R^{N} is a measurable set then |B| will denote its Lebesgue measure.

A modulus of continuity will be any continuous function w : [0, +∞) → [0, +∞) such that w(0) = 0. For every u ∈ W^{1,∞}(Ω) we denote by û the set

û := {x ∈ Ω : x is a differentiability point of u and a Lebesgue point of Du} .

2 Necessary and sufficient conditions for the w* lower semicontinuity

Before stating the main results of this paper we introduce the following definitions.

Definition 2.1 A function f : Ω × R^{N} → \overline{R} is said to be

(a) a normal supremand if:

(i) f is a Borel function;

(ii) for a. a. x ∈ Ω the function ξ ↦ f(x, ξ) is lower semicontinuous in R^{N};

(b) a Carathéodory supremand if:

(i) for every ξ ∈ R^{N} the function x ↦ f(x, ξ) is measurable in Ω;

(ii) for a. a. x ∈ Ω the function ξ ↦ f(x, ξ) is continuous in R^{N}; (c) a level convex normal (respectively, a level convex Carathéodory) supremand if f is a normal

(respectively, a Carathéodory) supremand and f(x, ·) is level convex on R^{N} for almost every x ∈ Ω.

Definition 2.2 A functional F : X → \overline{R} defined on a topological vector space X is said to be level convex if for every t ∈ R the level set {u ∈ X : F (u) ≤ t} is convex.

Now we are in a position to enunciate the main theorems of this paper. Theorems 2.3, 2.4 and 2.5 consider functionals F : W^{1,∞}(Ω) → R of the form

F (u) = ess sup_{Ω} f(x, Du(x))  (2.6)

where Ω is an open subset of R^{N}.

First we show that for a large class of situations the level convexity of the functional F is a consequence of its weak* lower semicontinuity.

Theorem 2.3 Let Ω ⊂ R^{N} be a connected open set with Lipschitz continuous boundary. Let f : Ω × R^{N} → R be a Carathéodory supremand satisfying the following assumption: for any M > 0 there exists some modulus of continuity ω_{M} such that

|f(x, ξ) − f(x, η)| ≤ ω_{M}(|ξ − η|)  (2.7)

for a.e. x ∈ Ω and for every ξ, η ∈ B_{M}(0). If the functional F defined by (2.6) is weakly* l.s.c. on W^{1,∞}(Ω) then F is a level convex functional.

As shown in Remark 3.1 of [16], in the general case the above result does not give as consequence the level convexity of f(·, ξ). However, we can prove that there exists at least a level convex supremand φ for a level convex supremal functional F. We notice, as shown in Section 8, that φ may not coincide neither with the level convex envelope of f.

Theorem 2.4 Let Ω ⊂ R^{N} be an open set. Let f : Ω × R^{N} → R be a normal supremand and let F be the functional defined by (2.6). Then F is level convex if and only if there exists a level convex normal supremand φ : Ω × R^{N} → R such that

F (u) = ess sup_{Ω} φ(x, Du(x))

for all u ∈ W^{1,∞}(Ω). In particular if F is level convex then F is weakly* l.s.c. on W^{1,∞}(Ω).

In the previous theorem, if f is globally Lipschitz continuous then it is possible to show that the function φ is Lipschitz continuous too (see Proposition 5.2). But when f is a Carathéodory supremand it is not clear if φ is a Carathéodory supremand too. However if f satisfies (2.7) and a further coercivity condition we may put together the previous results and obtain the following characterization.

Theorem 2.5 Let Ω ⊂ R^{N} be a connected open set with Lipschitz continuous boundary. Let f : Ω × R^{N} → R be a Carathéodory supremand satisfying (2.7) and the following assumption: there exists an increasing continuous function α : R^{+} → R^{+}such that lim_{t→+∞} α(t) = +∞ and

f(x, ξ) ≥ α(|ξ|)  for a.e x ∈ Ω, for every ξ ∈ R^{N}.  (2.8)

Let F be the functional defined by (2.6). The following facts are equivalent:

1) F is weakly* l.s.c. on W^{1,∞}(Ω);

2) F is a level convex functional;

3) there exists a level convex Carathéodory supremand φ : Ω × R^{N} → R given by

φ(x, ξ) := inf {ess sup_{Ω} f(y, Du(y)) | u ∈ W^{1,∞}(Ω) s.t. x ∈ û, with Du(x) = ξ }  (2.9)

such that

F (u) = ess sup_{Ω} φ(x, Du(x))

for all u ∈ W^{1,∞}(Ω). Moreover φ satisfies (2.8), (2.7) (for a suitable family (ω′_{M})_{M} of moduli of continuity) and for a.e. x ∈ Ω φ(x, ·) ≥ f(x, ·).

The following result concerns the class of the 1-homogeneous supremal functional studied in [16]. Note that compared with the assumptions done in Theorem 2.5, in the next result we do not require that f satisfies assumption (2.7).

Theorem 2.6 Let Ω ⊂ R^{N} be a connected open set with Lipschitz continuous boundary. Let f : Ω × R^{N} → R be a Carathéodory supremand satisfying the following assumptions

α |ξ| ≤ f(x, ξ) ≤ β |ξ|  (2.10)

and

f(x, tξ) = |t|f(x, ξ)  (2.11)

for every ξ ∈ R^{N} , for a.e. x ∈ Ω and for every t ∈ R and for some positive constants α, β > 0. Let F be the functional defined by (2.6), let d : Ω × Ω → R be the distance defined by

d(x, y) = sup {u(x) − u(y), u ∈ W^{1,∞}(Ω) : F (u) ≤ 1}.  (2.12)

and let φ_{d} be the metric derivative of d defined as

φ_{d}(x, η) := lim sup_{t→0^{+}} \frac{d(x, x + tη)}{t}.  (2.13)

Then the following facts are equivalent:

1) F is weakly* l.s.c. on W^{1,∞}(Ω);

2) F is a convex functional;

3) for all u ∈ W^{1,∞}(Ω)

F (u) = ess sup_{Ω} φ_{d}^{0}(x, Du(x))

where

φ_{d}^{0}(x, ξ) := sup {ξ · η : φ_{d}(x, η) ≤ 1}.

Moreover if φ is given by (2.9) there exists a negligible set H ⊂ Ω such that

φ_{d}^{0}(x, ξ) = φ(x, ξ)

for every x ∈ Ω \ H and for every ξ ∈ R^{N}.

Finally, through a method of localization and an appropriate choice of the supremand, the results above can be extended to the class of the supremal functionals F : W^{1,∞}(Ω) × A → R of the form

F (u, A) = ess sup_{A} f(x, Du(x)).  (2.14)

In particular we give the following result:

Theorem 2.7 Let Ω be an open subset of R^{N}. Let f : Ω × R^{N} → R be a Carathéodory supremand satisfying (2.7) and (2.8). Let F (·, A) be the functional defined by (2.14). The following facts are equivalent:

1) F (·, A) is weakly* l.s.c. on W^{1,∞}(Ω) for every A ∈ A;

2) F (·, A) is a level convex functional for every A ∈ A; 3) there exists a level convex normal supremand f̃: Ω × R^{N} → R given by

f̃(x, ξ) := inf {ess sup_{B_{r}(x)} f(y, Du(y)) | r > 0, u ∈ W^{1,∞}(Ω) s.t. x ∈ û, with Du(x) = ξ } (2.15)

such that

F (u, A) = ess sup_{A} f̃(x, Du(x))

for all u ∈ W^{1,∞}(Ω) and for all A ∈ A. Moreover f̃ satisfies (2.8), (2.7) (for a suitable family (ω′_{M})_{M} of moduli of continuity) and for a.e. x ∈ Ω f̃(x, ·) ≥ f(x, ·).

Notice that in the previous theorem when f(·, ξ) is continuous for every ξ ∈ R^{N} then for a.e. x ∈ Ω f̃(x, ·) = f(x, ·) (see Theorem 5.4) and therefore we obtain that if F is weakly* l.s.c. then f is a level convex supremand.

In order to show all the results above we need to introduce some tools, to recall some known facts and to prove further preliminary results. For these reasons the proofs of the previous theorems will be given later in Section 6.

3 The class of the difference quotients

In order to show that a weakly* lower semicontinuous supremal functional is level convex, we recall some results and some tools given for the 1-homogeneous supremal functionals in [16] with the aim to extend them to more general supremal functionals. Consider a supremal functional (2.6) represented through a Carathéodory supremand f : Ω × R^{N} → R^{+} satisfying (2.7) and (2.8). With every λ ∈ R such that the sub level set E_{λ} := {u ∈ W^{1,∞}(Ω) : F (u) < λ} is nonempty we can associate a distance d_{λ} in the following way:

d_{λ}(x, y) := sup {|u(x) − u(y)| : u ∈ W^{1,∞}(Ω) : F (u) ≤ λ}.  (3.16)

Notice that if B_{r}(x) ⊂ Ω then, from (2.8), we have that d_{λ}(x, y) ≤ α^{−1}(λ)r for every y ∈ B_{r}(x). In general if Ω is a connected open set, then for every x, y ∈ Ω it holds

d_{λ}(x, y) ≤ α^{−1}(λ)|x − y|_{Ω}  (3.17)

where

|x − y|_{Ω} = inf{L(γ) : γ ∈ Γ_{x,y}(Ω)} ,

Γ_{x,y}(Ω) being the set of Lipschitz curves in Ω with end-points x and y, and L(γ) the Euclidean length of γ. In particular if ∂Ω is Lipschitz continuous then there exists a constant C > 0 such that

d_{λ}(x, y) ≤ |x − y|_{Ω} ≤ C|x − y|.  (3.18)

Moreover with every λ ∈ R there exists δ = δ(λ) such that for every x, y ∈ Ω

d_{λ}(x, y) ≥ δ|x − y|.  (3.19)

In fact since E_{λ} is nonempty, then ε > 0 there exists ε > 0 and u ∈ W^{1,∞}(Ω) such that F (u) < λ − ε. Now fix x, y ∈ Ω and without loss of generality, assume u(x) ≥ u(y). Chosen M > ||u||_{1,∞} there exists 0 < σ < 1 such that w_{M+2}(t) ≤ ε for every 0 < t ≤ σ. Thus if 0 < δ < min\{\frac{1}{diamΩ+1}, σ\} then the function

v(z) := u(z) + δ \frac{(x − y)}{|x − y|} · z

is such that

||v||_{1,∞} ≤ M + δ(diamΩ) + σ ≤ M + 2

and

F (v) ≤ F (u) + w_{M+2}(δ) < λ − ε + ε = λ.

This implies

d_{λ}(x, y) ≥ |v(x) − v(y)| = |δ|x − y| + u(x) − u(y)| = δ|x − y| + u(x) − u(y) ≥ δ|x − y|.

Now for every λ such that E_{λ} is nonempty, we consider the functional R_{λ} : W^{1,∞}(Ω) → R̄ given by

R_{λ}(u) := sup_{x,y∈Ω, x≠y} \frac{|u(x) − u(y)|}{d_{λ}(x, y)}.  (3.20)

The functional R_{λ} is referred to as the difference quotient associated with level set E_{λ} of F.

Proposition 3.1 For every λ s.t. E_{λ} ≠ ∅ the difference quotient R_{λ} is a convex lower semicontinuous functional with respect to the strong convergence in L^{∞}. Moreover R_{λ}(u + v) ≤ R_{λ}(u) + R_{λ}(v) for every u, v ∈ W^{1,∞}(Ω).

Proof. Let u ∈ W^{1,∞}(Ω) and let {u_{n}} ⊂ W^{1,∞}(Ω) be a sequence converging to u in L^{∞}(Ω). We have that for every x, y ∈ Ω such that 0 < d_{λ}(x, y) < +∞

\frac{|u(x) − u(y)|}{d_{λ}(x, y)} = lim_{n} \frac{|u_{n}(x) − u_{n}(y)|}{d_{λ}(x, y)} ≤ lim inf_{n} R_{λ}(u_{n}).

Taking the supremum for x, y ∈ Ω, x ≠ y we get the thesis. The convexity and the sublinearity of R_{λ} are trivial. ⊓ ⊔

The key tool we will use in the sequel is the following lemma. It is an adaptation of Lemma 3.4 in [16]: We report its revised proof for sake of completeness.

Lemma 3.2 Let Ω ⊂ R^{N} be a connected open set with Lipschitz continuous boundary. Let F be a supremal functional on W^{1,∞}(Ω) represented by a Carathéodory supremand f : Ω × R^{N} → R satisfying (2.7) and (2.8). Let v ∈ W^{1,∞}(Ω) be such that R_{λ}(v) < 1.Then there exists a sequence {v_{n}} ⊂ W^{1,∞}(Ω) converging to v in L^{∞}(Ω) with F (v_{n}) ≤ λ for n ∈ N.

Proof. Let us fix r > 0. By the fact that R_{λ}(v) < 1 and thanks to (3.19), for every x, y ∈ Ω with |x − y| = r

|v(y) − v(x)| < d_{λ}(x, y) − γ,  (3.21)

for a positive constant γ depending on r. Let us fix 0 < ε < \frac{γ}{3} . For every x ∈ Ω and for every y ∈ ∂B_{r}(x) ∩ Ω, by the definition of d_{λ} there exists a function w_{r}^{x,y} ∈ W^{1,∞}(Ω) such that

1) F (w_{r}^{x,y}) ≤ λ;

2) |w_{r}^{x,y}(y) − w_{r}^{x,y}(x)| ≥ d_{λ}(x, y) − ε;

3) w_{r}^{x,y}(x) = v(x);

the third property is possible thanks to the translation invariance of the first two. By properties 2), 3) and by (3.21), for every y ∈ ∂B_{r}(x) ∩ Ω

|w_{r}^{x,y}(y) − v(x)| ≥ d_{λ}(x, y) − ε > |v(y) − v(x)| + γ − ε.  (3.22)

Note that by (2.8) we have that ess sup_{Ω} |Dw_{r}^{x,y}| < α^{−1}(λ), and hence there exists δ > 0 (depending only on ε) such that

|w_{r}^{x,y}(z)−v(x)| > |v(z)−v(x)|+γ−2ε > |v(z)−v(x)|+ε  for every z ∈ ∂B_{r}(x)∩Ω : |z−y| ≤ δ. (3.23)

Moreover, since w_{r}^{x,y}(x) = v(x), there exists 0 < r′ < r (depending only on ε) such that

|w_{r}^{x,y}(z) − v(x)| < |v(z) − v(x)| + ε  for every z ∈ B_{r′}(x) ∩ Ω.  (3.24)

For every x ∈ Ω, let us fix a finite set of points {y_{1}, …, y_{N}} on ∂B_{r}(x) ∩ Ω such that

∂B_{r}(x) ∩ Ω ⊂ ⋃_{i=1}^{N} B_{δ}(y_{i}),

and let us set the function w_{r}^{x} : B_{r}(x) ∩ Ω → R defined by

w_{r}^{x}(z) := max_{i} w_{r}^{x,y_{i}}(z)  for every z ∈ B_{r}(x) ∩ Ω.  (3.25)

By construction and by (3.23) and (3.24), we have

1) ess sup_{B_{r}(x)∩Ω} f(z, Dw_{r}^{x}) ≤ λ;

2) |w_{r}^{x}(z) − v(x)| > |v(z) − v(x)| + ε for every z ∈ ∂B_{r}(x) ∩ Ω;

3) |w_{r}^{x}(z) − v(x)| < |v(z) − v(x)| + ε for every z ∈ B_{r′}(x) ∩ Ω.

Now let Z_{r} be a finite set of points of Ω such that

Ω ⊂ ⋃_{z∈Z_{r}} B_{r′}(z),

and consider the function w_{r} : Ω → R defined by

w_{r}(x) := min_{z∈Z_{r}∩B_{r}(x)} w_{r}^{z}(x).  (3.26)

From properties 2) and 3) above it follows that w_{r} is continuous. Moreover, for almost every x in Ω, Dw_{r}(x) coincides with Dw_{r}^{z}(x) for some z ∈ Z_{r} and this implies that w_{r} ∈ W^{1,∞}(Ω) and F (w_{r}) ≤ λ.

Now let us prove that ∥w_{r} − v∥_{∞} → 0 as r → 0^{+}. To this aim, let us fix x ∈ Ω, and let z ∈ B_{r}(x) be such that w_{r}(x) = w_{r}^{z}(x). Recalling that by construction w_{r}^{z}(z) = v(z), and by using (3.17) and (3.18) we conclude

|w_{r}(x) − v(x)| ≤ |w_{r}^{z}(x) − w_{r}^{z}(z)| + |w_{r}^{z}(z) − v(x)|
= |w_{r}^{z}(x) − w_{r}^{z}(z)| + |v(z) − v(x)|
≤ ||Dw_{r}^{z}||_{∞}|x − z|_{Ω} + d_{λ}(x, z)
≤ 2α^{−1}(λ)|x − z|_{Ω} ≤ 2α^{−1}(λ)Cr

Therefore, for every {r_{n}} → 0, the sequence v_{n} := w_{r_{n}} does the job. ⊓ ⊔

In the following version of the Lemma 3.2 we remove the assumption that Ω has Lipschitz continuous boundary but we require that the supremal functional is weakly* l.s.c. on every open subset of Ω.

Lemma 3.3 Let Ω ⊂ R^{N} be an open set. Let F be a supremal functional on W^{1,∞}(Ω) of the form (2.14) represented by a Carathéodory supremand f : Ω × R^{N} → R satisfying (2.7) and (2.8). Assume that F (·, A) is weakly* l.s.c. on W^{1,∞}(A) for every A ∈ A. If v ∈ W^{1,∞}(Ω) is such that R_{λ}(v) < 1 then F (v, Ω) ≤ λ.

Proof. Fix r̄ ∈ R^{+}. If x ∈ Ωr̄ = {x ∈ Ω : d(x, ∂Ω) > r̄} then B_{r}(x) ⊂ Br̄(x) ⊂ Ω for every 0 < r < r̄. In particular for every x, y ∈ Ωr̄ with |x − y| = r we have that d_{λ}(x, y) ≤ 2α^{−1}(λ)r. By repeating the proof of the previous lemma, for every r ∈ R^{+} we can construct a function w_{r} ∈ W^{1,∞}(Ω) such that F (w_{r}, Ωr̄) ≤ λ and satisfying

|w_{r}(x) − v(x)| ≤ 2α^{−1}(λ)r

for every x ∈ Ωr̄. If {r_{n}} → 0, the sequence v_{n} := w_{r_{n}} weakly* converges to v in Ωr̄ and thanks to the weak* lower semicontinuity of F (·, Ωr̄) we have that

F (v, Ωr̄) ≤ lim inf_{n} F (w_{r_{n}}, Ωr̄) ≤ lim inf_{n} F (w_{r_{n}}, Ω) ≤ λ.

This easily implies that F (v, Ω) ≤ λ. ⊓ ⊔

Finally, we cite the following result obtained in [16] as a corollary of Lemma 3.2. This will be useful when we will be interested to the 1-homogeneous supremal functionals represented by a Carathéodory function.

Proposition 3.4 (Proposition 3.5 in [16]) Let f : Ω × R^{N} → R be a Carathéodory supremand satisfying (2.10) and (2.11). Let d : Ω × Ω → R^{+} be the distance defined by

d(x, y) := sup {u(x) − u(y), u ∈ W^{1,∞}(Ω) : ess sup_{Ω} f(x, Du(x)) ≤ 1}.  (3.27)

If the functional F (u) = ess sup_{Ω} f(x, Du(x)) is a weakly* l.s.c. on W^{1,∞}(Ω) then

F (u) = sup_{x,y∈Ω,x≠y} \frac{u(x) − u(y)}{d(x, y)}

for every u ∈ W^{1,∞}(Ω).

4 Approximation for supremal functionals

4.1 Moreau-Yosida Transform

A key tool we shall use is a modification of the Moreau-Yosida Transform. More precisely, the following approximation result holds.

Theorem 4.1 Let F : W^{1,∞}(Ω) → [0, +∞] be a (strongly) l.s.c. functional such that F (u + c) = F (u) for every u ∈ W^{1,∞}(Ω) and for every c ∈ R. If we set

F_{λ}(u) := inf{F (v) ∨ λ||Du − Dv||_{∞} : v ∈ W^{1,∞}(Ω)}

for every λ > 0 then we have

F (u) = sup_{λ} F_{λ}(u).

Moreover, the functional F_{λ} satisfies the condition

F_{λ}(u) ≤ F_{λ}(v) + λ||Du − Dv||_{∞}  for every u, v ∈ W^{1,∞}(Ω).  (4.28)

Therefore if F is finite in at least one point then

|F_{λ}(u) − F_{λ}(v)| ≤ λ||Du − Dv||_{∞} for every u, v ∈ W^{1,∞}(Ω).

Proof. Fix u ∈ W^{1,∞}(Ω). By taking v = u in the definition of F_{λ}(u) we obtain the inequality

F_{λ}(u) ≤ F (u).

Take now t < F (u); since F is lower semicontinuous there exists δ > 0 such that

t < inf {F (w) : w ∈ W^{1,∞}(Ω), ||u − w||_{W^{1,∞}(Ω)} < δ}.

Let 0 < δ′ < \frac{δ}{1+diamΩ}. Then

t < inf {F (v) : v ∈ W^{1,∞}(Ω), ||Du − Dv||_{∞} < δ′}.  (4.29)

In fact, let v ∈ W^{1,∞}(Ω) be such that

||Du − Dv||_{∞} < δ′.

Fix x_{0} ∈ Ω and define w(x) := v(x) + u(x_{0}) − v(x_{0}). Then

||Du − Dw||_{∞} = ||Du − Dv||_{∞} ≤ δ′

and for a.e. x ∈ Ω

|u(x) − w(x)| ≤ |(u(x) − u(x_{0})) − (w(x) − w(x_{0})| ≤ ||Du − Dw||_{∞}|x − x_{0}| ≤ δ′ · diamΩ.

Therefore

||u − v||_{W^{1,∞}(Ω)} ≤ δ′ + δ′ · diamΩ < δ.

This implies F (v) = F (w) ≥ inf {F (w) : w ∈ W^{1,∞}(Ω), ||u − w||_{W^{1,∞}(Ω)} < δ} and thus (4.29) follows. Now let λ > 0 be such that λ · δ′ > t. For every v ∈ W^{1,∞}(Ω) with ||Du − Dv||_{∞} < δ′ we have

F (v) ∨ λ||Du − Dv||_{∞} ≥ F (v) > t

whereas for every v ∈ W^{1,∞}(Ω) with ||Du − Dv||_{∞} ≥ δ we have

F (v) ∨ λ||Du − Dv||_{∞} ≥ λδ′ > t.

Hence F_{λ}(u) ≥ t and, since t was arbitrary, this proves the inequality

F (u) ≤ sup {F_{λ}(u) : λ > 0}.

Finally take u, v ∈ W^{1,∞}(Ω), let ε > 0 be fixed, and let w ∈ W^{1,∞}(Ω) be such that

F_{λ}(v) ≥ F (w) ∨ λ||Dv − Dw||_{∞} − ε.

Taking into account that for every a, b, c ∈ R we have

a ∨ b ≤ a ∨ c + |b − c|

we obtain

F_{λ}(u) ≤ F (w) ∨ λ||Du − Dw||_{∞}
≤ F (w) ∨ λ||Dv − Dw||_{∞} + λ|||Du − Dw||_{∞} − ||Dv − Dw||_{∞}|
≤ F_{λ}(v) + ε + λ||Du − Dv||_{∞}.

Since ε was arbitrary, inequality (4.28) follows and the proof is achieved. ⊓ ⊔

Analogously one can show the following result:

Proposition 4.2 Let f : Ω × R^{N} → [0, +∞) be a normal supremand. If we set for every λ > 0

f_{λ}(x, ξ) = inf {f(x, η) ∨ λ|ξ − η| : ξ ∈ R^{N}}  (4.30)

then we have that f_{λ} is a normal supremand such that

|f_{λ}(x, ξ) − f_{λ}(x, η)| ≤ λ|ξ − η|

and

f(x, ξ) = sup {f_{λ}(x, ξ) : λ > 0}

for every x ∈ Ω and for every ξ, η ∈ R^{N}.

4.2 An approximation through coercive functionals

Here we approximate a non negative supremal functional through a sequence of coercive supremal functionals.

Proposition 4.3 Let Ω be an open subset of R^{N}. Let g : Ω × R^{N} → R^{+} be a normal supremand. Let

g_{n}(x, ξ) := g(x, ξ) ∨ \frac{1}{n}|ξ|  (4.31)

and let G, G_{n} : W^{1,∞}(Ω) → R be the functionals defined respectively by

G(u) = ess sup_{Ω} g(x, Du(x))  (4.32)

and by

G_{n}(u) = ess sup_{Ω} g_{n}(x, Du(x))  (4.33)

where n ∈ N. Then

(i) for every n ∈ N and for every u ∈ W^{1,∞}(Ω)

G_{n}(u) = G(u) ∨ \frac{1}{n}||Du||_{L^{∞}(Ω)};  (4.34)

(ii) the sequence (G_{n})_{n} pointwise converges to G;

(iii) If G_{n} is a level convex functional for every n ∈ N then G is a level functional.

Proof. In order to show (i), fix n ∈ N and u ∈ W^{1,∞}(Ω). The inequality G_{n}(u) ≤ G(u) ∨ \frac{1}{n}||Du||_{L^{∞}(Ω)} is trivial. For the converse inequality, for fixed δ there exists B_{δ} ⊂ Ω with |B_{δ}| > 0 such that g_{n}(x, Du(x)) ≥ G_{n}(u) − δ for every x ∈ B_{δ}. Set B_{δ}^{+} = {x ∈ B_{δ} : g_{n}(x, ξ) = g(x, ξ)} and B_{δ}^{−} = \{x ∈ B_{δ} : g_{n}(x, ξ) = \frac{1}{n}|ξ|\}. If |B_{δ}^{+}| > 0 then G(u) ≥ G_{n}(u) − δ while if |B_{δ}^{−}| > 0 then \frac{1}{n}||Du||_{L^{∞}(Ω)} ≥ G_{n}(u) − δ. In both cases \frac{1}{n}||Du||_{L^{∞}(Ω)} ∨ G(u) ≥ G_{n}(u) − δ for every δ > 0. Then \frac{1}{n}||Du||_{L^{∞}(Ω)} ∨ G(u) ≥ G_{n}(u). As consequence for every u ∈ W^{1,∞}(Ω)

lim_{n} G_{n}(u) = lim_{n} G(u) ∨ \frac{1}{n}||Du||_{L^{∞}(Ω)} = G(u) ∨ 0 = G(u).

Now let u ∈ W^{1,∞}(Ω) Then Finally, concerning (iii), fix u, v ∈ X and λ ∈ (0, 1). Then

G(λu + (1 − λ)v) = lim_{n} G_{n}(λu + (1 − λ)v) ≤ lim_{n} G_{n}(u) ∨ G_{n}(v) = G(u) ∨ G(v)

i.e. G is a level convex functional. ⊓ ⊔

5 Some representation results

This section is devoted to the construction of admissible supremands for supremal functionals of the form (2.6) and (2.14). Note that, as shown in the example given in the Section 1 of [11], if F is a supremal functional of the form (2.14), it is not possible to represent F through the function h defined by

h(x, ξ) := inf_{r>0} F (w_{x,ξ}, B_{r}(x))

where w_{x,ξ}(y) := u + ξ · (y − x).

The following proof is inspired by the proof of Lemma 3.3 in [11].

Proposition 5.1 Let Ω be an open subset of R^{N}. Let F : W^{1,∞}(Ω) → \overline{R} be a functional such that for any M > 0 there exists a modulus of continuity ω_{M} such that

|F (u) − F (v)| ≤ ω_{M}(||Du − Dv||_{∞})

for every u, v ∈ W^{1,∞}(Ω) s.t. ||Du||_{∞}, ||Dv||_{∞} ≤ M. Let φ : Ω × R^{N} → R be defined by

φ(x, ξ) := inf {F (u) | u ∈ W^{1,∞}(Ω) s.t. x ∈ û, with Du(x) = ξ } .  (5.35)

Then for every ξ ∈ R^{N} the function x ↦ φ(x, ξ) is measurable in Ω. Moreover if there exists an increasing continuous function α : R^{+} → R^{+} such that lim_{t→+∞} α(t) = +∞ and F (u) ≥ α(||Du||_{∞}) then (i) φ is a Carathéodory supremand satisfying (2.8) and (2.7) for a suitable family (ω′_{M})_{M} of moduli of

continuity;

(ii) for any u ∈ W^{1,∞}(Ω)

F (u) ≥ ess sup_{Ω} φ(x, Du(x)).

Proof. Let ξ ∈ R^{N} and λ ∈ R be fixed. Define the sets

A(x) := {u ∈ W^{1,∞}(Ω) : x ∈ û with Du(x) = ξ, },  (5.36)

and

K_{λ} := {x ∈ Ω : ∀u ∈ A(x) F (u) ≥ λ} = {x ∈ Ω : φ(x, ξ) ≥ λ}.  (5.37)

If we prove that K_{λ} is measurable for every λ ∈ R, then φ(·, ξ) is measurable. Suppose that K_{λ} is not measurable. Then there is a set C with K_{λ} ⊂ C s.t. C is measurable and of minimal measure. Let x_{0} ∈ Ĉ \ K_{λ} where Ĉ the set of the points of density 1 of C. From the definition of K_{λ}, there is some u ∈ A(x_{0}) such that F (u) < λ. Now, fix ε > 0 such that F (u) < λ − ε. Since the functional F is strongly continuous in W^{1,∞}(Ω) then, there exists δ > 0 such that F (v) < λ for every v ∈ W^{1,∞}(Ω) such that ||v − u||_{W^{1,∞}(Ω)} ≤ δ. Set

A_{1} = \{ x ∈ Ω | x ∈ û, |u(x) − u(x_{0})| ≤ δ/2, |Du(x) − ξ| ≤ \frac{δ}{2diam(Ω)} \} .  (5.38)

Note that A_{1} is measurable and since x_{0} is a Lebesgue point of Du, then |A_{1}| > 0. We claim that A_{1} ∩ K_{λ} = ∅. In fact, if x ∈ A_{1}, then the function v_{x} ∈ W^{1,∞}(Ω) defined by

v_{x}(y) := u(y) + (u(x_{0}) − u(x)) + ⟨Du(x_{0}) − Du(x), y − x⟩

belongs to A(x) and ∥v_{x} −u∥_{W^{1,∞}(Ω)} ≤ δ. Thus it easily follows that F (v_{x}) < λ. So x ∉ K_{λ} which implies that K_{λ} ⊂ C \ A_{1}. Moreover the set C\A_{1} is still measurable. If we show that x_{0} is a point of density 1 of A_{1} then there exists r > 0 such that

|A_{1} ∩ B_{r}(x_{0})| ≥ \frac{1}{2}|B_{r}(x_{0})|

and

|C ∩ B_{r}(x_{0})| ≥ \frac{3}{4}|B_{r}(x_{0})|.

Therefore

|A_{1} ∩ C| ≥ |A_{1} ∩ B_{r}(x_{0}) ∩ C| = |A_{1} ∩ B_{r}(x_{0})| − |A_{1} ∩ B_{r}(x_{0},) \ C|
≥ |A_{1} ∩ B_{r}(x_{0})| − |B_{r}(x_{0}) \ C|
≥ |A_{1} ∩ B_{r}(x_{0})| − |B_{r}(x_{0})| + |B_{r}(x_{0}) ∩ C|
≥ \frac{3}{4}|B_{r}(x_{0})| − |B_{r}(x_{0})| + \frac{1}{2}|B_{r}(x_{0})|
= \frac{1}{2}|B_{r}(x_{0})|.

In particular |C \ A_{1}| < |C| and since K ⊂ (C\A_{1}), we have contradicted the minimality of C. In order to show that x_{0} is a point of density 1 of A_{1} note that there exists r_{0} = r_{0}(δ) > 0 such that for every r < r_{0}

B_{r}(x_{0}) ∩ A_{1} = \{ x ∈ B_{r}(x_{0}); x ∈ û, |Du(x) − ξ_{0}| ≤ \frac{δ}{2diam(Ω)} \} .  (5.39)

Now

\frac{|B_{r}(x_{0}) ∩ A_{1}|}{|B_{r}(x_{0})|} = 1 − \frac{|B_{r}(x_{0}) \ A_{1}|}{|B_{r}(x_{0})|} = 1 − \frac{∫_{B_{r}(x_{0})\A_{1}} dx}{|B_{r}(x_{0})|} ≥ 1 − \frac{2diam(Ω)}{δ|B_{r}(x_{0})|} ∫_{B_{r}(x_{0})} |Du(x) − ξ_{0}|dx

and since x_{0} is a Lebesgue point of Du with Du(x_{0}) = ξ_{0} we obtain

lim_{r→0^{+}} \frac{|B_{r}(x_{0}) ∩ A_{1}|}{|B_{r}(x_{0})|} = 1

i.e. x_{0} is a point of density 1 of A_{1}. Now we assume that F satisfies also the coercivity assumption F (u) ≥ α(||Du||_{∞}) for every u ∈ W^{1,∞}(Ω). We show that φ satisfies assumption (2.7) for a suitable family of moduli of continuity. Let M > 0 be fixed. Then there is some constant K = K(M)) such that, for any (x, ξ) ∈ Ω × R^{N} with |ξ| ≤ M and for any v ∈ W^{1,∞}(Ω),

[F (v) ≤ φ(x, ξ) + 1 ] ⇒ ∥Dv∥_{∞} ≤ M′.

In fact from the continuity assumption on F,

F (v) ≤ φ(x, ξ) + 1 ≤ F (φ_{ξ}) + 1 ≤ F (0) + ω_{M}(|ξ|) + 1

where φ_{ξ}(x) := ξ · x and from the coercivity condition on F we have

∥Dv∥_{∞} ≤ α^{−1}(F (0) + ω_{M}(M) + 1) .

In particular if ξ, η ∈ B_{M}(0) and u_{ξ} ∈ W^{1,∞}(Ω) is such that F (u_{ξ}) ≤ φ(x, ξ) + ε where 0 < ε < 1 then

φ(x, η) ≤ F (u_{ξ} + φ_{η−ξ}) ≤ F (u_{ξ}) + ω_{M′}(|ξ − η|) ≤ φ(x, ξ) + ω_{M′}(|ξ − η|) + ε

where M′ = α^{−1}(ω_{M}(M) + F (0) + 1) + 2M. As ε → 0^{+} the last inequality implies

φ(x, η) ≤ φ(x, ξ) + ω_{M′}(|ξ − η|)

and by changing the roles of ξ and η it follows

|φ(x, η) − φ(x, ξ)| ≤ ω_{M′}(|ξ − η|).

Then it is sufficient to define ω′_{M} := ω_{M′}. Finally by the definition of φ it follows that

ess sup_{Ω} φ(x, Du(x)) ≤ F (u)

for every u ∈ W^{1,∞}(Ω). ⊓ ⊔

Under the stronger assumption that F is a Lipschitz continuous functional, it is not necessary to require a coercivity assumption in order to show that φ is a Carathéodory supremand.

Proposition 5.2 Let Ω be an open subset of R^{N}. Let F : W^{1,∞}(Ω) → \overline{R} be a functional. Assume that there exists L > 0 such that F is L-Lipschitz continuous functional i.e.

|F (u) − F (v)| ≤ L||u − v||_{W^{1,∞}(Ω)} for every u, v ∈ W^{1,∞}(Ω).

Then

(i) the function φ defined by (5.35) is a Carathéodory supremand such that

|φ(x, ξ) − φ(x, η)| ≤ L|ξ − η|  (5.40)

for every x ∈ Ω and for every ξ, η ∈ Ω;

(i) for any u ∈ W^{1,∞}(Ω)

F (u) ≥ ess sup_{Ω} φ(x, Du(x)).

Proof. Thanks to Proposition 5.1 for every ξ ∈ R^{N} the function x ↦ φ(x, ξ) is measurable in Ω. In order to show that for every x ∈ Ω φ(x, ·) is L-Lipschitz continuous, fix x ∈ Ω and ξ, η ∈ R^{N}. Let u_{ξ} ∈ W^{1,∞}(Ω) be such that F (u_{ξ}) ≤ φ(x, ξ) + ε. Then

φ(x, η) ≤ F (u_{ξ} + φ_{η−ξ}) ≤ F (u_{ξ}) + L|ξ − η| ≤ φ(x, ξ) + L|ξ − η| + ε.

As ε → 0^{+} the last inequality implies

φ(x, η) ≤ φ(x, ξ) + L|ξ − η|

and by changing the roles of ξ and η it follows

|φ(x, η) − φ(x, ξ)| ≤ L|ξ − η|.

⊓ ⊔

Lemma 5.3 Let Ω be an open subset of R^{N}. Let f : Ω × R^{N} → R^{+} be a Carathéodory supremand satisfying assumption (2.7) and let φ and f̃ the functions given respectively by (2.9) and (2.15). Then there exists a negligible set H ⊂ Ω such that

φ ≥ f̃ ≥ f on (Ω \ H) × R^{N}.

Proof. By definition, it easily follows that φ ≥ f̃. Now, for every ξ ∈ Q^{N} let N_{ξ} be the negligible set such that Ω \ N_{ξ} is the set of the Lebesgue points of f(·, ξ). Then ⋃_{ξ∈Q^{N}} N_{ξ} is a negligible set and if x ∈ Ω′ = Ω \ ( ⋃_{ξ∈Q^{N}} N_{ξ}) then x is a Lebesgue point of f(·, η). In fact, let (ξ_{n})_{n} ⊂ Q^{N} be such that ξ_{n} → η. Fix ε > 0 and n_{0} ∈ N such that |ξ_{n} − η| ≤ ε for every n ≥ n_{0}. Then for every n ≥ n_{0} and for every ρ > 0

∫_{B_{ρ}(x)} |f(y, η) − f(x, η)|dy
≤ ∫_{B_{ρ}(x)} |f(y, η) − f(y, ξ_{n})|dy + ∫_{B_{ρ}(x)} |f(y, ξ_{n}) − f(x, ξ_{n})|dy + ∫_{B_{ρ}(x)} |f(x, ξ_{n}) − f(x, η)|dy
≤ ∫_{B_{ρ}(x)} 2w_{M}(|η − ξ_{n}|)dy + ∫_{B_{ρ}(x)} |f(y, ξ_{n}) − f(x, ξ_{n})|dy
≤ 2|B_{ρ}|w_{M}(ε) + ∫_{B_{ρ}(x)} |f(y, ξ_{n}) − f(x, ξ_{n})|dy.

Letting ρ → 0 we obtain

lim_{ρ→0} \frac{1}{|B_{ρ}|} ∫_{B_{ρ}(x)} |f(y, η) − f(x, η)|dy ≤ 2w_{M}(ε)

and from the arbitrariness of ε we can conclude. Now we will show that for every x ∈ Ω′ and for every ξ ∈ R^{N} it holds

f̃(x, ξ) ≥ f(x, ξ).  (5.41)

Note that Ω \ Ω′ is a negligible set. Fix x ∈ Ω′ and ξ ∈ R^{N}. Then there exists B_{r}(x) and u_{x} ∈ W^{1,∞}(Ω) such that x ∈ û_{x}, Du_{x}(x) = ξ and f̃(x, ξ) ≥ ess sup_{B_{r}(x)} f(y, Du_{x}(y))−ε. If we show that x is a Lebesgue point of h = f(·, Du_{x}(·)) then

f̃(x, ξ) ≥ ess sup_{B_{r}(x)} f(y, Du_{x}(y)) − ε ≥ f(x, ξ) − ε

for every ε > 0 and thus we can conclude. Let us show that x is a Lebesgue point of the function h(y) := f(y, Du_{x}(y)). For M = ||Du_{x}||_{∞} we have

∫_{B_{ρ}(x)} |f(y, Du_{x}(y)) − f(x, Du_{x}(x))|dy ≤
≤ ∫_{B_{ρ}(x)} |f(y, Du_{x}(y)) − f(y, Du_{x}(x)|dy + ∫_{B_{ρ}(x)} |f(y, Du_{x}(x)) − f(x, Du_{x}(x))|dy
≤ ∫_{B_{ρ}(x)} ω_{M}(|Du_{x}(y) − Du_{x}(x)|)dy + ∫_{B_{ρ}(x)} |f(y, Du_{x}(x)) − f(x, Du_{x}(x))|dy.

Since x ∈ Ω′ we have that

\frac{1}{|B_{ρ}|} ∫_{B_{ρ}(x)} |f(y, Du_{x}(x)) − f(x, Du_{x}(x))|dy → 0  (5.42)

when ρ → 0. Then for fixed ε > 0 there exists r_{0} = r_{0}(ε) such that for every ρ ≤ r_{0}

∫_{B_{ρ}(x)} |Du_{x}(y) − Du_{x}(x)|dy ≤ ε|B_{ρ}|.

By Cebishev Theorem we have that

|(\{y ∈ B_{ρ}(x) : |Du(y) − Du(x)| ≥ \sqrt{ε}\}| ≤ \frac{1}{\sqrt{ε}} ∫_{B_{ρ}(x)} |Du_{x}(y) − Du_{x}(x)|dy ≤ \sqrt{ε}|B_{ρ}|.

Thus for every ρ ≤ r_{0}, we have

∫_{B_{ρ}(x)} ω_{M}(|Du_{x}(y) − Du_{x}(x)|)dy = ∫_{\{y∈B_{ρ}(x):|D(y)−Du(x)|≥\sqrt{ε}\}} ω_{M}(|Du_{x}(y) − Du_{x}(x)|)dy +
+ ∫_{\{y∈B_{ρ}(x):|Du(y)−Du(x)|≤\sqrt{ε}\}} ω_{M}(|Du_{x}(y) − Du_{x}(x)|)dy ≤
≤ C\sqrt{ε}|B_{ρ}| + ω_{M}(\sqrt{ε})|B_{ρ}|

where C = max{ω_{M}(ξ) : |ξ| ≤ 2M}. Thus

lim_{ρ→0^{+}} \frac{1}{|B_{ρ}|} ∫_{B_{ρ}(x)} ω_{M}(|Du_{x}(y) − Du_{x}(x)|)dy ≤ C\sqrt{ε} + ω_{M}(\sqrt{ε})

for every ε > 0 and thus

lim_{ρ→0^{+}} \frac{1}{|B_{ρ}|} ∫_{B_{ρ}(x)} ω_{M}(|Du_{x}(y) − Du_{x}(x)|)dy = 0.  (5.43)

In particular from (5.42) and (5.43) it follows that

\frac{1}{|B_{ρ}|} ∫_{B_{ρ}(x)} |f(y, Du_{x}(y)) − f(x, Du_{x}(x))|dy → 0

when ρ → 0. This completes the proof of (5.41). ⊓ ⊔

We underline the difference between the following theorem and the representation result shown in Theorem 2.2. in [11]. We prove that the function f̃ is an admissible supremand for a localized supremal functional F (u, A) = ess sup_{A} f(x, Du(x)) without requiring that for every open set A ⊂ Ω F (·, A) is weakly* lower semicontinuous.

Theorem 5.4 Let Ω be an open subset of R^{N}. Let f : Ω × R^{N} → R^{+} be a Carathéodory supremand satisfying assumptions (2.7) and (2.8). Then (i) the functions φ and f̃ given respectively by (2.9) and (2.15) are Carathéodory supremands satisfying

(2.8) and (2.7) for a suitable family (ω′_{M})_{M} of moduli of continuity;

(ii) for every u ∈ W^{1,∞}(Ω) and for every A ∈ A

ess sup_{Ω} f(x, Du(x)) = ess sup_{Ω} φ(x, Du(x))

and

ess sup_{A} f(x, Du(x)) = ess sup_{A} f̃(x, Du(x));

(iii) if f(·, ξ) is continuous on Ω for every ξ ∈ R^{N} then there exists a negligible set H such that f = f̃ on (Ω \ H) × R^{N}.

Proof. By applying Proposition 5.1 to the functional F : W^{1,∞}(Ω) → \overline{R} defined by

F (u) = ess sup_{Ω} f(x, Du(x))

we obtain that the function φ is a Carathéodory supremand satisfying (2.8) and (2.7) for a suitable family (ω′_{M})_{M} of moduli of continuity and such that

ess sup_{Ω} f(x, Du(x)) ≥ ess sup_{Ω} φ(x, Du(x))

for any u ∈ W^{1,∞}(Ω). Thanks to Lemma 5.3 the converse inequality follows.

Now let us choose a countable base (A_{n})_{n∈N} of open subsets of Ω and for every n ∈ N and for every u ∈ W^{1,∞}(Ω) define F_{n}(u) := ess sup_{A_{n}} f(x, Du(x)). By applying Proposition 5.1 to the functional F_{n} we obtain that for every n ∈ N

φ_{n}(x, ξ) := inf {ess sup_{A_{n}} f(x, Du(x)) | u ∈ W^{1,∞}(A_{n}) s.t. x ∈ û, with Du(x) = ξ }

is a Carathéodory supremand such that

ess sup_{A_{n}} f(x, Du(x)) ≥ ess sup_{A_{n}} φ_{n}(x, Du(x))  (5.44)

for any u ∈ W^{1,∞}(A_{n}). Since

f̃ = inf φ_{n},

we have that f̃ is a Borel function. Moreover by applying Proposition 5.1 for any M > 0 there exists some modulus of continuity ω′_{M} such that

|φ_{n}(x, ξ) − φ_{n}(x, η)| ≤ ω′_{M}(|ξ − η|)

for a.e. x ∈ Ω and for every ξ, η ∈ B_{M}(0) and for every n ∈ N. This implies that

|f̃(x, ξ) − f̃(x, η)| ≤ ω′_{M}(|ξ − η|)

for a.e. x ∈ Ω and for every ξ, η ∈ B_{M}(0). Therefore f̃ is a Carathéodory supremand. Finally thanks to (5.44) we have that for every n ∈ N

ess sup_{A_{n}} f(x, Du(x)) ≥ ess sup_{A_{n}} f̃(x, Du(x))

for any u ∈ W^{1,∞}(A_{n}). Thanks to Lemma 5.3 the converse inequalities follow and since (A_{n})_{n∈N} is a countable base of open subsets of Ω it follows that

ess sup_{A} f(x, Du(x)) = ess sup_{A} f̃(x, Du(x))

for any u ∈ W^{1,∞}(Ω) and for every A ∈ A.

Finally in order to show (iii) thanks to (5.41) it is sufficient to show that f(x_{0}, ξ) ≥ f̃(x_{0}, ξ) for every x_{0} ∈ Ω′. Fix x_{0} ∈ Ω′. By the definition of f̃

ess sup_{B_{r}(x_{0})} f(x, ξ) ≥ f̃(x_{0}, ξ).

By letting r → 0 and by using the continuity of f(·, ξ) it easily follows f(x_{0}, ξ) ≥ f̃(x_{0}, ξ). ⊓ ⊔

Finally we consider the particular case in which f(x, ·) is a 1-homogeneous function. Note that inequality (5.45) cannot be proved directly by applying Lemma 5.3 since f does not satisfy (2.7).

Theorem 5.5 Let Ω be an open subset of R^{N}. Let f : Ω × R^{N} → R^{+} be a Carathéodory supremand satisfying assumption (2.10) and (2.11). Then the function φ given by (2.9) is a Carathéodory supremand satisfying assumption (2.10) and (2.11) and such that

(i) for a.e. x ∈ Ω

φ(x, ξ) ≥ f(x, ξ) ∀ξ ∈ R^{N};  (5.45)

(ii) for every u ∈ W^{1,∞}(Ω)

ess sup_{Ω} f(x, Du(x)) = ess sup_{Ω} φ(x, Du(x)).

Proof. Let F be the functional defined by

F (u) = ess sup_{Ω} f(x, Du(x))

for all u ∈ W^{1,∞}(Ω). In order to show (i) of definition (2.1), let ξ ∈ R^{N} and λ ∈ R be fixed. Consider the sets A(x) and K_{λ} defined respectively by (5.36), (5.37). If we prove that K_{λ} is measurable for every λ ∈ R, then φ(·, ξ) is measurable. Suppose that K_{λ} is not measurable. Then there is a set C with K_{λ} ⊂ C s.t. C is measurable and of minimal measure. In fact, let x_{0} ∈ Ĉ \ K_{λ} where Ĉ the set of the points of density 1 of C. From the definition of K_{λ}, there is some u ∈ A(x_{0}) such that F (u) < λ. Now, fix ε > 0 such that F (u) < λ − ε and set δ := \frac{ε}{β} . By (2.10) we have that F (v) ≤ ε for every v ∈ W^{1,∞}(Ω) such that ||v||_{W^{1,∞}(Ω)} ≤ δ. Now consider the set A_{1} defined by (5.38). We claim that A_{1} ∩ K_{λ} = ∅. Now, if x ∈ A_{1}, then the function v_{x} ∈ W^{1,∞}(Ω) defined by

v_{x}(y) := u(y) + (u(x_{0}) − u(x)) + ⟨Du(x_{0}) − Du(x), y − x⟩

belongs to A(x) and ∥v_{x} − u∥_{W^{1,∞}(Ω)} ≤ δ. By Proposition 3.4 we can write F as

F (u) = sup_{x,y∈Ω, x≠y} \frac{u(x) − u(y)}{d_{1}(x, y)}

where d_{1} is given by (3.27). In particular, by Proposition 3.1

F (u + v) ≤ F (u) + F (v)  (5.46)

for every u, v ∈ W^{1,∞}(Ω). This implies

F (v_{x}) ≤ F (v_{x} − u) + F (u) < ε + λ − ε = λ.

So x ∉ K_{λ}. This implies that K_{λ} ⊂ C \ A_{1}. Moreover the set C\A_{1} is still measurable. Repeating the proof of Proposition 5.1 one can show that

|A_{1} ∩ C| ≥ \frac{1}{2}|B_{r}(x_{0})|

which implies that |C \ A_{1}| < |C| and since K ⊂ (C\A_{1}), we have contradicted the minimality of C. Now we show (ii) of definition (2.1). Let us fix x ∈ Ω, ε > 0 and ξ_{1}, ξ_{2} ∈ R^{N} such that |ξ_{1} − ξ_{2}| ≤ min\{\frac{ε}{2βdiam(Ω)}, \frac{ε}{2β} \}. From the definition of φ we can find some u ∈ W^{1,∞}(Ω) such that Du(x) = ξ_{1} and φ(x, ξ_{1}) ≥ F (u) − ϵ. Then, defined w_{x,ξ_{i}}(y) := ξ_{i} · (y − x), we have

∥w_{x,ξ_{1}} − w_{x,ξ_{2}}∥_{W^{1,∞}(Ω))} ≤ \frac{ε}{β}

and thus

φ(x, ξ_{2}) ≤ F (u + w_{x,ξ_{2}} − w_{x,ξ_{1}}) ≤ F (u) + F (w_{x,ξ_{2}} − w_{x,ξ_{1}}) ≤ φ(x, ξ_{1}) + ε.

By changing the roles of ξ_{1} and ξ_{2} we obtain that φ(x, ·) is uniformly continuous on R^{N}. Concerning (5.45), for every n ∈ N one can define

f_{n}(x, ξ) = inf_{η∈R^{N}}{f(x, η) ∨ n|ξ − η|}.  (5.47)

Then, thanks to Proposition 4.2 for every n ∈ N f_{n} is a Lipschitz continuous function with Lipschitz constant equal to n, f_{n} ≤ f_{m} if n ≤ m and since f(x, ·) is continuous, we have

f(x, ξ) = sup {f_{n}(x, ξ) : n ∈ N}.

Now for every n ∈ N let ψ_{n} be defined by

ψ_{n}(x, ξ) := inf {ess sup_{Ω} f_{n}(x, Du(x)) | u ∈ W^{1,∞}(Ω) s.t. x ∈ û, with Du(x) = ξ } .  (5.48)

By definition, we have that ψ_{n} ≤ φ for every n ∈ N and, thanks to Theorem 5.4, ψ_{n} is a Carathéodory supremand such that there exists a negligible set H_{n} ⊂ Ω such that

ψ_{n} ≥ f_{n} on (Ω \ H_{n}) × R^{N}.

In particular, by defining H = ⋃_{n=1}^{∞} H_{n} we obtain that

φ(x, ξ) ≥ ⋁_{n∈N} ψ_{n}(x, ξ) ≥ ⋁_{n∈N} f_{n}(x, ξ) = f(x, ξ) on (Ω \ H) × R^{N}

which implies that for every u ∈ W^{1,∞}(Ω)

ess sup_{Ω} φ(x, Du(x)) ≥ ess sup_{Ω} f(x, Du(x)).

By the definition of φ the converse inequality follows and thus (ii). ⊓ ⊔

6 The proofs

Now we are in position to show the main theorems of this paper. In their proofs it is fundamental the application of Lemma 3.2 in order to deduce the convexity of the sub level sets of a weakly* lower semicontinuous functional.

Proof of Theorem 2.3. Suppose that f is a Carathéodory supremand satisfying also the coercivity assumption (2.8). Let λ ∈ R be such that the sub level set E_{λ} := {u ∈ W^{1,∞}(Ω) : F (u) ≤ λ} is nonempty. If K_{λ} := {u ∈ W^{1,∞}(Ω) : F (u) < λ} is nonempty too, then let R_{λ} be the corresponding difference quotient defined by (3.20). Now we show that

E_{λ} = {u ∈ W^{1,∞}(Ω) : R_{λ}(u) ≤ 1}.  (6.49)

In fact, if R_{λ}(u) ≤ 1 then for every 0 < δ < 1 we have that R_{λ}(δu) ≤ δ < 1. By Lemma 3.2 there exists a sequence {u_{n}} ⊂ W^{1,∞}(Ω) converging to δu in L^{∞}(Ω) with F (u_{n}) ≤ λ. Since F is weakly* lower semicontinuous it follows F (δu) ≤ λ and then, for the same reason, when δ → 1 we can conclude F (u) ≤ λ. Vice versa, if F (u) ≤ λ then for every x, y ∈ Ω we have d_{λ}(x, y) ≥ u(x) − u(y) by definition of d_{λ}. This implies R_{λ}(u) ≤ 1. By (6.49), by applying Proposition 3.1, it follows that E_{λ} is a convex set. Finally, if K_{λ} is empty, note that K_{λ+ε} is nonempty for every ε > 0 since E_{λ} ⊂ K_{λ+ε} and from the first part of this proof it follows that E_{λ+ε} is a convex set. Since

E_{λ} = ⋂_{ε>0} E_{λ+ε}

then E_{λ} is a convex set too. Now we remove the previous coercivity assumption on f. Define

g(x, ξ) := arctan(f(x, ξ)) + \frac{π}{2} )

and for every n ∈ N consider the function g_{n} given by (4.31) and the functionals G and G_{n} given respectively by (4.32) and (4.33). Then it holds:

(i) g_{n}(x, ξ) ≥ \frac{1}{n}|ξ| for every n ∈ N, for a.e. x ∈ Ω and for every ξ ∈ R^{N};

(ii) for every n ∈ N, for a.e. x ∈ Ω and for every ξ, η ∈ B_{M}(0)

|g_{n}(x, ξ) − g_{n}(x, η)| ≤ ω′_{M,n}(|ξ − η|)  (6.50)

where ω′_{M,n}(s) := ω_{M}(s) ∨ \frac{1}{n}|s|.

Property (i) is trivial; concerning (ii), fix n ∈ N and let x ∈ Ω, ξ, η ∈ B_{M}(0). If g_{n}(x, ξ) ≥ \frac{1}{n}|ξ| and g_{n}(x, η) ≥ \frac{1}{n}|η| then (6.50) is trivial. If g_{n}(x, ξ) ≤ \frac{1}{n}|ξ| and g_{n}(x, η) ≤ \frac{1}{n}|η| then (6.50) is trivial. It remains to show (6.50) when g_{n}(x, ξ) < \frac{1}{n}|ξ| and g_{n}(x, η) > \frac{1}{n}|η|. In this case

g_{n}(x, ξ) − g_{n}(x, η) = \frac{1}{n}|ξ| ∨ g(x, ξ) − g(x, η)
≤ (\frac{1}{n}|ξ − η| + \frac{1}{n}|η|) ∨ (g(x, η) + ω_{M}(|ξ − η|) − g(x, η)
≤ \frac{1}{n}|ξ − η| ∨ ω_{M}(|ξ − η|)

and

g_{n}(x, ξ) − g_{n}(x, η) = \frac{1}{n}|ξ| − g(x, η) ∨ \frac{1}{n}|η|
≥ \frac{1}{n}|ξ| − (g(x, ξ) + ω_{M}(|ξ − η|) ∨ (\frac{1}{n}|ξ − η| + \frac{1}{n}|ξ|)
≥ −(\frac{1}{n}|ξ − η| ∨ ω_{M}(|ξ − η|)).

Now, since the function h(s) = arctan s is uniformly continuous and increasing on R, then the supremal functional G is weakly* lower semicontinuous on W^{1,∞}(Ω). Thus, thanks to (4.34), for every n ∈ N the supremal functional G_{n} is a weakly* lower semicontinuous on W^{1,∞}(Ω). Since G_{n} is represented through a coercive function, from the first part of this proof it follows that (G_{n})_{n} is a sequence of level convex functionals. Since (G_{n})_{n} pointwise converges to G, then G is a level convex functional too. As easy consequence F is a level convex functional. ⊓ ⊔

Proof of Theorem 2.4. One implication is trivial. Now assume that F is a level convex functional. With every n ∈ N let F_{n} : W^{1,∞}(Ω) → R be the functional defined by

F_{n}(u) := inf {F (v) ∨ n||Dv − Du||_{∞} : v ∈ W^{1,∞}(Ω)}.

Then thanks to Proposition 4.1 for every n ∈ N F_{n} is an n-Lipschitz continuous functional such that

F = ⋁_{n} F_{n}.

Moreover for every n ∈ N F_{n} is a level convex functional. In fact let u_{1}, u_{2} ∈ W^{1,∞}(Ω) and θ ∈ (0, 1). For fixed ε > 0 there exist v_{1}, v_{2} ∈ W^{1,∞}(Ω) such that

F_{n}(u_{i}) ≥ F (v_{i}) ∨ n||Dv_{i} − Du_{i}||_{∞} − ε

for every i ∈ {1, 2}. Then

F_{n}(θu_{1} + (1 − θ)u_{2}) ≤ F (θv_{1} + (1 − θ)v_{2}) ∨ n||D(θu_{1} + (1 − θ)u_{2}) − D(θv_{1} + (1 − θ)v_{2})||_{∞}
≤ F (v_{1}) ∨ F (v_{2}) ∨ n||Du_{1} − Dv_{1}||_{∞} ∨ n||Du_{2} − Dv_{2})||_{∞}
≤ F_{n}(u_{1}) ∨ F_{n}(u_{2}) + ε.

Since ε is arbitrary, it follows that

F_{n}(θu_{1} + (1 − θ)u_{2} ≤ F_{n}(u_{1}) ∨ F_{n}(u_{2}),

i.e. F_{n} is a level convex functional. Now define

φ_{n}(x, ξ) := inf {F_{n}(u) | u ∈ W^{1,∞}(Ω) s.t. x ∈ û, with Du(x) = ξ } .  (6.51)

Thanks to Proposition 5.2, for every n ∈ N φ_{n} is a Carathéodory supremand, n-Lipschitz continuous w.r.t. ξ and such that

ess sup_{Ω} φ_{n}(x, Du(x)) ≤ F_{n}(u).  (6.52)

Moreover for every n ∈ N φ_{n} is a level convex function. Fix x ∈ Ω, ξ, η ∈ R^{N} and λ ∈ (0, 1). By the definition of φ_{n} there exist B_{r}(x), u_{ε}, v_{ε} ∈ W^{1,∞}(Ω), differentiable at x such that Du_{ε}(x) = ξ, Dv_{ε}(x) = η and φ_{n}(x, ξ) ≥ F_{n}(u_{ε}) − ε and φ_{n}(x, η) ≥ F_{n}(v_{ε}) − ε. Since F_{n} is a level convex functional then

φ_{n}(x, λξ + (1 − λ)η) ≤ F_{n}(λu_{ε} + (1 − λ)v_{ε}) ≤ F_{n}(u_{ε}) ∨ F_{n}(v_{ε}) ≤ (φ_{n}(x, ξ) − ε) ∨ (φ_{n}(x, η) − ε)

By ε → 0 we obtain the thesis. The sequence {φ_{n}}_{n} is not decreasing and thus

∃ lim_{n} φ_{n}(x, ξ) = ⋁_{n} φ_{n}(x, ξ) =: φ(x, ξ).

It is easy to verify that φ is a level convex normal supremand and getting n → ∞ in (6.52) we obtain

ess sup_{Ω} φ(x, Du(x)) ≤ F (u).

In order to show the converse inequality, for every n ∈ N we re-introduce the functions f_{n}, ψ_{n} defined by (5.47) and (5.48). Now for every n ∈ N and for every u ∈ W^{1,∞}(Ω) it holds

ess sup_{Ω} f_{n}(x, Du(x)) ≤ F_{n}(u).  (6.53)

In fact, by definition, for fixed ε > 0 there exists u_{ε} ∈ W^{1,∞}(Ω) such that

F_{n}(u) ≥ ess sup_{Ω} f(x, Du_{ε}(x)) ∨ n||Du_{ε} − Du||_{∞} − ε.

In particular for a.e. x ∈ Ω

F_{n}(u) ≥ f(x, Du_{ε}(x)) ∨ n|Du_{ε}(x) − Du(x)| − ε ≥ f_{n}(x, Du(x)) − ε

which implies

F_{n}(u) ≥ ess sup_{Ω} f_{n}(x, Du(x)) − ε.

Since ε is arbitrary, then (6.53) follows. From (6.53) and by the definitions of f_{n}, ψ_{n}, φ_{n} we obtain

f_{n}(x, ξ) ≤ ψ_{n}(x, ξ) ≤ φ_{n}(x, ξ) on (Ω \ H) × R^{N}

where H ⊂ Ω is a negligible set. By passing to the limit when n → ∞ in the previous inequality and by applying the fact that

f(x, ξ) = ⋁_{n∈N} f_{n}(x, ξ)

we obtain that

f(x, ξ) ≤ φ(x, ξ) on (Ω \ H) × R^{N}

which implies that

F (u) ≤ ess sup_{Ω} φ(x, Du(x)).

Proof of Theorem 2.5. 1)⟹ 2). It follows by Theorem 2.3. 2)⟹ 3). Thanks to Lemma 5.3 and Theorem 5.4 it remains to show that φ is level convex w.r.t. ξ. Fix x ∈ Ω, ξ, η ∈ R^{N} and λ ∈ (0, 1). By definition of φ there exist u_{ε}, v_{ε} ∈ W^{1,∞}(Ω), differentiable at x such that Du_{ε}(x) = ξ, Dv_{ε}(x) = η and φ(x, ξ) ≥ F (u_{ε}) − ε and φ(x, η) ≥ F (v_{ε}) − ε. Since F is a level convex functional, we have that

φ(x, λξ + (1 − λ)η) ≤ F (λu_{ε} + (1 − λ)v_{ε}) ≤ F (u_{ε}) ∨ F (v_{ε}) ≤ (φ(x, ξ) − ε) ∨ (φ(x, η) − ε)

By ε → 0 we obtain the thesis.

3)⟹ 1) It follows by Theorem 1.1. ⊓ ⊔

Proof of Theorem 2.6. 1)⟹ 2) By applying Proposition 3.4 we have that

F (u) = sup_{x,y∈Ω, x≠y} \frac{u(x) − u(y)}{d(x, y)}.

Then F is convex (see for instance the proof of Proposition 3.1).

2)⟹ 3) Thanks to Theorem 5.5 we can represent F through the function φ given by (2.9), i.e. F (u) = ess sup_{Ω} φ(x, Du(x)) for every u ∈ W^{1,∞}(Ω). If we show that φ is convex w.r.t. ξ then by applying Proposition 2.4 and Proposition 2.5 in [16] it follows that F can be represented also through the function φ_{d}^{0}. Fix x ∈ Ω, ξ, η ∈ R^{N} and λ ∈ (0, 1). By definition of φ there exist u_{ε}, v_{ε} ∈ W^{1,∞}(Ω), differentiable at x such that Du_{ε}(x) = ξ, Dv_{ε}(x) = η and φ(x, ξ) ≥ F (u_{ε}) − ε and φ(x, η) ≥ F (v_{ε}) − ε. Since F is a convex functional, we have that

φ(x, λξ + (1 − λ)η) ≤ F (λu_{ε} + (1 − λ)v_{ε}) ≤ λF (u_{ε}) + (1 − λ)F (v_{ε}) ≤ λ(φ(x, ξ) − ε) + (1 − λ)(φ(x, η) − ε),

and getting ε → 0 we obtain that φ is convex w.r.t. ξ. Finally , thanks to Lemma 5.3, we have that there exists a negligible set H ⊂ Ω such that

φ_{d}^{0}(x, ξ) ≤ φ(x, ξ) for every x ∈ Ω \ H, for ξ ∈ R^{N}.

The converse inequality follows by applying Proposition 1.6 in [16].

3)⟹ 1) It follows by Theorem 1.1. ⊓ ⊔

Proof of Theorem 2.7. 1) ⟹ 2) For every open subset A ⊂ Ω with Lipschitz continuous boundary it is sufficient to apply Theorem 2.3 to the functional F (u) := F (u, A). Now let A ⊂ Ω be a generic open set and u_{1}, u_{2} ∈ W^{1,∞}(Ω) and θ ∈ (0, 1). Then there exists a countable family of open sets (A_{n})_{n∈N} with Lipschitz continuous boundaries such that A = ⋃_{n} A_{n}. For every n ∈ N we have that

F (θu_{1} + (1 − θ)u_{2}, A_{n}) ≤ F (v_{1}, A_{n}) ∨ F (v_{2}, A_{n}).

This implies

F ((θu_{1}+(1−θ)u_{2}, A) = ⋁_{n∈N} F (θu_{1}+(1−θ)u_{2}, A_{n}) ≤ ⋁_{n∈N} F (v_{1}, A_{n})∨ ⋁_{n∈N} F (v_{2}, A_{n}) = F (v_{1}, A)∨F (v_{2}, A).

2)⟹ 3) Since f satisfies all the assumptions of Theorem 5.4, F can be represented also by the Carathéodory supremand f̃. It remains to show that f̃ is a level convex function. Fix x ∈ Ω, ξ, η ∈ R^{N} and λ ∈ (0, 1). By the definition of f̃ there exist B_{r}(x), u_{ε}, v_{ε} ∈ W^{1,∞}(Ω), differentiable at x such that Du_{ε}(x) = ξ, Dv_{ε}(x) = η and f̃(x, ξ) ≥ F (u_{ε}, B_{r}(x))−ε and f̃(x, η) ≥ F (v_{ε}, B_{r}(x))−ε. Since F (·, B_{r}(x)) is a level convex functional then

f̃(x, λξ +(1−λ)η) ≤ F (λu_{ε} +(1−λ)v_{ε}, B_{r}(x)) ≤ F (u_{ε}, B_{r}(x))∨F (v_{ε}, B_{r}(x)) ≤ (f̃(x, ξ)−ε)∨(f̃(x, η)−ε)

By ε → 0 we obtain the thesis.

3)⟹ 1) It follows by Theorem 1.1. ⊓ ⊔

7 An existence theorem of AMLs and a principle of comparison

In general a minimizer for a supremal functional is not necessarily a local minimizer (see Example 1.2. in [12]). Then by analogy with the case of integral functionals, G. Aronsson introduced the following notion of local minimizers for a supremal functional of the form

F (u, A) = ess sup_{A} f(x, Du(x))  (7.54)

(for instance see [3], [4], [5]).

Definition 7.1 Let g be a Lipschitz function defined on ∂Ω and let us denote by W_{g}^{1,∞}(Ω) the space of functions such that (u − g) ∈ W^{1,∞}(Ω) ∩ C_{0}(Ω). A absolute minimizer or an AML for the problem

min_{v∈W_{g}^{1,∞}(Ω)} F (v, Ω)  (7.55)

(where F is given by (7.54)) is a minimizer u such that for all open subset V ⊂⊂ Ω one has

F (u, V) ≤ F (v, V)

for all v in W^{1,∞}(V) such that v = u on ∂V.

With different techniques Barron, Jensen and Wang in [7] and Champion, De Pascale, Prinari in [12] have proved an existence theorem of AML for a supremal functional F by assuming that it is represented by a level convex function f. Now, thanks to Theorem 2.7 we can give the following:

Theorem 7.2 Let f : Ω×R^{N} → R be a Carathéodory supremand satisfying (2.7) and (2.8). Let F (·, A) be the functional defined by (7.54). Let g be a Lipschitz function defined on ∂Ω. If F (·, A) is lower semicontinuous with respect to the weak* convergence of W^{1,∞}(Ω) for every A ∈ A then there exists at least an absolute minimizer v ∈ W_{g}^{1,∞}(Ω).

Proof. By applying Theorem 2.7, we can represent F through a level convex Carathéodory supremand satisfying (2.7) and (2.8). Then we can conclude by applying Theorem 4.1 in [12]. ⊓ ⊔

Remark 7.3 In [12] (see Theorem 4.7) the authors give another existence theorem for AML based on a Perron-like method. They show that if the functional (7.54) is weakly* l.s.c. and coercive and satisfies the additional hypothesis that for any A ∈ A, w ∈ W^{1,∞}(A) ∩ C(Ā) and y ∈ A, the image set

A_{y,w} = {u(y) : u ∈ W_{w}^{1,∞}(A), u is an AML}

is connected then then there exists at least an absolute minimizer v ∈ W_{g}^{1,∞}(Ω). Now if the functional (7.54) is weakly* lower semicontinuous for every A ∈ A and f satisfies (2.7) and (2.8) then the last assumption is trivially satisfied. In fact, thanks to Theorem 2.3 F is a level convex functional and then the sets A_{y,w} are convex.

In Theorem 3.5 of the paper [13] Champion and De Pascale characterize the absolute minimizers of a wide class of supremal functionals by extending the principle of comparison with cones introduced by Crandall, Evans and Gariepy in [1] for the minimizing Lipschitz Extension Problem. Their characterization relies on the fact that when f = f(x, ξ) is a lower semicontinuous function, satisfying (2.8) and level convex in the ξ-variable, then for every open set V ⊂ R^{N} the pseudo-distances

ρ_{λ}^{V} (x, y) = sup {u(x) − u(y), u ∈ W^{1,∞}(V) : ess sup_{V} f(x, Du(x)) ≤ λ}  (7.56)

coincide with the following distances:

δ_{λ}^{V} (x, y) = inf_{γ∈Γ_{x,y}(V)} ∫_{0}^{1} f^{0}(γ, γ′, λ) dt

where f^{0} is defined by

f^{0}(x, ξ, λ) := sup {ξ · η : f(x, η) ≤ λ}

(see Lemma B.3 and Proposition A.2 in [13]). Now if f is not lower semicontinuous this equality could fail. In fact let Ω = (−2, 2)^{2} and consider the segment S = (−1, 1) × {0}. Consider the Carathéodory supremand defined by

f(x, ξ) = \{ \begin{matrix}β|ξ| & if x ∈ Ω \ S \\ α|ξ| & if x ∈ S \end{matrix}

with 0 < β < α. Then f is not globally lower semicontinuous on Ω and for every λ ≥ 0 and for every x, y ∈ V = (−1, 1)^{2} it holds

δ_{λ}^{V} (x, y) = \frac{λ}{β} |x − y|.

If x̄ = (−1, 0) e ȳ = (0, 1) then

δ_{λ}^{V} (x̄, ȳ) = \frac{2λ}{α} < \frac{2λ}{β} = ρ_{λ}^{V} (x̄, ȳ).

This inequality is due to the fact that when one modifies the values of the supremand f on a negligible subset of Ω the distances δ_{λ}^{V} can change while the distances ρ_{λ}^{V} does not depend on the supremand f chosen to represent F.

Now when f is not globally l.s.c. it is an open problem if it is possible to characterize the AMLs of the problem (7.55) through a comparison principle with distance functions. However thanks to the results shown in Sections 3 and 6, we can give a partial result. We consider a supremal functional F represented by a Carathéodory supremand f : Ω × R^{N} → R^{+}, level convex in the ξ-variable, satisfying (2.7) and (2.8). For any open set V ⊂⊂ Ω, for every x, y ∈ V and for every λ ∈ R we define

d_{λ}^{V} (x, y) := \{ \begin{matrix}sup \{|u(x) − u(y)| : u ∈ W^{1,∞}(Ω), F (u, V) ≤ λ\} if λ > inf_{v∈W^{1,∞}(V)} F (v, V) \\ inf_{ε>0} d_{λ+ε}^{V}(x, y) if λ = inf_{v∈W^{1,∞}(V)} F (v, V) \\ −∞ if λ < inf_{v∈W^{1,∞}(V)} F (v, V). \end{matrix}

(7.57)

Moreover for every x, y ∈ V̄ and for every λ ∈ R we define

d_{λ}^{V} (x, y) := inf{lim inf_{n} d_{λ}^{V} (x_{n}, y_{n}) : (x_{n})_{n}, (y_{n})_{n} ⊂ V, x_{n} → x, y_{n} → y}.  (7.58)

We point out that since the boundary of V may be not regular, it may happen that d_{λ}^{V} (x̄, ȳ) = +∞ for some λ ∈ R and for some x̄ ∈ ∂V and ȳ ∈ V. In this case it is easy to show that d_{λ}^{V} (x̄, y) = +∞ for every y ∈ V.

Definition 7.4 We shall say that a continuous function u : Ω̄ → R satisfies the Comparison with the Distance Functions d_{λ}^{V} from above in Ω if and only if for any connected open subset V ⊂⊂ Ω, any x_{0} ∈ V̄, any λ ∈ R and α ∈ R the inequality

u ≤ d_{λ}^{V} (x_{0}, .) + α on ∂(V \ {x_{0}})

implies

u ≤ d_{λ}^{V} (x_{0}, .) + α on V̄ .

Now we can easily show that

Theorem 7.5 Under the assumption of Theorem 7.2 if u ∈ W^{1,∞}(Ω) satisfies the Comparison with the Distance Functions d_{λ}^{V} on Ω from above then u is an absolute minimizer of F.

Proof. Assume that u satisfies the comparison principle with the Distance Functions d_{λ}^{V}. Let V ⊂⊂ Ω be an open set. We will show that

F (u, V) = inf{F (v, V) : v ∈ W^{1,∞}(V), v = u on ∂V }.

In fact since F (·, V) is weakly* l.s.c. and coercive there exists w ∈ W^{1,∞}(V), such that

F (w, V) = λ = min{F (v, V) : v ∈ W^{1,∞}(V), v = u on ∂V }.

If λ > inf{F (v, V) : v ∈ W^{1,∞}(V)} then

|w(x) − w(y)| ≤ d_{λ}^{V} (x, y)  (7.59)

for every x, y ∈ V. Now w − u = 0 on ∂V means that there exists a sequence w_{n} ∈ C_{0}^{∞}(V) weakly* converging to w − u. In particular the sequence (w_{n} + u)_{n} is bounded in W^{1,∞}(Ω) and thus it admits a subsequence which weakly* converges to a function v ∈ W^{1,∞}(Ω). In particular v = w on V and therefore

|v(x) − v(y)| ≤ d_{λ}^{V} (x, y)

for every x, y ∈ V. Since v ∈ C(V̄) and v = u on ∂V, by continuity we obtain that

|u(x) − u(y)| ≤ d_{λ}^{V} (x, y)  (7.60)

for every x, y ∈ ∂V. Since u satisfies the principle of comparison, (7.60) yields to u(x) ≤ u(y) + d_{λ}^{V} (x, y) for every y ∈ ∂V and for every x ∈ V. By applying again the the principle of comparison we obtain that u(x) ≤ u(y) + d_{λ}^{V} (x, y) for every y, x ∈ V i.e. R_{λ}(u, V) := sup_{x,y∈V,x≠y} \frac{|u(x)−u(y)|}{d_{λ}^{V} (x,y)} ≤ 1 and thus for every 0 < δ < 1 R_{λ}(δu, V) ≤ δ < 1. Now we can notice that thanks to Theorem 2.7 F (·, V) is weakly* l.s.c. in W^{1,∞}(V). Thus by applying Lemma 3.3 we have that F (δu, V) ≤ λ. By letting δ → 1 we can conclude that F (u, V) ≤ λ. If λ = min{F (v, V) : v ∈ W^{1,∞}(V)} it is sufficient to note that

|w(x) − w(y)| ≤ d_{λ+ε}^{V}(x, y)

for every x, y ∈ V and for every ε > 0. By repeating the first part of this proof, it is easy to show that F (u, V) ≤ λ + ε. By letting ε → 0^{+} we obtain the thesis. ⊓ ⊔

8 A counterexample

In this section we will show that in general even if F (u) = ess sup_{Ω} f(x, Du(xf)) is weakly* lower semicontinuous on W^{1,∞}(Ω) it cannot be represented by the level convex envelope f^{lc} of f i.e. in general ess sup_{Ω} f^{lc}(x, Du(x)) < F (u).

In fact, with some suitable changes to Example 3.2 in [16], we can construct the following example.

Example 8.1 Let us call G the set of all continuous functions g : R^{N} → R, positively 1-homogeneous and satisfying α|ξ| ≤ g(ξ) ≤ β|ξ| for all ξ ∈ R^{N}, and let

C := {C ⊆ R^{N} : C = {ξ ∈ R^{N} : g(ξ) ≤ 1} for some g ∈ G}.  (8.61)

Note that the sets in C are closed, star-shaped (with respect to the origin), and that by definition to every C ∈ C is associated a function g ∈ G, which we denote by g_{C}. Moreover C is closed for intersection and union.

Let now B be the unit closed ball in R^{N} centered at 0. Then B ∈ C with g_{B}(ξ) = |ξ|. Let H ∈ C be satisfying the following properties:

1) H is not convex;

2) H ∩ B is convex;

3) H \ B ≠ ∅ and B \ H ≠ ∅;

4) B is contained in the convex hull of H .

Finally let us construct an open and dense set A ⊂ Ω with 0 < |A| < |Ω| as follows. Let {v_{i}}_{i∈N} be a dense subset of ∂B and let {p_{j}}_{j∈N} be dense in Ω. For a given positive constant δ > 0 we define

A := ⋃_{i,j∈N} \{x ∈ Ω : dist(x, \{p_{i} + sv_{j}, s ∈ R\}) < \frac{δ}{2^{ij}} \} .

Clearly, if δ is small enough, we have that 0 < |A| < |Ω|. Roughly speaking the set A is given by a countable union of thin strips along a dense set of directions.

We consider the functions f, f_{+} : Ω × R^{N} → R defined by

f(x, ξ) := \{ \begin{matrix}g_{B}(ξ) & if x ∈ A; \\ g_{H}(ξ) & if x ∈ Ω \ A. \end{matrix} f_{+}(x, ξ) := \{ \begin{matrix}g_{B}(ξ) & if x ∈ A; \\ g_{H∩B}(ξ) & if x ∈ Ω \ A. \end{matrix}

The associated supremal functionals are

F (u) := ess sup_{Ω} f(x, Du(x)) and F_{+}(u) := ess sup_{Ω} f_{+}(x, Du(x)).

Now we will show the following facts:

1) F = F_{+} and therefore F is weakly* lower semicontinuous (in fact F_{+} is weakly* lower semicontinuous being represented by a (level) convex function);

2) ∃u ∈ W^{1,∞}(Ω) such that F (u) > ess sup_{Ω} f^{lc}(x, Du(x)).

In fact, by construction we have that F_{+} ≥ F, and so let us assume by contradiction that for some u ∈ W^{1,∞}(Ω) we have

F_{+}(u) > 1  while  F (u) < 1.  (8.62)

This will imply that Du ∈ H \ \overline{B} on a set of positive measure. Therefore there exists a point x ∈ Ω of differentiability for u with |Du(x)| > 1. To simplify the notation we can assume x = 0 and u(0) = 0. Let {ρ_{n}} be a sequence converging to zero, and for every n let us consider the function u_{n} : B → R defined by

u_{n}(x) := \frac{1}{ρ_{n}} u(ρ_{n}x) for every x ∈ B.

By the definition of A, for every n and for every ε > 0 we can find an open strip L_{n}^{ε} in B such that ρ_{n}L_{n}^{ε} ⊂ A and such that L_{n}^{ε} contains two points a_{n}^{ε} and b_{n}^{ε} with

|a_{n}^{ε} − \frac{Du(0)}{|Du(0)|}| + |b_{n}^{ε} − (−\frac{Du(0)}{|Du(0)|})| ≤ ε.  (8.63)

By Proposition 3.4, we have that

ess sup_{L_{n}^{ε}} |Du_{n}(x)| = sup_{x,y∈L_{n}^{ε}} \frac{u_{n}(x) − u_{n}(y)}{|x − y|} ≥ \frac{u_{n}(a_{n}^{ε}) − u_{n}(b_{n}^{ε})}{|a_{n}^{ε} − b_{n}^{ε}|}.

Using that, by the differentiability of u at 0, {u_{n}} converges to Du(0) · x uniformly, by (8.63) we deduce that, for n big enough,

ess sup_{L_{n}^{ε}} |Du_{n}(x)| ≥ |Du(0)| + o(ε),

where o(ε) → 0 as ε → 0. Therefore, recalling that |Du(0)| > 1, we can find ε and n such that ess sup_{L_{n}^{ε}} |Du_{n}(x)| > 1. We conclude that

F (u) ≥ ess sup_{A} |Du(x)| ≥ ess sup_{ρ_{n}L_{n}^{ε}} |Du(x)| = ess sup_{L_{n}^{ε}} |Du_{n}(x)| > 1,

which is in contradiction with (8.62).

Finally let ξ ∈ B such that ξ is not in H ∩ B. In particular ξ belongs to the convex hull of H and therefore

f^{lc}(x, ξ) ≤ 1 a.e. on Ω.

On the other hand by the definition of f_{+} and the choice of ξ we have

f_{+}(x, ξ) > 1 a.e. on Ω \ A.  (8.64)

Therefore if we define ū(x) = x · ξ we obtain

F (ū) > 1 ≥ ess sup_{Ω} f^{lc}(x, ξ) = ess sup_{Ω} f^{lc}(x, Dū(x))

Acknowledgements. I wish to thank Prof. G. Buttazzo, Prof. D. Pallara, Prof. A. Garroni and Prof. M. Ponsiglione for several helpful discussions and suggestions.

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