A sufficient condition for the lower semicontinuity of nonlocal supremal functionals in the vectorial case

We present a sufficient condition ensuring lower semicontinuity for nonlocal supremal functionals of the type where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} is a bounded open subset of RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^N$$\end{document} and W:Ω×Ω×Rd×N×Rd×N→R.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W:\Omega \,{\times }\, \Omega \,{\times }\, {\mathbb {R}}^{d \times N}{\times }\, {\mathbb {R}}^{d \times N} \rightarrow {\mathbb {R}}.$$\end{document}

Motivated by the Direct Methods in the Calculus of Variations the study of necessary and sufficient conditions ensuring lower semicontinuity of such functionals has been conducted in many papers, see [8,10,21,22,23,24,25].
In particular, given a bounded open set Ω ⊂ R N , in [22] characterizing conditions for the sequential lower semicontinuity in L ∞ (Ω; R d ) of the functional G : v ∈ L ∞ (Ω; R d ) → R defined as G(v) := ess sup (x,y)∈Ω×Ω W (v(x), v(y)) (1.1) have been provided.Furthermore in [21] necessary and sufficient assumptions on the supremand W have been determined to ensure that, in absence of lower semicontinuity, the sequentially weakly * lower semicontinuous envelope of G has the same form, i.e. it can be expressed as a double supremal functional.We also emphasize that [21] contains a powerlaw approximation result for functionals as in (1.1), which, on the other hand, also appear in their inhomogeneous version in the context of image denoising (cf.[14]).
Unfortunately, analogous results are not available in the context where the fields v satisfy some differential constraint, in particular when v(x) = ∇u(x), with u ∈ W 1,∞ (Ω; R d ).In this paper we will show that a sufficient condition for the functional to be weakly * sequentially lower semicontinuous in W 1,∞ (Ω; R d×N ) is the separate curl Young quasiconvexity in the second set of variables.The notion of curl Young quasiconvexity has been introduced in [2], as a sufficient condition for the sequential weak * lower semicontinuity of functionals of the type ess sup x∈Ω f (x, ∇u(x)) in W 1,∞ (Ω; R d ), (see also [13] for a similar notion suited for L p approximation of supremal functionals, and [28] for the setting adopted in this paper).
Let Q be the unit cube ]0, 1[ N , and let f : R d×N −→ R be a lower semicontinuous function, bounded from below.f is curl Young quasiconvex, if [20] for the introduction, [29] and [17] for a comprehensive description).For the readers' convenience we just say that Young measures encode information on the oscillation behaviour of weakly converging sequences.For a more detailed introduction to the topic, we refer to the broad literature, e.g.[16,Chapter 8], [23], [29,Section 4].
In general dimensions l, m, n ∈ N, we denote by M(R l ) the set of bounded Radon measures and by Pr(R l ) its subsets of probability measures.
Let U ⊂ R n be a Lebesgue measurable set with finite measure.By definition, a Young measure ν = {ν x } x∈U is an element of the space L ∞ w (U ; M(R m )) of essentially bounded, weakly * measurable maps defined in U → M(R m ), which is isometrically isomorphic to the dual of We also recall that if (z j ) j ⊂ L p (U ; R m ) (p ∈ (1, +∞]) generates a Young measure ν and converges weakly( * ) in L p (U ; R m ) to a limit function u, then [ν x ] = ν x , id = ´Rm ξdν x (ξ) = u(x) for L n -a.e.x ∈ U .In the sequel we will mainly restrict to gradient Young measure, namely with U := Ω ⊂ R N a bounded open set, and m = N ×d, a W 1,∞ -gradient Young measure (see [20]) is a Young measure generated by a sequence of (∇u j ) j with u j ∈ W 1,∞ (Ω; R d ).
For our purposes, we also recall that in [28, Proposition 4.3 and Remark 4.4] curl Young quasiconvexity has been characterized as follows.
f is curl Young quasiconvex if and only if it verifies whenever ν is a W 1,∞ -gradient Young measure.

Lower semicontinuity
The notion which will play a crucial role for us is the separate curl Young-quasiconvexity.
Definition 2.1.Let W : R d×N × R d×N → R be a lower semicontinuous function.W is said to be separately curl Young quasiconvex if A key tool for the proof of our result is the following lemma, first stated in [3] in the continuous and homogeneous case, and, then proved in its current version in [28].With the aim of analyzing nonlocal problems, in [22] to any function u ∈ L 1 (Ω; R m ) it has been associated the vector field w u (x, y) := (u(x), u(y)) for (x, y) ∈ Ω × Ω. (2.2) In the sequel we will consider nonlocal fields w ∇u (x, y) = (∇u(x), ∇u(y)) for (x, y) ∈ Ω × Ω.
Then Λ is the Young measure generated by the sequence (w uj Indeed for any fixed η or ξ the function W (•, η) or W (ξ, •) turns out to be curl Young quasiconvex but not generally level convex.
We are in position to establish our main result Theorem 2.5.
Then the functional F is sequentially weakly * lower semicontinuous in W 1,∞ (Ω; R d ).
Remark 2.6.We observe that this result extends to the non-homogeneous and differential setting [22,Proposition 3.6].
The same proof could be used to show that separate level convexity of W (x, y, •, •) for L N ⊗ L N -a.e.(x, y) ∈ Ω × Ω is sufficient to guarantee the sequential weak* lower semicontinuity in L ∞ (Ω; R d ) of ess sup (x,y)∈Ω×Ω W (x, y, u(x), u(y)).
Nevertheless, as proven in the latter setting, under homogeneity assumptions, we conjecture that separate curl Young quasiconvexity is not 'really' necessary for the sequential lower semicontinuity of the functional in (2.3), since from one hand some symmetry of W should be taken into account, (cf. the notions of Cartesian separate level convexity in [22,21]), but also it is worth to observe that even in the local setting it is currently an open question the necessity of curl Young quasiconvexity for the sequential weak* lower semicontinuity of ess sup x∈Ω f (∇u(x)), namely it is not known, in general, if curl Young quasiconvexity is equivalent to the Strong Morrey quasiconvexity introduced in [4], except some particular case as those considered in [2] and [26].
Finally, we also point out that, under suitable continuity conditions on the second set of variables for W , our arguments could be successfully employed to prove the lower semicontinuity of nonlocal supremal functionals under more general differential constraints than curl.
Proof.The result follows from Lemmas 2.2 2.3, and Definition 2.1.Without loss of generality we can assume that W is non negative.

Conclusions
In this paper we provide a sufficient condition for the lower semicontinuity of nonlocal supremal functionals depending on the gradients of suitable Lipschitz fields.We conjecture that this notion is also suitable to provide an L p approximation result in the spirit of what is proven for L ∞ fields in [21].This latter study and the search for necessary conditions will be the subject of future research.We conclude observing that analogous results in the case of nonlocal integral functionals, depending on the gradient of scalar fields, can be found in [10].

Lemma 2 . 2 .
Let U ⊂ R n be an open set with finite measure and let f : U × R m → R be a normal integrand bounded from below.Further, let (u k ) be a uniformly bounded sequence of functions in L ∞ (U ; R m ) generating a Young measure ν = {ν x } x∈U .Then,lim inf k→∞ ess sup x∈U f (x, u k (x)) ≥ ess sup x∈U f (x),where f (x) := ν x -ess sup ξ∈R m f (x, ξ) for x ∈ U .