Lower semicontinuity and relaxation of nonlocal $L^\infty$-functionals

We study variational problems involving nonlocal supremal functionals $L^\infty(\Omega;\mathbb{R}^m) \ni u\mapsto {\rm ess sup}_{(x,y)\in \Omega\times \Omega} W(u(x), u(y)),$ where $\Omega\subset \mathbb{R}^n$ is a bounded, open set and $W:\mathbb{R}^m\times\mathbb{R}^m\to \mathbb{R}$ is a suitable function. Motivated by existence theory via the direct method, we identify a necessary and sufficient condition for $L^\infty$-weak$^\ast$ lower semicontinuity of these functionals, namely, separate level convexity of a symmetrized and suitably diagonalized version of the supremands. More generally, we show that the supremal structure of the functionals is preserved during the process of relaxation. Whether the same statement holds in the related context of double-integral functionals is currently still open. Our proof relies substantially on the connection between supremal and indicator functionals. This allows us to recast the relaxation problem into characterizing weak$^\ast$ closures of a class of nonlocal inclusions, which is of independent interest. To illustrate the theory, we determine explicit relaxation formulas for examples of functionals with different multi-well supremands.


Introduction
Nonlocal functionals in the form of double integrals appear naturally in different applications; examples include peridynamics [12,25,37], image processing [14,22] or the theory of phase transitions [18,20,36]. In the homogeneous case, separate convexity of the integrands has been identified as a necessary and sufficient condition for the weak lower semicontinuity of such functionals [13,27,29]. When it comes to relaxation, meaning the characterization of weak lower semicontinuous envelopes, though, the problem is still largely open. The difficulty lies in the fact that, counterintuitively, relaxation formulas can in general not be obtained via separate convexification of the integrands, as explicit examples in [11,13,31] show. It is hence unclear whether relaxed double integral functions can be at all represented in the form of double integrals.
Inspired by these recent developments, this article addresses a related problem by discussing homogeneous supremal (or L ∞ -)functionals in the nonlocal setting, i.e., L ∞ (Ω; R m ) ∋ u → J(u) := esssup (x,y)∈Ω×Ω W (u(x), u(y)), (1.1) where Ω ⊂ R n is a bounded, open set and W : R m × R m → R is a given Borel function satisfying suitable further assumptions regarding continuity and coercivity. We contribute answers to two key questions, which are motived by the existence theory for solutions to variational problems in form of the direct method in the calculus of variations: (Q1) What are necessary and sufficient conditions on the supremand W for the (sequential) lower semicontinuity of J with respect to the natural topology, that is, the L ∞ -weak * topology? (Q2) If J fails to satisfy the conditions resulting as an answer to (Q1), can we find an explicit representation of its relaxation, that is, of its L ∞ -weak * (sequential) lower semicontinuous envelope?
It is useful to observe that functionals of the type (1.1) share key features with two different classes of functionals that have been studied intensively in the literature, namely double integral functionals mentioned already at the beginning, i.e., L p (Ω; R m ) ∋ u → Ω Ω W (u(x), u(u)) dx dy with p ∈ [1, ∞), and supremal functionals, i.e., with a suitable function f : R m → R; for more details and background on these two branches of research, including a list of references, we refer to Sections 2.3 and 2.4. Borrowing and combining methods and techniques from these two fields, which are largely based on Young measure theory, equip us with quite a rich tool box for analyzing nonlocal supremal functionals. However, it will become clear in the following that, in order to settle the questions (Q1) and (Q2), new ideas are needed in addition.
A crucial realization is that the functional J in (1.1) remains unaffected by certain changes of W , beyond mere symmetrization. Indeed, replacing W with its diagonalized and symmetrized version W (see (7.1) along with Section 4 for the precise definition) still gives the same functional.
To understand better the role of diagonalization, it helps to take a different perspective on our nonlocal supremal functionals and to exploit their connection with socalled nonlocal indicator functionals. These are double integrals over the characteristic function χ K for a compact set K ⊂ R m × R m , i.e., By modification of a result due to Barron, Jensen & Wang [9, Lemma 1.4], we find that (Q1) and (Q2) for J in (1.1) are equivalent to studying the same questions for all indicator functionals associated with the sublevel sets of W , cf. Proposition 7.1. Then again, (1.2) is closely tied to nonlocal inclusions of the form (u(x), u(y)) ∈ K for a.e. (x, y) ∈ Ω × Ω, (1.3) and (Q2) comes down to identifying the asymptotic behavior of L ∞ -weakly * converging sequences subject to this type of constraint, which is also of independent interest. If we denote by A K the set of all functions in L ∞ (Ω; R m ) satisfying (1.3), the task is to characterize the L ∞ -weak * closure of A K . In the classical local setting, that is, when (1.3) is changed into u(x) ∈ A for a.e. x ∈ Ω with A ⊂ R m compact, (1.4) it is well known that the L ∞ -weak * limits of sequences with this property correspond to essentially bounded functions with values in the convex hull of A. In the nonlocal case, where one expects the separate convexification to take over the role of convexification in the local problem, things turn out to be a bit more subtle.
Then, the L ∞ -weak * closure of A K is given by A K sc , where K sc is the separately convex hull of K. Remark 1.2. a) The above theorem implies that A K is weakly * closed if and only if K sc = K, (1.6) which, in the scalar case m = 1, is equivalent to the separate convexity of K, cf. Corollary 5.9. Notice that this necessary and sufficient condition is strictly weaker than requiring that K is separately convex. b) As an immediate corollary of Theorem 1.1, we obtain that the relaxation of the indicator functional (1.2) is given by , u(y)) dx dy; in particular, (1.2) is L ∞ -weak * lower semicontinuous if and only if (1.6) holds, cf. Corollary 6.1.
The proof of Theorem 1.1 relies on a series of auxiliary results. With (1.5) established in Proposition 5.1, an argument based on pointwise approximation by piecewise affine functions allows us to deduce a refined representation of elements A K , saying that for each u ∈ A K there exists a Cartesian product A × A ⊂ K with A ⊂ R m such that u ∈ A A×A , see Proposition 5.5. Another important ingredient in the case m = 1 is a characterization of the separately convex hull of K, which can be shown to have a particularly simple form. In fact, K sc is the union of all squares in R × R whose corners are extreme points (in the sense of separate convexification of) K, for details see Corollary 4.12. In higher dimensions, the analogous statement, which could be viewed as a Caratheodory type formula, is in general false (cf. Remark 4.8 c)); the required extra assumptions on K sc if m > 1 are introduced to compensate for this. Combining all the previous arguments reduces the proof of Theorem 1.1 to the case when K takes the form of a Cartesian product in R m × R m . Under this assumption, the desired L ∞ -weak * approximation of u ∈ A K sc follows from an explicit construction of periodically oscillating sequences, see Lemma 5.7. Alternatively, one could use a more abstract approach via Young measures generated by sequences that satisfy an approximate nonlocal constraint, together with a projection step to enforce the exact nonlocal inclusion (1.3), cf. Proposition 5.10.
Conceptually, the study of nonlocal inclusions as in (1.3) shows close parallels with the field of differential inclusions, dealing with problems such as ∇u ∈ M a.e. in Ω and M ⊂ R m×n compact for u ∈ W 1,∞ (Ω; R m ) (see e.g. [19,34] and the references therein), and compensated compactness theory [28,38]; notice that the latter deal with problems that are all local in nature. The overall challenge is to capture the interplay between pointwise constraints and the structural properties of the vector fields, whether they are gradients, or more generally, A-free fields with some differential operator A, or, like here, nonlocal vector fields of the form (2.4). Yet, besides these conceptual parallels, nonlocality creates effects that are not typically encountered in local problems, as for instance (1.5) indicates.
In generalization of Theorem 1.1, we characterize the set of Young measures generated by nonlocal vector fields associated with uniformly bounded sequences (u j ) j ⊂ L ∞ (Ω; R m ), cf. (2.4); indeed, if (u j ) j generates the Young measure ν = {ν x } x∈Ω , the sought-after set consists of all the product measures Λ = {Λ (x,y) } (x,y)∈Ω×Ω = {ν x ⊗ ν y } (x,y)∈Ω×Ω with supp Λ contained almost everywhere in a Cartesian subset of K, see Theorem 5.11 for the precise statement. Interpreted in the context of indicator functionals, the latter yields a Young measure relaxation result for a class of unbounded functionals (defined precisely in (6.6)), extending part of a recent work by Bellido & Mora-Corral [11, Section 6], cf. Section 6.2.
The next theorem collects the main results of this paper regarding nonlocal supremal functionals. In contrast to the theory of double-integral functionals, where the question about the qualitative structure of relaxations, i.e., whether they are again of double-integral form or not, is still unsolved, we show here that relaxation of nonlocal supremal functionals is structure preserving, in the sense that it is again of nonlocal supremal type. For simplicity, we formulate the result here in the scalar case; for the extension to the vectorial setting (under additional conditions), we refer to Corollary 7.2 and Remark 7.6. Theorem 1.3. Let J be as in (1.1) and W : R × R → R be lower semicontinuous and coercive.
(i) The functional J is L ∞ -weakly * lower semicontinuous if and only if W is separately level convex, where W , defined in (7.1), is the density resulting from diagonalization and symmetrization of W . (ii) The relaxation J rlx of J is given by the nonlocal supremal functional of the form (1.1) with supremand W slc , which is the separately level convex envelope of W .
To close the loop with the beginning of the introduction, we stress the link between nonlocal supremal functionals and nonlocal double-integral functionals via L p -approximation; if W = W is separately level convex, this can be made rigorous by imitating the arguments by Champion, Pascale & Prinari in [17,Theorem 3.1].
As an outlook on interesting future research beyond the scope of this work, we would like to mention in particular the proof of a characterization result for the L ∞ -weak * closure of A K in general dimensions without extra assumptions on K, or the extension to our theory to inhomogeneous nonlocal functionals.
The paper is organized as follows. First, we collect some preliminaries in Section 2; these include subsections on frequently used notation, auxiliary results for Young measures, as well as background on the theories of both supremal and nonlocal double-integral functionals. After introducing and discussing the notion of separate level convexity in Section 3, we investigate the interaction of separate convexification of sets with their diagonalization and symmetrization in Section 4. In Section 5, we turn to the analysis of nonlocal inclusions; more precisely, Subsection 5.1 provides alternative representations of A K , Subsection 5.2 contains the proof of Theorem 1.1, and Subsection 5.3 is concerned with the characterization of Young measures generated by sequences of nonlocal vector fields. In Section 6, we reformulate the insights about nonlocal inclusions in terms of nonlocal indicator functionals (see Subsections 6.1 and 6.2), and discuss the connection between different notions of nonlocal convexity for extended-valued functionals (see Subsection 6.3). The main theorems on lower semicontinuity and relaxation of nonlocal supremal functionals, which address the questions (Q1) and (Q2), are established in Section 7. To illustrate the theory, we finally present a few examples of nonlocal supremal functionals with different multiwell supremands in Subsection 7.2, and determine explicitly the corresponding relaxation formulas.

Preliminaries
In this section, we fix notations and recall some well-known results that will be exploited in the remainder of the paper.
2.1. Notation. In the following, m and n are natural numbers. For any vector ξ ∈ R m , let ξ i , i = 1, . . . , m, denote its components, and |ξ| = ( m i=1 ξ 2 i ) 1 2 its Euclidean norm. By B r (ξ), we denote the closed (Euclidean) ball centered in ξ ∈ R m with radius r > 0. For two vectors ξ, ζ ∈ R m , the relation ξ < ζ is to be understood componentwise, meaning ξ i ≤ ζ i for i = 1, . . . , m, and we define the generalized closed interval Our notation for the complement of a set A ⊂ R m is A c = R m \ A, whilst A co stands for the convex hull of A. Moreover, we denote the characteristic function of A ⊂ R m in the sense of convex analysis by χ A and the indicator function of A by ½ A , i.e.
The distance from a point β ∈ R m to a set A ⊂ R m is dist(β, A) := inf α∈A |α − β|, and the Hausdorff distance between two non-empty sets A, B ⊂ R m is given by Further, we denote by R ∞ the set R∪{∞}. For every c ∈ R and every function f : Let E ⊂ A × A with A ⊂ R m ; then π 1 (E) and π 2 (E) stand for the the projection of E onto the first and second component, respectively, that is To denote the sections of E in the first and second argument at β ∈ A, we use a notation with letters in Frakture, precisely, for all β ∈ A, and we simply write π(E) and E β . Notice that throughout the manuscript, we use the identification R m × R m ∼ = R 2m without explicit mention.
Let C 0 (R m ) be the closure with respect to the maximum norm of the space of smooth, real-valued functions on R n with compact support. By the Riesz representation theorem (see e.g. [2, Theorem 1.54]), the dual space of C 0 (R m ) can be identified via the duality pairing µ, ϕ = R m ϕ(ξ) dµ(ξ) with the space M(R m ) of finite signed Radon measures on R m .
For the class of probability measures defined on the Borel sets of R m , we write Pr(R m ). The barycenter of µ ∈ Pr(R m ) is defined by and supp µ stands for the support of µ. If f : R m → R and µ is a probability measure, or more generally, a positive measure, on the Borel sets of R m , the µ-essential supremum of f over the set A ⊂ R m is defined as We use the notation ν ⊗ µ to denote the product measure of two measures ν and µ. By U we denote a generic measurable (Lebesgue or Borel) subset of R m . The Lebesgue measure of a Lebesgue measurable set U ⊂ R n is denoted by L n (U ). We skip the Lebesgue measure symbol L n whenever it is clear from the context, for example, we often write simply 'a.e. in U ' instead of 'L n -a.e. in U '. Unless mentioned otherwise, Ω is always a non-empty, open and bounded subset of R n . We use standard notation for L p -spaces with p ∈ [1, ∞]; in particular, for a sequence of functions (u j ) j ⊂ L p (Ω; R m ) and u ∈ L p (Ω; R m ), we write u j ⇀ u in L p (Ω; R m ) with p ∈ [1, ∞) and u j ⇀ * u in L ∞ (Ω; R m ) to express weak and weak * convergence of (u j ) j to u, respectively. In the following, we often deal with functions u ∈ L p (Ω; R m ) and their composition with Borel measurable functions f : R m → R. The L n -essential supremum of f (u), whenever f is nonnegative, corresponds to the L ∞ -norm of f (u). Depending on the context, we write either L n -ess sup x∈Ω f (u(x)), f (u) L ∞ (Ω) , or simply, ess sup x∈Ω f (u(x)).

2.2.
Young measures. Young measures are an important technical tool in nonlinear analysis, as they encode refined information on the oscillation behavior of weakly converging sequences. To make this article self-contained, we briefly recall some basics from this theory, focusing on what will be used in the sequel. For a more detailed introduction to the topic, we refer to the broad literature, e.g. [21,Chapter 8], [30], [34,Section 4].
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 U → M(R m ), which is isometrically isomorphic to the dual of L 1 (U ; C 0 (R m )), such that ν x := ν(x) ∈ Pr(R m ) for L n -a.e. x ∈ U . One calls ν homogeneous if there is a measure ν 0 ∈ Pr(R m ) such that ν x = ν 0 for L n -a.e. x ∈ U .
A sequence (z j ) j of measurable functions z j : U → R m is said to generate a Young measure ν ∈ L ∞ w (U ; Pr(R m )) if for every h ∈ L 1 (U ) and ϕ ∈ C 0 (R m ), The following result is often referred to as the fundamental theorem for Young measures, see e.g. , [5], [ be a uniformly bounded sequence. Then there exists a subsequence of (z j ) j (not relabeled) and a Young measure ν ∈ L ∞ w (U ; M(R m )) such that z j Y M −→ ν. Moreover, (i) for any continuous integrand f : R m → R with the property that f (z j ) j ⊂ L 1 (U ) is uniformly bounded and equiintegrable, it holds that (ii) for any continuous f : R m → R ∞ bounded from below, In particular, if (z j ) j ⊂ L p (U ; R m ) generates a Young measure ν and converges weakly( * ) in L p (U ; R m ) to a limit function u, then With the aim of analyzing nonlocal problems, we associate with any function u ∈ L 1 (Ω; R m ) the vector field v u (x, y) := (u(x), u(y)) for (x, y) ∈ Ω × Ω. Then Λ is the Young measure generated by the sequence (v u j ) j ⊂ L p (Ω × Ω; R m × R m ) defined according to (2.4) if and only if supp ν x ⊂ K for L n -a.e. x ∈ Ω with a fixed compact set K ⊂ R m , if p = ∞.

2.3.
Supremal functionals and level convexity. Next, we collect some basic properties and useful results from the theory of supremal functionals, i.e., functionals F : L ∞ (Ω; R m ) → R ∞ given by where f : R m → R ∞ is a Borel measurable function bounded from below.
Barron & Jensen in [7] and Barron & Liu in [10] were the first to study necessary and sufficient conditions of supremal functionals as F in (2.5). Assuming that Ω ⊂ R is an interval, they proved that F is sequentially L ∞ -weakly * lower semicontinuous if and only if the supremand f is level convex. The same statement holds for general Ω ⊂ R n ; see [1,Theorem 4.1], as well as [9] and [32].
Note that level convexity is known in the literature on operational research and convex analysis as quasiconvexity, see e.g. [24]. To avoid ambiguity with the notion introduced by Morrey [26] in the context of integral functionals, we have chosen here to use the same terminology as in [1].
The following lemma provides different characterizations of level convexity, in particular, in terms of a supremal Jensen type inequality. It can be found e.g. in [ (i) f is level convex; (ii) for every ξ, ζ ∈ R m and t ∈ [0, 1] it holds that (iii) for any open set Ω ⊂ R n with L n (Ω) < ∞ and every ϕ ∈ L 1 (Ω; R m ) one has that The following auxiliary result is a slight modification of [6,Theorem 34] and is based on L p -approximation in combination with the lower semicontinuity type result for Young measure in Theorem 2.1.
Proof. We give the details here for the reader's convenience, referring to [6] for the original proof. Up to a translation argument, there is no loss of generality in assuming that f is non-negative.
Let ε > 0 be fixed. and choose a set S ⊂ Ω with positive Lebesgue measure such that for all x ∈ S ′ and p > 1 sufficiently large. Indeed, with We take the inequality in (2.6) to the pth power and integrate over S ′ . Along with Theorem 2.1 (ii), it follows that for p > 1 sufficiently large. Letting p → ∞ and recalling that ε > 0 is arbitrary concludes the proof.
2.4. Double-integral functionals and separate convexity. This subsection presents some preliminaries on nonlocal integral functionals, see also [31] for a recent overview article. For p > 1, consider a double-integral functional I : L p (Ω; R m ) → R, where W : R m × R m → R is a continuous function that is bounded from below and has standard p-growth.
In 1997, Pedregal [29] gave the first necessary and sufficient condition for L p -weak lower semicontinuity of I in the scalar case m = 1. This condition was quite implicit, but could be shown to be equivalent to the separate convexity of the integrand W a decade later by Bevan & Pedregal [13]. Also in the vectorial case, W being separately convex is the characterizing property to ensure weak lower semicontinuity of I, as Muñoz proved in [27]; the latter is formulated in the gradient setting, using W 1,p -weak convergence of scalar valued functions, but the statement and the ideas of the proof carry over to functionals of the form (2.7), cf. [31]. Results about inhomogeneous double-integral functionals, meaning with integrands W depending also explicitly on x, y ∈ Ω, can be found e.g. in [11,27,31]. Definition 2.6. We call a function W : R m × R m → R ∞ separately convex (with vectorial components) if for every ξ ∈ R m , the functions W (·, ξ) and W (ξ, ·) are convex.
Besides our terminology, which is inspired by [19], other names for separate convexity are common in the literature, such as orthogonal convexity, directional convexity or bi-convexity; see [4], for the first detailed treatment of the subject.
As discussed recently in [11], there are different 'nonlocal' definitions of convexity related to the weak lower semicontinuity of I, which coincide under suitable assumptions. In Section 6, we extend the discussion of these notions to the context of unbounded functionals.
It was observed in [29, p. 1383] that for W : R × R → R continuous and bounded from below, separate convexity of W can equivalently be characterized by a separate Jensen's inequality. In view of [16,Theorem 4.1.4], this statement can easily be generalized to extended-valued, lower semicontinuous functions defined on R m × R m as follows.

lower semicontinuous and bounded from below, then W is separately convex if and only if
for any µ, ν ∈ Pr(R m ).
The fact that (2.8) yields separate convexity of W follows after choosing µ and ν to be convex combinations of Dirac measures.
The question of relaxation of functionals I as in (2.7) for which the density W fails to be separately convex is still mostly open. It may seem counter-intuitive, but there are examples [11,13,31] indicating that separate convexification of W does in general not give rise to the right candidate for the weakly lower semicontinuous envelope of I. In the context of Young measure, we refer to [11] for a relaxation result with respect to the narrow convergence.

Separate level convexity
In this section, we introduce the notion of separate level convexity, and show that it provides a sufficient condition for the L ∞ -weak * lower semicontinuity of nonlocal supremal functionals as in (1.1).
Before doing so, let us specify what we mean by separate convexity with vectorial components (in the sequel, just referred to as separate convexity) of subsets of R m × R m .
For m = 1, this definition reduces to classical separate convexity in the sense of [19, Proposition 7.5 and Definition 7.13].
Definition 3.1 (Separate convexity (with vectorial components) of sets). A set E ⊂ R m × R m is called separately convex, if for every t ∈ (0, 1) and every (ξ 1 , ζ 1 ), (ξ 2 , ζ 2 ) ∈ E such that ξ 1 = ξ 2 or ζ 1 = ζ 2 it holds that The separately convex hull of E, denoted by E sc , is defined as the smallest separately convex set in R m × R m containing E.
The separately convex hull of E ⊂ R m × R m can be characterized by Furthermore, W slc stands for the separately level convex envelope of W , that is, the largest separately level convex function below W .

Remark 3.4. a) An equivalent way of expressing separate level convexity of
b) In view of the above definitions, we observe that In general, equality in (3.3) is not true as the example The following lemma collects a number of different representations of separate level convexity.
Then the following statements are equivalent: (iv) for every ν, µ ∈ Pr(R m ) it holds that Proof. These equivalences follow as an immediate corollary of Lemma 2.4. Indeed, we apply the characterizations therein twice in each of the two variables of W , fixing the other.
The sufficiency of separate level convexity of W for ensuring L ∞ -weak * lower semicontinuity of J in (1.1) follows from the next proposition. The proof relies on combining elements from both theories of supremal and double integral functionals, cf. Sections 2.2 and 2.3, respectively.
As we show later in Section 7.1, separate level convexity of W is not necessary for J being sequentially L ∞ -weakly * lower semicontinuous, cf. Corollary 7.2.

Diagonalization, symmetrization and separately convex hulls
be the diagonalization and symmetrization of E. Accordingly, we call E symmetric, if E = E sym , and diagonal if E = E diag . By combining these two operations, we introduce As an immediate consequence of these definitions, one observes that if E is closed (compact), then E sym and E diag , and consequently, also E, are closed (compact).
This section is devoted to the study of characterizing properties of diagonal and symmetric sets. For illustration, we start with a few simple examples in the scalar case m = 1.
For the points sets one obtains that K 5 = ∅ and K 6 = K 6 , respectively.
Notice the following equivalent way of expressing E in (4.1),  Let E ⊂ R m × R m . We call a set P ⊂ E a maximal Cartesian subset of E if P = A × A with A ⊂ R m and if for any B ⊂ R m with A ⊂ B and B × B ⊂ E it holds that B = A. We denote the set of all maximal Cartesian subsets of E by P E .
On the other hand, we know for (ξ, ζ) ∈ E that {ξ, ζ}×{ξ, ζ} ⊂ E ⊂ E, and hence B ×B ⊂ E with B = {ξ, ζ}. Due to the Cartesian structure of B × B, there is a maximal Cartesian subset of E containing B × B, which proves the statement. Lemma 4.5. If E ⊂ R × R is separately convex, then E is also separately convex.
Proof. Let (ξ 1 , ζ), (ξ 2 , ζ) ∈ E. By Lemma 4.3 we know that there are P 1 , P 2 ∈ P E such that   (4.4) and hence, in view of E = E, we find that E sc = E. Since E is strictly contained in E sc , however, E is not separately convex.
The next lemma gives a characterization of the separate convex hull of symmetric and diagonal sets in the scalar case m = 1.
Remark 4.8. a) As a consequence of Lemma 4.7, the properties of a symmetric and diagonal set E ⊂ R × R carry over to its separate convexification E sc . b) In view of (4.5), a Caratheodory type formula holds for separate convex hulls of sets as in Lemma 4.7. In general, this cannot be expected, see e.g. [ Indeed, if (ξ, ζ) ∈ E sc , then (4.5) implies that (ξ, ζ) ∈ Q α,β for some (α, β) ∈ E, and there are t, s ∈ [0, 1] such that ξ = tα + (1− t)β and ζ = sα + (1− s)β. Thus, (ξ, ζ) = t(α, ζ)+ (1− t)(β, ζ), or equivalently, c) We emphasize that the representation formula (4.5) is in general not true in the vectorial case, that is, for symmetric and diagonal subsets of R m × R m with m > 1. To see this, consider the example of Remark 4.6 b), where E is the union of two Cartesian products generated by convex sets A 1 , A 2 ⊂ R m with m > 1 whose union is not convex. Then, due to the convexity of A 1 and A 2 and the fact that E is not separately convex, we conclude that After diagonalization (and symmetrization), however, we observe that d) It remains an open question at this point to find an explicit representation for E sc , or E sc , with general E ⊂ R m × R m symmetric and diagonal.
In the special case when at most two of the separately convex hulls of the maximal Cartesian subsets of E intersect, we can derive a formula for E sc based on (4.4). Precisely, suppose that E = P =A×A∈P E P and that there are P 1 = A 1 × A 1 ∈ P E and P 2 = A 2 × A 2 ∈ P E with A 1 , A 2 ⊂ R m such that P sc ∩ Q sc = ∅ for all P ∈ P E and Q ∈ P E \ {P, P 1 , P 2 }. Along with the observation that (B × B) sc = B co × B co for any B ⊂ R m , it follows that Hence, where we have used that the diagonalization and symmetrization of We continue with a lemma that will be used later on in Section 7.1 to give a characterization of the sublevel sets of W slc . Lemma 4.9. For j ∈ N, let K j ⊂ R × R be compact, symmetric and diagonal. If the sets K j are nested, i.e. K j ⊃ K j+1 for all j ∈ N, then Proof. One inclusion follows directly from the definition of separately convex hulls. For the other one, let (ξ, ζ) ∈ j∈N K sc j . Then for each j ∈ N, there exists according to (4.5) an element (α j , β j ) ∈ K j with (ξ, ζ) ∈ Q α j ,β j , and therefore with s j , t j ∈ [0, 1]. By compactness, we know that after passing to subsequences, we can assume that s j → s ∈ [0, 1], t j → t ∈ [0, 1], and (α j , β j ) → (α, β) ∈ j∈N K j as j → ∞. Finally, taking j → ∞ in (4.6) shows that (ξ, ζ) ∈ Q α,β ⊂ ( j∈N K j ) sc .
Inspired by the definition of extreme points in the separately convex sense, see e.g. [19,Definition 7.30], we introduce here directional extreme points for subsets of R m × R m . These can be used to refine the characterization formula (4.5), see Corollary 4.12 below.
For general E ⊂ R m × R m , we say that (ξ, ζ) ∈ R m × R m is a directional extreme point if (ξ, ζ) is a directional extreme point for E sc in the above mentioned sense.
We denote the set of all directional extreme points of a set E by E dex . The representation formula (4.5) can be simplified by considering only unions of squares whose vertices are directional extreme points of E.
We close this section with a representation of separately convex hulls in terms of measures. For K ⊂ R m ×R m non-empty and compact, one obtains the following alternative characterization of K sc , which is essentially a reformulation of (3.1) and (3.2): In general, the measures whose barycenters yield elements in K sc cannot be expected to be of product form. If m = 1, however, this is the case, as the next lemma shows. Proof. The second implication is trivial, the proof of the first is a simple consequence of Corollary 4.12. Indeed, if (ξ, ζ) ∈ K sc , then by (4.7) there is (α, β) ∈ K dex such that (a, b) ∈ Q α,β . We choose t, s ∈ [0, 1] such that ξ = tα + (1 − t)β and ζ = sα + (1 − s)β and set which is a measure has the desired properties.

Nonlocal inclusions
For a set E ⊂ R m × R m , we consider The main focus of this section is to prove the characterization result for the limits of weakly converging sequences in A K with compact K ⊂ R × R stated in Theorem 1.1. In the first subsection, we lay important groundwork by investigating the role of the set E in A E . This gives important structural insight into the interplay between nonlocality effects and pointwise constraints, which are also interesting per se.

5.1.
Alternative representations of A E . The next result shows that the set E \ E has no influence on the solutions to the nonlocal inclusion (u(x), u(y)) ∈ E for a.e. (x, y) ∈ Ω × Ω. In particular, Proof. To show that equality of A E and A F implies that E = F , it suffices to prove that E ⊂ F . In fact, the reverse inclusion follows then from interchanging the roles of E and F . The case E = ∅ is trivial. Otherwise, let (ξ, ζ) ∈ E, and consider the piecewise constant function where Ω ξ ⊂ Ω measurable with L n (Ω ξ ) > 0 and L n (Ω \ Ω ξ ) > 0. By definition, u ∈ L ∞ (Ω; R m ), and since (ξ, ζ) ∈ E ⊂ E, it holds that also (ξ, ξ), (ζ, ζ), (ζ, ξ) ∈ E. Hence, u ∈ A E = A F , and therefore (ζ, ξ), (ξ, ζ), (ξ, ξ), (ζ, ζ) ∈ F . This shows (ξ, ζ) ∈ F . Notice that the converse implication, i.e. A E = A F if E = F , follows immediately, if one knows (5.2). To prove the latter, we start by observing that A E = A E sym . Indeed, if u ∈ A E , then also u ∈ A E T , and therefore u ∈ A E sym , because E sym = E ∩ E T . Thus, from now we assume E to be symmetric.
Next, we will show that a specific class of subsets of E can be removed without affecting To see this, let B ⊂ R m × R m satisfy the first condition in (5.3) (the reasoning in case the second condition holds is analogous), and consider u ∈ A E , assuming to the contrary that u / ∈ A E\B . Then there exists an (L n ⊗ L n )-measurable set N ⊂ Ω × Ω with positive measure such that (u(x), u(y)) ∈ B for all (x, y) ∈ N . By Tonelli's theorem or Cavalieri's principle, there existsȳ ∈ Ω with L n (Nȳ 1 ) > 0; recall that Nȳ 1 stands for the section in the first variable of N at y, cf. Subsection 2.1. Hence, (u(x), u(ȳ)) ∈ B for all x ∈ Nȳ 1 , or equivalently, using projections, u(x) ∈ π 1 (B) for x ∈ Nȳ 1 . This leads to (u(x), u(y)) ∈ π 1 (B) × π 1 (B) for all (x, y) ∈ Nȳ 1 × Nȳ 1 .
Next we apply (5.4) to suitable sets whose union amounts to E \ E. Owing to the fact that the complement E c of E in R m × R m is open, one can find for any vector of rational numbers ξ ∈ Q m with (ξ, ξ) / Finally, accounting for (4.3) along with the observation that B ∪ = B E yields that E \ B ∪ = E, which implies (5.2).
To see this, let n = m and Ω = (0, 1) m , and consider Then, E = ∅, and hence, A E = ∅. On the other hand, the identity map The next lemma is the basis for a useful approximation result, which is formulated below in Corollary 5.4. For shorter notation, we write S ∞ (Ω; R m ) for the subspace of L ∞ (Ω; R m ) of simple functions, i.e., u ∈ S ∞ (Ω; R m ) if ..,k a partition of Ω into L n -measurable sets and ξ (i) ∈ R m for i = 1, . . . , k. By possibly choosing a different representative, one may assume without loss of generality that L n (Ω (i) ) > 0 for all i = 1, . . . , k.
Proof. The proof follows along the lines of standard arguments for approximating unconstrained bounded functions uniformly by simple ones. Yet, particular care is needed here when choosing the function values to guarantee that the nonlocal inclusion defining A E is not violated. This last step critically exploits the assumption that E = E. For clarification regarding notations throughout this proof, we refer the reader to Subsection 2.1.
After choosing a suitable representative of u ∈ A E , we may assume that z ≤ u(x) ≤ z for all x ∈ Ω with z, z ∈ R m . For j ∈ N, we partition the set [ for all i = 1, . . . , k, (5.6) and define the L n -measurable sets . . , k} be the index set defined by Possibly after rearranging, one may assume without loss of generality that I j = {1, . . . , l} for some l ∈ N with l ≤ k.
Consider the simple function j ∩ N (this set is indeed non-empty by (5.9) and (5.7)) and iteratively for i = 2, . . . , l, Notice that the set on the right-hand side in (5.10) has positive L n -measure and is therefore in particular not empty. Indeed, this follows from (5.9) and (5.7) in combination with In view of (5.8), it holds therefore that which implies that u j ∈ A E for any j ∈ N. Moreover, together with (5.6), so that u j → u in L ∞ (Ω; R m ) as j → ∞. This shows that (u j ) j is an approximating sequence for u with the stated properties.
The following density statement for A E with a closed set E is an immediate consequence of Lemma 5.3 and Proposition 5.1.
Based on this approximation result and the special properties of simple functions in A E , there is another way to represent A E , namely in terms of Cartesian products (cf. Definition 4.2).
Proof. For the proof of the nontrivial inclusion, consider any u ∈ A E . We will show that there exists A ⊂ R m with A × A ⊂ E such that u ∈ A A×A . Then, A × A ⊂ P for some P ∈ P E , and therefore u ∈ A P .
First, we observe that (5.11) holds for simple functions. In fact, if u ∈ S ∞ (Ω; R m ) ∩ A E , then it is of the form (5.5) with (ξ (i) , ξ (i ′ ) ) ∈ E for all i, i ′ = 1, . . . , k. Here we use in particular that the sets Ω (i) can be chosen to have positive L n -measure. Consequently, which yields the statement in the case when u is simple.
To prove (5.11) in the general case, let (u j ) j be an approximating sequence resulting from Lemma 5.3, so that u j → u in L ∞ (Ω; R m ). (5.12) Due to the uniform boundedness of (u j ) j in L ∞ (Ω; R m ), we may assume without loss of generality that E is bounded, and hence compact. Since each u j is simple, one can thus find for every j ∈ N a compact set A j ⊂ R m with P j := A j × A j ⊂ E such that u j ∈ A P j .
Next, we exploit the fact that the metric space of closed subsets of a compact set in R m endowed with the Hausdorff distance d m H in (2.2) is compact, see e.g. [35] or [2, Theorem 6.1] for Blaschke selection theorem. Hence, there is a subsequence of (A j ) j (not relabelled) and A ⊂ R m compact such that  13) and since P j ⊂ E for all j ∈ N, it follows that A × A ⊂ E.
Moreover, by (5.12) in combination with dominated convergence and (5.13), Hence, v u ∈ A × A a.e. in Ω × Ω or u ∈ A A×A , which finishes the proof.
Remark 5.6. Note that Proposition 5.5 fails if E is not closed. For the example in Remark 5.2, it holds that P E = ∅, whereas A E = ∅.
For general functions u ∈ A Q α,β , we argue via approximation. Let (ũ k ) k ⊂ A Q α,β ∩S ∞ (Ω; R m ) be a sequence of simple functions such thatũ k → u in L ∞ (Ω; R m ) as k → ∞, see Lemma 5.3.The previous construction allows us to find for each k ∈ N a sequence (ũ k,j ) j ⊂ A K withũ k,j ⇀ * ũ k in L ∞ (Ω; R m ) as j → ∞. By a version of Attouch's diagonalization lemma [3, Lemma 1.15, Corollary 1.16] (exploiting in particular that L ∞ (Ω; R m ) is the dual of a separable space), we can select k(j) → ∞ as j → ∞ such that for u j :=ũ k(j),j ∈ A K , This shows that u ∈ A ∞ K and completes the proof.
Proof of Theorem 1.1. We prove separately the two inclusions that make up (5.15). First, let u ∈ A ∞ K . Then, in view of Proposition 5.1, there exists a sequence (u j ) j ⊂ L ∞ (Ω; R m ) with v u j ∈ K a.e. in Ω × Ω such that u j ⇀ * u in L ∞ (Ω; R m ). Since K, and hence also K, is compact, so is K sc in the case m = 1 according to Remark 3.2. For m > 1, the compactness of K sc is guaranteed directly by assumption. As a result, the map is coercive and continuous with quadratic growth, and we infer from lower semicontinuity theorems in the nonlocal integral setting (see e.g. [ To prove the reverse inclusion, recall that the second assumption on K sc in the case m > 1 says that Now, we combine Lemma 4.7 if m = 1, or the previous assumption (5.16) if m > 1, with Proposition 5.5 and Lemma 5.7 to infer that This finishes the proof.  For m = 1, the condition (5.18) is equivalent with the separate level convexity of K.

Characterization of Young measures generated by sequences in
K be the set of Young measures generated by a sequence of nonlocal vector fields associated with (u j ) j ⊂ A K ; more precisely, Regarding barycenters, we observe that As a consequence of Proposition 5.1, Lemma 2.2 and Theorem 2.1 (iii), where for any compact C ⊂ R m × R m , Pr(R m )) and supp Λ (x,y) ⊂ C for a.e. (x, y) ∈ Ω × Ω}, and Y ∞ C is a modification of Y ∞ C in the sense that the exact inclusion is weakened to an approximate version, i.e., In the simple special case, when K has the form of a Cartesian product (then clearly, K = K), we are able to show that equality holds in (5.21). The proof combines well-known results from the theory of Young measures with a projection argument. Note that for more general K the projection result fails due to non-trivial interactions between the different variables.
Proof. In view of (5.21), it remains to show that Y ∞ K ⊂ Y ∞ K . To this end, we project the sequences generating the Young measures in Y ∞ K onto K. Let Λ ∈ Y ∞ K be generated by (vũ j ) j with (ũ j ) j ⊂ L ∞ (Ω; R m ) such that dist(vũ j , K) = dist(vũ j , A × A) → 0 in measure as j → ∞. By measurable selection [21, Section 6.1.1, Theorem 6.10], one can find a measurable and essentially bounded function u j : Ω → R m with u j (x) ∈ argmin ξ∈A dist(ũ j (x), ξ) for a.e. x ∈ Ω.
Then by construction, v u j ∈ A × A = K a.e. in Ω × Ω, and v u j − vũ j → 0 in measure as j → ∞. The latter implies in particular that (v u j ) j generates the same Young measure as (vũ j ) j , namely Λ. Hence, Λ ∈ Y ∞ K . With these prerequisites at hand, we can derive the following characterization of Young measures generated by sequences with nonlocal constraints.
Proof. Owing to the fact that any set in P K is a subset of K with the form of a Cartesian product in R m × R m , the inclusion P ∈P K Y P ⊂ Y ∞ K follows immediately from Proposition 5.10. For the proof of reverse inclusion, consider (v u j ) j as in (5.19), generating the Young measure Λ ∈ Y ∞ K . Then, Proposition 5.5 implies for every j ∈ N the existence of A j ⊂ R compact such that v u j ∈ P j := A j × A j ⊂ P K a.e. in Ω × Ω.
Arguing similarly to Proposition 5.5, we conclude (possibly after passing to a non-relabelled subsequence of ( in Ω × Ω, that dist(v u j , A × A) L ∞ (Ω×Ω;R m ×R m ) → 0 as j → ∞. Then, by the fundamental theorem of Young measures in Theorem 2.1 (iii), supp Λ ⊂ P a.e. in Ω × Ω for some P ∈ P K with A × A ⊂ P , or in other words, Λ ∈ Y P , which finishes the proof.

Nonlocal indicator functionals
The aim of this section is to relate the previous results with the theory of nonlocal unbounded functionals, in particular, with indicator functionals. 6.1. Lower semicontinuity and relaxation. For K ⊂ R m × R m , we define the indicator functional I K : L ∞ (Ω; R m ) → {0, ∞} by recall the notations from (2.1) and (5.1). It is clear from the second equality in (6.1) that the lower semicontinuity and relaxation of I K regarding the weak * topology in L ∞ (Ω; R m ) are closely related to the asymptotic behaviour of sequences in A K with respect to the same topology. In fact, the L ∞ -weak * lower semicontinuity of I K corresponds to the weak * closedness of A K , while determining its relaxation, i.e., for all u ∈ L ∞ (Ω; R m ), is equivalent to the characterizing the L ∞ -weak * closure of A K , denoted by A ∞ K in (5.14).
Formulated here again for the readers' convenience, the counterparts of Corollary 5.9 and Theorem 1.1 in terms of indicator functionals are the following. for m = 1, this is the same as K (or equivalently, χ K ) being separately convex.
(ii) Moreover, I rlx K = I K sc , where the latter is the functional in (6.1) associated with the separately convex hull K sc .
6.2. Young measure relaxation. As an application of Theorem 5.11, we determine the relaxation in the Young measure setting of a class of extended-valued double-integral functionals. This result can be viewed as a generalization of [11,Theorem 6.1].
For K ⊂ R m × R m , let the functional I Y K : L ∞ w (Ω; Pr(R m )) → {0, ∞} be defined by for ν ∈ L ∞ w (Ω; Pr(R m )). The follwing reformulation of Theorem 5.11 states a Young measure relaxation for nonlocal indicator functionals in general dimensions.
Remark 6.3. If K ⊂ R m × R m is compact as in Theorem 1.1, i.e. K sc is compact and satisfies (5.16), we can directly verify the expected relations between the functionals arising from classical and Young measure relaxation of I K . For any ν ∈ L ∞ w (Ω; Pr(R m )), moreover, for every u ∈ L ∞ (Ω; R m ), there exists a Young measure ν ∈ L ∞ w (Ω; Pr(R m )) with [ν] = u such that To see (6.5), it is enough to invoke Theorem 1.1 and the characterizion in Theorem 5.11.
As regards the justification of (6.4), we may assume without loss of generality that I Y K (ν) = 0; thus, there exists P = A × A ∈ P K with A ⊂ R m such that ν x ⊗ ν y ∈ P for a.e. (x, y) ∈ Ω × Ω. By Theorem 5.11, one can find a sequence (u j ) j ⊂ A P generating ν and converging weakly * to u = [ν] in L ∞ (Ω; R m ), with u ∈ A co for a.e. in Ω. These observations, together with Lemma 2.7 and A co × A co = (A × A) sc ⊂ K sc , imply that as stated.
As a consequence of Corollary 6.2 and the results in [11,Section 6], one can deduce a Young measure representation for the relaxation of constrained nonlocal integral functionals of the type Ω Ω w((x, y, u(x), u(y)) dx dy + I K (u), (6.6) where w : Ω×Ω×R m ×R m → R ∞ is exactly as in [11,Theorem 6.1]. Indeed, the superadditivity of lim inf, (6.3), and [11, Theorem 6.1] entail for every sequence ( On the other hand, if ν ∈ L ∞ w (Ω; Pr(R m )), we choose (u j ) j to be a sequence as in Corollary 6.2 (ii), and apply the version of the fundamental theorem on Young measures in [11,Proposition 3.6] to conclude that lim j→∞ Ω Ω w(x, y, u j (x), u j (y)) dx dy + I K (u j ) = Ω Ω R m R m w(x, y, ξ, ζ) dν x (ξ) dν y (ζ) dx dy + I Y K (ν).
6.3. Notions of nonlocal convexity. In [11] and the references therein, the authors introduce and analyze different notions of nonlocal convexity for inhomogeneous finite-valued doubleintegral functionals, including nonlocal convexity, nonlocal convexity for Young measures, and a nonlocal Jensen inequality. Here, we transfer these notions to our context of homogeneous indicator functionals in the scalar setting, i.e. functionals I K and I Y K as in (6.1) and (6.2) with K as in Theorem 1.1, and discuss their relation.
Let us first define the condition referred to as nonlocal convexity (NC): For every w ∈ L ∞ (Ω; R m ), the function A generalization of condition (NC) is the following nonlocal convexity for Young measures (NY), which requires that for every ν ∈ L ∞ w (Ω; Pr(R m )), the function The next proposition establishes the equivalence of all these notions. In particular, in view of Corollary 5.9, they are all necessary and sufficient for L ∞ -weak * lower semicontinuity of I K .
Proposition 6.4. If K ⊂ R m × R m is as in Theorem 1.1, then Proof. For the proof of (NJ) ⇔ (SC), we make use of (6.4) and (6.5), together with the fact that I K = I K sc implies K sc = K sc = K due to Proposition 5.1 and Lemma 4.5.
The arguments behind the other implications are straight-forward. The implication (SC) ⇒ (NY) follows right from the definition of separate convexity of χ K . Via the identification of u ∈ L ∞ (Ω; R m ) with the family of Dirac measures {δ u(x) } x∈Ω , the condition (NY) is clearly at least as strong as (NC). To see (NC) ⇒ (SC), it suffices to restrict (NC) to constant functions and exploit the symmetry of K.

Nonlocal supremal functionals
The main focus of this section is the proof of Theorem 1.3, which is based on the results established previously. In what follows, W : R m × R m → R is always assumed to be lower semicontinuous and coercive, i.e., W (ξ, ζ) → ∞ as |(ξ, ζ)| → ∞. In terms of the level sets of W , this means that L c (W ) are compact for any c ∈ R.
We start with a characterization result for L ∞ -weak * lower semicontinuity of functionals as in (1.1) that exploits the relations with nonlocal indicator functionals and nonlocal inclusions. It is a nonlocal version of the analogous statement in the local setting pointed out first by Acerbi  (i) J is L ∞ -weakly * lower semicontinuous; (ii) A Lc(W ) is L ∞ -weakly * closed for all c ∈ R; (iii) I Lc(W ) L ∞ -weakly * lower semicontinuous for all c ∈ R.
Proof. The equivalence of (ii) and (iii) follows immediately from (6.1). It remains to prove that (i) and (ii) are equivalent. Assuming that (i) holds, consider any c ∈ R and any sequence (u j ) j ⊂ A Lc(W ) and u ∈ L ∞ (Ω; R m ) such that u j ⇀ * u in L ∞ (Ω; R m ). Since the L ∞ -weak * lower semicontinuity of J ensures that ess sup we conclude that (u(x), u(y)) ∈ L c (W ) for a.e. (x, y) ∈ Ω × Ω, meaning u ∈ A Lc(W ) . This proves (ii).
For the reverse implication, we take u j ⇀ * u in L ∞ (Ω; R m ) with lim j→∞ J(u j ) = lim inf j→∞ J(u j ) < ∞ for all j ∈ N.
Remark 7.3. Notice that the sufficiency of the separate convexity of the symmetrized and diagonalized sublevel sets of W to ensure L ∞ -weak * lower semicontinuity of J as in (1.1) holds without any further assumptions also in the vectorial case m > 1. The argument employs Proposition 3.6 under consideration of (7.3) and (7.2) below.
Our next goal is to establish a representation formula for the relaxation of J. Inspired by the previous corollary, we define W : Then, for any c ∈ R, Since the sublevel sets of W are compact, this shows in particular that the level sets of W are compact as well, and hence, that W is lower semicontinuous. Moreover, W is coercive due to W ≥ W , and symmetric, i.e., W (ξ, ζ) = W (ζ, ξ) for every (ξ, ζ) ∈ R m × R m , by definition, cf. (4.1).
It is crucial to realize that a functional J as in (1.1) has a uniquely determined supremand W only up to symmetrization and diagonalization in the sense of (7.1). To be precise, it holds that J(u) = ess sup (x,y)∈Ω×Ω W (u(x), u(y)) = ess sup (x,y)∈Ω×Ω W (u(x), u(y)) =:Ĵ(u) (7.3) for u ∈ L ∞ (Ω; R m ); indeed, along with Proposition 5.1 and (7.2), ess sup In light of Definition 3.3 for the separate convex envelope of a function and Definition 3.1 for the separately convex hull of a set, it is immediate to see that for every c ∈ R. If m = 1, one can show that even equality holds in (7.5). In particular, if we recall the properties of W and Remark 3.2, this implies that W slc : R × R → R is lower semicontinuous and coercive.
Since all sublevel sets of W are compact, symmetric and diagonal, Lemma 4.9 entails that for any c ∈ R, which shows that V is separately level convex. Due to W ≥ V , we conclude that W slc ≥ V , and consequently L c ( W slc ) ⊂ L c (V ) = L c ( W ) sc for all c ∈ R. Considering that the other inclusion is immediate in view of the definition of the separately level convex envelope W slc completes the proof.
With these preparations, we can now prove Theorem 1.3 (ii), namely the relaxation result for supremal nonlocal functionals in the scalar case. Then there exists a sequence of real numbers (c k ) k with c k ց c as k → ∞ such that owing to (7.4) and (7.2), u ∈ A Lc k ( W slc ) = A Lc k ( W ) sc = A Lc k (W ) sc for all k ∈ N. Now, Theorem 1.1 applied to A Lc k (W ) sc for every k ∈ N guarantees the existence of a sequences (u k,j ) j ⊂ A Lc k (W ) with u k,j ⇀ * u in L ∞ (Ω) as j → ∞. Via diagonalization (see [3, Lemma 1.15, Corollary 1.16]), one can select a diverging subsequence k(j) → ∞ as j → ∞ such that the sequence (u j ) j with u j := u k(j),j ∈ A Lc k(j) (W ) for j ∈ N satisfies u j ⇀ * u in L ∞ (Ω).
Then, Under additional assumptions, we can generalize Proposition 7.5 to the vectorial case.
Remark 7.6. Let W : R m × R m → R with m > 1 such that for any c ∈ R, the sublevel set L c (W ) is compact and satisfies both (5.16) and (7.6). Then, the L ∞ -weak * lower semicontinuous envelope of J is then given by the nonlocal supremal functional with density W slc , which may in general be different from W slc , as Remark 4.6 b) indicates.

7.2.
Explicit examples of lower semicontinuous functionals and relaxations. To illustrate the general results of Section 7.1, we present a few examples of nonlocal L ∞ -functionals whose supremands have multiwell structure.
In the scalar setting, we determine explicit relaxation formulas for two nonlocal four-well supremands. Even though the sets of wells can be transformed into each other via rotation and scaling, their relaxations feature qualitative differences.
The next examples show the L ∞ -weak * lower semicontinuity of two types of supremal functionals with symmetric two-well supremands in the vectorial setting. recalling that B r (ξ) = {ζ ∈ R m : |ζ − ξ| ≤ r} for r > 0 and ξ ∈ R m , cf. Section 2.1. Note that W is not separately level convex, since L c (W ) fails to be separately convex for c ≥ |α|; in particular, Proposition 3.6 is not applicable here. However, as the union of Cartesian products of convex sets, all level sets of W are clearly symmetric and diagonal, meaning W = W , and we can infer in light of Remark 4.6 b) and (7.2) that L c (W ) sc = L c (W ) sc = L c (W ) = L c (W ). By Remark 7.6, this condition is sufficient for L ∞ -weakly * lower semicontinuity for J as in (1.1).