Vector-valued obstacle problems for non-local energies

We investigate the asymptotics of obstacle problems for non-local energies in a 
vector-valued setting. Motivations arise, in particular, in phase field models 
for ferroelectric materials and variational theories for dislocations.


1.
Introduction. The homogenization of obstacle problems for non-local energies has been object of recent researches. Interesting applications can be found in several fields such as fractional diffusion, contact mechanics, theories of Markov processes, and stock options pricing (see [9] and [16] for an exhaustive list of references).
In a simplified setting, the problem consists in understanding the asymptotic behaviour of the (global) minimizers of the energies in the sequel supplemented by appropriate boundary conditions if u ∈ W s,p (U, R m ), u ∈ E cap s,p q.e. on T j ∩U , +∞ otherwise in L p (U, R m ). Here, U ⊂ R n , n ≥ 1, is a Lipschitz open set, W s,p (U, R m ) is the Sobolev-Slobodeckij space for s ∈ (0, 1), p ∈ (1, +∞) and sp ∈ (0, n], cap s,p is the related variational (p, s)-capacity, and u denotes the precise representative of u ∈ W s,p (U, R m ) which is defined except on a cap s,p -negligible set (see subsections 2.3 and 2.4). In addition, with fixed a set E in R m , a bounded subset T of R n , and a discrete and homogeneous distribution of points Λ = {x i } i∈Z n (see Definition 2.1), the confinement condition u ∈ E cap s,p q.e. is imposed on the obstacle set T j ⊆ R n defined by T j := ∪ i∈Z n (ε j x i + λ j T ), j ∈ N, where (ε j ) j∈N and (λ j ) j∈N are positive infinitesimal sequences. The scalar framework with bilateral or unilateral conditions on the obstacles, corresponding basically to the choices m = 1 and E = {0} or E = (0, +∞) respectively, has been analyzed by means of different approaches (cp. with [8], [9], [15], [16]). In particular, in the Hilbertian framework, i.e. select p = 2 in (1), the asymptotic analysis can be reduced to the case of energies defined on standard (weighted) Sobolev spaces building upon an extension result analogous to the classical harmonic extension of W 1/2,2 functions (see [10]). Very recently a direct proof working directly at the level of non-local energies has been proposed in [16] for values of the parameters s, p such that sp ∈ (1, n). Several possible generalizations are also highlighted there, for instance obstacles with random sizes and shapes or centred on random distribution of points are also analyzed (see [16,Section 4]).
In this paper we push forward the approach introduced in [16] into another direction by extending it to vector-valued problems motivated by phase fields models for ferroelectric solids and for dislocations. In a first model, the vector valued field u : U ⊆ R 2 → {z ∈ R 2 : z ≤ r} represents the spontaneous polarization of a ferroelectric material, the obstacles are then to be considered as zones where an insulator is present, and the non-local energy represents the electrostatic energy related to the electric field created by the charges induced by the spontaneuos polarization field on ∂U (see [14] for more details). In Theorem 3.2 we determine the homogenization limit of the model. Let us remark that the additional constraint that admissible fields take values into a disk is actually not affecting the asymptotics below (see Remark 5 for more comments).
Instead, in a second example we refer to the variational theory for dislocations in an elastic crystal under the action of an applied shear stress introduced in [21] (see also [11] for related analytical results). In this model, supposing that only one slip system is active, u : R 2 → R 2 is the slip field induced by the presence of dislocations, the obstacle sets can be interpreted as pinning sites modeling impurities in the material restraining the motion of those line defects. The free energy is given by the sum of two competing terms: a non-local term analogous to the W 1/2,2 -seminorm, representing the long-range elastic interaction energy related to dislocations, and a non-convex multiwell potential favouring vector-valued phase fields taking integer values, in order to penalize slips not compatible with the underlying crystalline structure. In the analysis below the latter extra energy contribution shall not be included in the spirit of the second order Γ-development performed in the onedimensional case in [17] (see [18] and [19] for the study of the full model in the scalar case with p = 2 and sp = 1; see also [22] if p = 2 and any s ∈ (0, 1) provided no obstacle constraint is considered).
The description of the asymptotics of the energies (F j ) j∈N is addressed in this paper via Γ-convergence. This variational theory is known to be particularly wellsuited to study the behaviour of the sequence of (global) minimizers corresponding to (F j ) j∈N under appropriate boundary conditions or by adding appropriate forcing terms (see [13], [4] and [5]). In what follows, we shall consider only the leading part of the energy since neither forcing terms nor Dirichlet boundary conditions imposed on ∂U , if properly formulated (cp. with Remark 2), change the asymptotics of the problem.
We will show that Γ(L p )-limits of the family (F j ) j∈N take the form if u ∈ W s,p (U, R m ), +∞ otherwise in L p (U, R m ). In the previous formula the prefactor ϑ is related to the mutual relationship between the scalings of the problem (see (36) or (57) for a precise definition according to the different ranges of sp in (0, n]), β describes the limit distribution of the points in Λ (see (30)); eventually the function ϕ is defined in formulas (35) if sp = n, and in (56) if sp ∈ (0, n).
More precisely, the previous result holds true upon the extraction of subsequences in the scaling-invariant framework. Actually, we will analyze energies as in (1) comparable to the fractional seminorms, that is defined through singular kernels K that are anisotropic versions of those above. In such a generality, Γ-convergence to a functional analogous to that in (2) holds only up to subsequences unless extraassumptions are imposed on the kernel K (cp. with Theorems 3.2 and 3.3 and Section 4). The energy density of the obstacle penalization term, the function ϕ in (2), describes the asymptotic behaviour of non-linear, vector-valued relative capacitary problems defined by the non-local energy under consideration.
We will focus our attention mainly on the scaling invariant case, corresponding to the choice sp = n in (1), since the subcritical framework sp ∈ (0, n) can be deduced from the results of the ensuing sections combined with those in [16] (see Subsection 3.2). Actually, we will slighlty improve upon [16] by including the range sp ∈ (0, 1] which was not covered there. From a technical point of view the main novelties of the paper are contained in Proposition 2 (see also Proposition 3 for related results) where instrumental properties of the mentioned non-linear, vector-valued (relative) capacitary problems are established. The latter, combined with a joining lemma in varying domains for non-local energies established by the Author in [16,Lemma 3.9], are relevant in the analysis developed in the subsequent sections.
Finally, we remark that the homogenization of obstacle problems for gradient energies in the local framework, i.e. defined on the standard Sobolev spaces W 1,p , has been investigated by several Authors. We mention [2], [5], [8], [9], [15], [16] and [24] for exhaustive references. Here, we only stress that such a problem was recently addressed by [2] in the subcritical case, i.e. p ∈ (1, n), and by [24] in the scaling-invariant case, p = n. Vector-valued problems for quasiconvex energies with E = {0} were analyzed in both papers. Extensions to more general constraint sets E using the methods of Proposition 2 are likely to be obtained in that setting, too.
A brief resume of the paper is as follows: Section 2 is devoted to introduce the notations adopted throughout the whole paper and the instrumental preliminary results. In particular, Subsection 2.4 deals with (relative) capacities for Sobolev-Slobodeckij spaces both in the scaling invariant and subcritical cases.
In Section 3 we prove the Γ-convergence statements contained in Theorems 3.2 and 3.3. Some possible generalizations are considered in Section 4.

Preliminaries and notations.
2.1. Basic notations. The Euclidean norm in R n shall be denoted by | · |, the maximum one by | · | ∞ . We will write B r (x) for the Euclidean ball in R n with centre x and radius r > 0, and simply B r in case x = 0. As usual, we will set ω n := L n (B 1 ).
Given any set D ⊂ R n , its complement will be indifferently denoted by D c or R n \ D; instead, its interior and closure will be denoted by int(D) and D, respectively.
Given an open set A ⊆ R n the collections of its open subsets will be indicated by A(A), the diagonal set in R n × R n by ∆, and for every δ > 0 its open δneighborhood by ∆ δ := {(x, y) ∈ R n × R n : |x − y| < δ}. Accordingly, for any set D ⊆ R n and for any δ > 0 In the following, U will always be an open and connected subset of R n whose boundary is Lipschitz regular. We shall use standard notations for Lebesgue and Hausdorff measures, and for Lebesgue and Sobolev function spaces. In addition, u O will denote the mean value of a summable function u on a L n -measurable set O with positive measure, i.e.
In several computations below the letter c shall generically denote a positive constant. We assume this convention since it is not essential to distinguish from one specific constant to another, leaving understood that the constant may change from line to line. The parameters on which each constant c depends will be explicitely highlighted.

2.2.
Non-periodic tilings. Aperiodic sets of points are considered in the ensuing sections. More precisely, the Voronoï tessellation related to a Delone set of points Λ will substitute the usual periodic lattice.
The book by M. Senechal [23] is the standard reference for all the results quoted below.
Definition 2.1. A point set Λ ⊂ R n is a Delone (or Delaunay) set if it satisfies (i) Discreteness: there exists r > 0 such that for all x, y ∈ Λ, x = y, |x − y| ≥ 2r; (ii) Homogeneity: there exists R > 0 such that Λ ∩ B R (x) = ∅ for all x ∈ R n .
It is then easy to show that Λ is countably infinite. Hence, from now on we use the notation Λ = {x i } i∈Z n . By the very definition the quantities are finite and strictly positive; R Λ is called the covering radius of Λ.
Definition 2.2. Let Λ ⊂ R n be a Delone set, the Voronoï cell of a point x i ∈ Λ is the set of points The Voronoï tessellation induced by Λ is the partition of R n given by {V i } i∈Z n .
The sets V i 's are closed, convex polytopes intersecting only along their boundaries. Several other interesting properties of Voronoï tessellations are collected in [23, Propositions 2.7, 5.2] (see also [16,Propositions 2.4 and 2.5]). Here, we will only recall some results that will be used in the subsequent analysis. We omit their proofs since they are justified by elementary counting arguments. For any A ∈ A(R n ) set then the following results hold true.
Proposition 1. Let Λ ⊂ R n be a Delone set and {V i } i∈Z n its induced Voronoï tessellation. Then, ω n r n Λ #(I Λ (A)) ≤ L n (A), There exists a constant c = c(n) > 0 such that for every i ∈ Z n and h ∈ N it holds 2.3. Sobolev-Slobodeckij spaces. Let A ⊆ R n be any bounded open Lipschitz set and let s ∈ (0, 1), p ∈ (1, +∞) be such that sp ∈ (0, n]. We will use several properties of fractional Sobolev spaces, giving precise references for those employed in the sequel in the respective places mainly referring to [1] and [25]. Here, we recall only the Poincaré-Wirtinger inequality in fractional Sobolev spaces, which follows from the usual argument by contradiction once the reflexivity of W s,p is ensured (for this see [25, Theorems 2.6.1 and 4.2.3]).
for a constant c P W = c P W (n, s, O, A).

Remark 1.
A scaling argument and Hölder inequality yield for any x ∈ R n and r > 0 and for some c = c(n, s, O, A) > 0 with the usual convention inf ∅ = +∞. It is also worth introducing relative capacities. To this aim, first localize (12) for open sets A ∈ A(R n ) setting and then extend it to all subsets of R n by outer regularity as Arguing as in [12,Section 3], the set functions above turn out to be Choquet capacities (see also [1, Chapter V] and [20, Theorem 2.2]). Recall that a set T in R n is said to be of (s, p)-capacity zero if for all ρ > 0. We also say that a property holds (s, p)-quasi everywhere, in short cap s,p q.e., if it holds up to a set of (s, p)-capacity zero. In particular, any function , has a precise representative u defined cap s,p q.e. and the following formula holds (see [1,Proposition 5.3] and [12, Section 4]), The behaviour of relative capacities can be distinguished according to whether sp = n or sp ∈ (0, n). Before proceeding into this direction let us enlarge the framework of interest to a vector-valued setting and also to singular kernels different from those defining the fractional seminorms. More generally, in the sequel we shall be concerned in the sequel with non-linear, vector-valued capacitary problems related to translation-invariant singular kernels K : R n \ {0} → (0, +∞) satisfying for some constant α ≥ 1 and for all x ∈ R n \ {0} (see Section 4 for generalizations). For every A ∈ A(R n ), the kernel K defines a functional K : if u ∈ W s,p (A, R m ), +∞ otherwise on L p (A, R m ). We shall drop the dependence on A if A = R n . A relevant notion related to K is that of locality defect: for any L n -measurable function w and any L n×n -measurable set D ⊆ R n × R n Clearly, K(w, A) = D K (w, A×A); the terminology is justified since given two disjoint subdomains A, B ⊆ R n , for C = A ∪ B we get The scaling-invariant case. Let us first focus our attention to parameters p ∈ (1, +∞) and s ∈ (0, 1) satisfying sp = n (for related results and references in the local case see [24]). The scaling invariance of the kernel yields that for any subset T of R n and any In addition, the following estimates can be obtained as in [12,Theorem 3.11]: for some constant c = c(n, p) > 0 for every x ∈ R n and for every pair of positive numbers t, ρ such that t < ρ/2. Hence, one can show that cap s,p (T ) = 0 for all sets T as in the standard (local) Sobolev setting. By formula (17) a logarithmic rescaling is then needed. We have not been able to prove that the function ρ → (ln ρ) p−1 C s,p (B t , B ρ ) has actually a limit as ρ diverges. Such a property is well-known for (standard) Sobolev relative n-capacities (for a homogenization-like proof see [24,Proposition 5.1]).
Despite this, (16) and (17) imply that for all bounded subsets T of R n with non-empty interior part we have for some constant c = c(n, p) ≥ 1.
With fixed subsets T ⊆ R n and E ⊆ R m , radii ρ and R such that where the set of admissible test functions is given by For the sake of simplicity we shall drop the dependence on T and K in (19) when there will be no risk of confusion, and write only φ ρ if in addition R = +∞.
With the same choices of E and K a similar characterization can be given in the vectorial setting, too.
In general, the properties enjoyed by (φ ρ,R ) ρ , as described in what follows, are not obtained by explicit characterizations.
In the next proposition we shall establish several results for (φ ρ,R ) ρ . We remark that the estimates below will always be uniform for all those kernels satisfying the growth conditions in (13). In addition, in what follows the letter α will always denote the constant introduced there. Proposition 2. Suppose T bounded. Then, for every 0 < ρ < R as above it holds (i) (φ ρ,R ) ρ is pointwise equi-bounded: there exists a positive constant c 2 depending on T and α such that for every z ∈ R m In addition, there exists a non-negative constant c 1 depending on T , m and α such that for every z ∈ R m (ii) (φ ρ,R ) ρ is locally equi-Lipschitz continuous: there exists a positive constant c depending only on E, T , m, p and α such that for all z 1 , (iii) there exists a positive constant c depending on m, n, p and α such that Proof. We start off with item (i). Given ε > 0, take a test function ξ for the capacitary problem of T in B ρ such that . For any point ζ ∈ E consider w := ξ ζ + (1 − ξ)z, then w ∈ AD z (T, B ρ ) and satisfies By letting first ε ↓ 0 + , and then passing to the infimum on ζ ∈ E we deduce the upper bound in (21) by (18) with c 2 := α lim sup ρ (ln ρ) p−1 C s,p (T, B ρ ).
To show (22), choose z ∈ R n with dist(z, E) > 0 and let δ : The second instance is completely analogous. The conclusion follows at once with We now turn to the proof of item (ii). To this aim we use special external variations. Clearly, by (21) it is not restrictive to assume R m \ E = ∅. Fix z 1 and z 2 ∈ R m , we shall estimate the oscillation of φ ρ,R in those points. Fix δ > 0 and first suppose dist(z 1 , E) ≥ 2δ. Note then that In addition, Kirszbraun's extension theorem supplies the existence of a map Ψ ∈ Lip(R m , R m ) such that Ψ| E δ = Id and Ψ(z 1 ) = z 2 , with .

VECTOR-VALUED OBSTACLE PROBLEMS FOR NON-LOCAL ENERGIES 495
Now take any w ∈ AD z1 (T, B ρ ), and note that Ψ • w ∈ AD z2 (T, B ρ ) since E δ is open and Ψ| E δ = Id. It is also immediate to check that In turn, the latter estimate, (21), (25), and the convexity of R t → |t| p imply Hence, if in addition dist(z 2 , E) ≥ 2δ, exchanging the roles of z 1 and z 2 we infer for a positive constant c 3 depending on E, T , p, m and α, and above all independent from δ. Hence, the arbitrariness of δ > 0 yields that (26) holds for all points z 1 , z 2 / ∈ E. Actually, (26) is still valid if z 1 , z 2 ∈ E by (21). Finally, if z 1 ∈ E and z 2 / ∈ E we have In conclusion, we have established (26) for all couple of points z 1 , z 2 in R m .
To prove item (iii), fix any test function w in AD z (T, B ρ ), for R > ρ we infer from the scaled Poincaré-Wirtinger inequality in (10) Multiplying by (ln ρ) p−1 the previous inequality and taking into account the arbitrariness of w imply the conclusion.
Furthermore, let us highlight some additional properties enjoyed by the families (φ ρ,R ) ρ and related to the geometry of E. Their proof is omitted since it is a straightforward consequence of the definition of φ ρ,R and Proposition 2.
Remark 4. The proofs of (21) and (22) show that the constants c 1 , c 2 depend on T through the inferior, superior limit of (ln ρ) p−1 C s,p (T, B ρ ), respectively. Hence, if T is bounded and has non-empty interior, those constants are both positive and do not actually depend on T itself by (18). Hence, if in addition the kernel K is (−2n)-homogeneous, under the same hypotheses on T , one can argue as in (18) and show that for all z ∈ R n the inferior and superior limits as ρ → +∞ of (ln ρ) p−1 φ ρ (z) do not depend on T either.
On the other hand, the functions φ ρ do depend on the set E as it follows, for instance, from item (i) in Proposition 3. Note also that E = R m if and only if φ ρ = 0 by (21) and (22), in case lim inf ρ (ln ρ) p−1 C s,p (T, B ρ ) > 0, that is c 1 > 0.

2.4.2.
The subcritical case. In case sp ∈ (0, n) the behaviour of (relative) capacities is rather different. Indeed, an elementary scaling argument shows that for all t > 0 and x ∈ R n cap s,p (B t (x)) = t n−sp cap s,p (B 1 ).
One can also show that the expression above is strictly positive. Consider a kernel K satisfying (13) and let AD z (T, B ρ ) given by (20). Following verbatim the arguments of Propositions 2 and 3 one can establish results completely analogous to those contained there, except for estimate (22). In case sp ∈ [1, n), the latter inequality can still be obtained by employing [6, Corollary 2, Proposition 2] (see also [3,Remark 4]) rather than the critical embedding into V M O exploited in Proposition 2. The proofs being even easier since in the current range no logarithmic rescaling is needed in the definition of φ K,T ρ,R (see also [16,Lemma 2.12]). Details will not be worked-out and left to the interested reader.
In addition, for all z ∈ R m the function ρ → φ K,T ρ (z) turns out to be nonincreasing, thus monotonicity implies the convergence As before, the function φ K,T ρ above equals φ K,T ρ,R for R = +∞. Actually, the pointwise convergence above is uniform on compact subsets of R m (for the full family) thanks to (23) in this setting.
3. Γ-convergence statement. Consider Delone sets Λ j = {x i j } i∈Z n , and let r j := r Λj , R j := R Λj , I j (A) := I Λj (A), for all A ∈ A(U ), dropping the dependence on the set if A = U , that is I j := I j (U ) (see (5) for the definition of I Λj (·)). Assume that the Λ j 's are such that for some β ∈ L 1 (U, (0, +∞)) with β L 1 (U ) = 1. It has been shown in [ Note that by (6), (7) and (29) we infer 0 < lim inf j r n j #I j ≤ lim sup j r n j #I j < +∞.
With fixed subsets E ⊆ R m , and T ⊂ R n bounded, for all j ∈ N define the obstacle set T j := ∪ i∈Z n T i j where T i j := x i j + λ j T, and λ j ∈ (0, r j ).
Subsections 3.1 and 3.2 contain the asymptotic analysis of the energies above in the scaling-invariant and subcritical cases, respectively. The proofs are strongly linked though some details are different. In particular, in the former case the energy does not concentrate at the same scale as the radii of the perforations.
The Γ-convergence statement established in Theorems 3.2 and 3.3 relies upon a technical result proved in [16,Lemma 3.9] in the scalar case. There is no difficulty in extending that result in the vector-valued setting currently under investigation, thus we limit ourselves to present its statement only.
On a technical side, it reduces the verification of liminf and limsup inequalities on sequences of functions almost matching the values of their limit on suitable annuli surrounding the obstacle sets. Following an early idea by De Giorgi, a clever slicing and averaging argument is exploited to change boundary values increasing the energy in a controlled and infinitesimal way (for more comments see [16,Subection 3.2]).
To recall the statement of [16, Lemma 3.9] we fix some more notation: for all i ∈ I j , N and h ∈ N let Lemma 3.1. Let (u j ) j∈N be converging to u in L p (U, R m ) and also satisfying sup j |u j | W s,p (U,R m ) < +∞. With fixed N ∈ N, for every j ∈ N there exists h j ∈ {1, . . . , N } and w j ∈ W s,p (U, R m ) such that for some c = c(m, n, p, s, α) > 0 it holds for every measurable set D in U × U and the sequence (w j ) j∈N converge to u in L p (U, R m ). In addition, if the functions u j ∈ L ∞ (U, R m ), then Eventually, if ζ j := i∈Ij (u j )

Remark 5.
A similar statement can be proved in case sequences (u j ) j∈N taking values into a convex set are considered. Indeed, the method to change boundary data performed in [16,Lemmata 3.8,3.9], is realized through a convexity argument, which is then compatible with the constraint on the target codomain. This clarification is necessary to fit the analysis of the phase-field model for ferroelectric solids mentioned in the Introduction in our setting.
3.1. The scaling-invariant case. In the sequel we will consider parameters s ∈ (0, 1) and p ∈ (1, +∞) fixed and such that sp = n. Notice then that p > n.
To describe the asymptotic behaviour of the sequence (K j ) j∈N we shall use the auxiliary functions in (19) given for any q ∈ Q + by In what follows we shall assume that the sequences (ϕ j,q ) j∈N converge uniformly on compact subsets of R m to functions ϕ q for every q ∈ Q + . Note that since ϕ q2 ≤ ϕ q1 if q 1 ≤ q 2 there exists the limit and the convergence is uniform on compact sets of R m . The convergence assumptions in (35) are not restrictive, they are always satisfied upon the extraction of subsequences thanks to (i) and (ii) in Proposition 2 since the family (λ 2n j K(λ j ·)) j∈N satisfies (13) uniformly. Theorem 3.2. Let U ∈ A(R n ) be bounded and connected with Lipschitz regular boundary.
Given sets of points Λ j satisfiying (29)-(30), and functions ϕ j,q satisfying (35), suppose that the following limit exists Then, the sequence (K j ) j∈N Γ-converges in the L p (U, R m ) topology to the functional K : L p (U, R m ) → [0, +∞] defined by 3.1.1. Proof of the Γ-convergence. In Proposition 4 below we show the lower bound inequality. By Lemma 3.1 we may consider only sequences assuming constant values around the obstacles, which are then approximately mean values of the target function close to the T i j 's (cp. with (ζ j ) j∈N in Lemma 3.1). Then, a separation of scale argument shows that the capacitary contribution is concentrated along any neighborhood of the diagonal set ∆. Instead, the remaining part of the energy provides the long range interaction term since the kernel is no longer singular far from ∆.
Proof. Without loss of generality we shall assume ϑ > 0 in what follows, the lower bound inequality being trivial otherwise.
Fix δ > 0, N ∈ N, and consider the sequence (w j ) j∈N provided by Lemma 3.1. Then, (w j ) j∈N converges to u in L p (U, R m ) and for some c = c(m, n, p, s, α) > 0 it holds Upon extracting a subsequence, not relabeled for convenience, we may assume that the right hand side above is actually a limit, and in addition the index h j ∈ {1, . . . , N } in Lemma 3.1 to be independent of j. Hence, from now on we shall denote it simply by h.
Note that for j sufficiently big ∪ i∈Ij (V i j × V i j ) ⊆ ∆ δ , and thus lim inf thanks to Fatou's lemma. We claim that for with N > 0 infinitesimal as N → +∞. Given this for granted, by (38) inequality (39) rewrites as The thesis then follows by passing to the limit first as N → +∞, and then as δ → 0 + in the last inequality.
To conclude we are left with proving (40). We keep the notations of Lemma 3.1 and formula (34) with q = N −(3h+2) ; note that B N qλj ρj (x i j ) ⊆ V i j for all i ∈ I j , and that λ j ρ j = r j .
A change of variables and item (iii) in Proposition 2 give where ζ j is defined in Lemma 3.1 and Ψ j (x) := (#I j ) −1 i∈Ij (L n (V i j )) −1 χ V i j (x). Note that Ψ j L ∞ (U ) ≤ (ω n r n j #I j ) −1 , so that (Ψ j ) j∈N is equi-bounded in L ∞ (U ) by (29). Furthermore, it is easy to show that (Ψ j ) j∈N converges to β weak * L ∞ (U ) (cp. with [16,Proposition 3.10]). Then, recalling that (ϕ j,q ) j∈N converges to ϕ q uniformly on compact subsets of R m , and (ζ j ) j∈N to u in L p (U, R m ) (see Lemma 3.1), it follows lim inf j i∈Ij Eventually, estimate (40) follows at once by letting A increase to U .
In the next proposition we prove that the lower bound established in Proposition 4 is sharp. Thanks to the insight provided by Proposition 4, we are able to construct a sequence for which there's no loss of energy asymptotically in all the estimates there.
Proof. To begin with we note that it is not restrictive to assume u ∈ W 1,∞ (U, R m ) by a standard density argument, the lower semicontinuity of Γ-lim sup K j , and the continuity of K with respect to strong convergence in W s,p as follows from item (ii) in Proposition 2.
In addition, we may also take u ∈ W 1,∞ (U , R m ) on an open and bounded smooth set U such that U ⊂⊂ U .
Fix N ∈ N, and let (w j ) j∈N be the sequence obtained from u by applying Lemma 3.1 on U . To simplify the notation introduced there set (cp. to (34)) γ j (z) := ϕ j,N −(3h j +2) (z), ψ j (z) := ϕ j,N −5 (z), j := N −(3hj +2) λ −1 j r j . Note that since h j ∈ {1, . . . , N }, then γ j ≤ ψ j . In addition, if i ∈ I j := I j ∪ I j (U ) let and notice then that C i,hj j ⊂ B i j (see (5) for the definition of I j (U )). For every j ∈ N the very definition of γ j and the fact that j = N −(3hj +2) ρ j yield the existence of a function ξ i j ∈ AD u i j (T, B j ) such that λ 2n Then, define For the sake of notational simplicity we have not highlighted the dependence of the sequence (u j ) j∈N on the parameter N ∈ N. Clearly, (u j ) j∈N converges strongly to u in L p (U, R m ), and moreover it satisfies the obstacle condition by construction. The rest of the proof is devoted to show that u j ∈ W s,p (U, R m ) with lim sup where δ → 0 + as δ → 0 + and N → 0 + as N → +∞. A recovery sequence as in the statement of Proposition 5 can be then constructed via a diagonal arument.
We first reduce ourselves to compute the energy of u j on a neighborhood of the diagonal ∆. Indeed, let δ > 0, Lebesgue dominated convergence and the stated convergence of (u j ) j∈N to u in L p (U, R m ) imply In addition, since u j = w j on U j by (33) in Lemma 3.1 we have for some positive constant c = c(m, n, p, s, α) The conclusion then follows provided we show that lim sup In order to prove this we introduce the following splitting of the left hand side above: Next we estimate separately each term I r j , r ∈ {1, 2, 3}. All the constants c appearing in the rest of the proof will depend only on m, n, p, s, α and u W 1,∞ (U ,R m ) . Hence, this dependence will no longer be indicated.
Step 1. Estimate of I 1 j : A straightforward change of variables, the very definition of u j and (21) yield where Ψ j (x) = (#I j ) −1 i∈Ij (L n (V i j )) −1 χ V i j (x) and ζ j is defined in Lemma 3.1 and it is related to the sequence with constant terms equal to u. Arguing as in Proposition 4, the convergences of (ψ j ) j∈N to ϕ N −5 uniform on compact subsets Therefore, a change of variables, Poincaré-Wirtinger inequality, and the very definition of j yield Clearly, (49) and (50) imply (47).
Step 3. Estimate of I 3 j : Being u j = w j on U j , we find Since , by a change of variables the first integral above can be bounded by To deal with the term I 3,2 j , we use the growth conditions on K in (13), integrate out y thanks to (11), and observe that w j | C i,h j j = u i j to apply the scaled Poincaré-Wirtinger inequality in (10) and infer: Finally, for what I 3,3 j is concerned we have By collecting (52)-(54) we infer (51).
Step 4: Conclusion. The conclusion follows at once from Step 1 -Step 3.
Remark 6. In Theorem 3.2 we can also allow for ϑ = +∞ provided E is assumed to be closed and we set +∞ · 0 := 0. For, the liminf inequality easily can be easily inferred by a comparison argument and Proposition 4 above without any restriction on E. In turn, this implies that a function u in L p (U, R m ) with K (u) < +∞ satisfies u(x) ∈ E for L n a.e on x ∈ U thanks to (22) in Proposition 2. Hence, u(x) ∈ E cap s,p q.e. on U , so that if E is closed the function u provides a trivial recovery sequence for itself to be employed in the proof of Proposition 5, i.e. K j (u) = K (u) for every j ∈ N.
3.2. The subcritical case. We now establish a result analogous to Theorem 3.2 in case the singular kernel K satisfies (13) with sp ∈ (0, n). We limit ourselves to state the result and comment on it since the scalar setting for homogeneous kernels has been investigated in details in [16] (for sp ∈ (1, n)) and few changes are needed to deal with the vector-valued one considered in this paper once Propositions 2 and 3 are at disposal.
In doing that we slightly extend the conclusions in [16,Theorem 3] by including the case sp ∈ (0, 1] that was originally excluded in that statement. The reason for that being the use of Hardy inequality in some estimates, which does not hold true if sp = 1 (see [ Hence, in what follows, we state an asymptotic result for the full subcritical range of values of p and s. To describe the limit behaviour of (K j ) j∈N we shall consider auxiliary functions as in (27) given for any q ∈ Q + by ϕ j,q (z) := φ λ n+sp j K(λj ·),T qρj (z), where ρ j := λ −1 j r j .
In addition, we shall assume that the sequences (ϕ j,q ) j∈N converge uniformly on compact subsets of R m to functions ϕ q for every q ∈ Q + . Note that since ϕ q2 ≤ ϕ q1 if q 1 ≤ q 2 there exists the limit ϕ(z) := lim q→0 + ϕ q (z) = lim q→0 + lim j→+∞ ϕ j,q (z) (56) and the convergence is uniform on compact sets of R m . Let us point out that in case the kernel K is (−n − sp)-homogeneous ϕ j,q reduces to φ K qρj , so that the convergence in (56) holds true for the whole sequence as noticed in (28). Otherwise, it is guaranteed only up to subsequences. Theorem 3.3. Let U ∈ A(R n ) be bounded and connected with Lipschitz regular boundary.
Given sets of points Λ j satisfiying (29)-(30), and functions ϕ j,q as in (55), suppose that (56) holds and that the following limit exists 4. Generalizations. Several generalizations are possible following the path in [16].