Non-local approximation of the Griffith functional

An approximation, in the sense of $\Gamma$-convergence and in any dimension $d\geq1$, of Griffith-type functionals, with $p-$growth ($p>1$) in the symmetrized gradient, is provided by means of a sequence of non-local integral functionals depending on the average of the symmetrized gradients on small balls.


Introduction
In this paper we provide a variational approximation by means of non-local integral energies of functionals of the form αˆΩ \K W (Eu(x)) dx + 2βH d−1 (K), (1.1) where Ω is a bounded subset of R d , K ⊆ Ω is closed, W is a non-negative and convex function with p−growth for some p > 1, u ∈ C 1 (Ω\K; R d ), Eu denotes the symmetric part of the gradient of u and H d−1 is the (d − 1)-dimensional Hausdorff measure. Functionals as in (1.1) are the core of many variational models of fracture mechanics, in the framework of Griffith's theory of brittle fracture under the small strain assumption (see, e.g., [31,29] and the references in [17,Introduction]). If, as usual, the set Ω denotes the reference configuration and u represents the displacement field of the body, then the total energy (1.1) is the sum of a bulk energy in Ω\K, where the material is supposed to have an elastic/elasto-plastic behavior (see, e.g., [29,Section 2], [32,Sections 10 and 11]), and a surface term accounting for the energy necessary to produce the fracture, proportional to the area of the crack surface K. It is only very recently that a rigorous weak formulation of the problem (1.1) has been provided [26,19]. Within this setting, u is a (vector-valued) generalized special function of bounded deformation, for which the symmetrized gradient Eu is defined almost everywhere in an approximate sense (see [26]), and the set K is replaced by the (d − 1)-rectifiable set J u , the jump set of u. This space is denoted by GSBD p (Ω), where the exponent p refers to the integrability of Eu. After that the existence of weak minimizers has been achieved, one can actually show that the jump set thereof is closed (up to a H d−1 -negligible set), and prove well-posedness of the minimization problem for (1.1) (see [21,16,18]).
However, the minimization of functionals of the type (1.1) may be a hard task in practice, mainly due to the presence of the surface term H d−1 (J u ). Such difficulties already appear for W (M ) = |M | 2 and in the case of antiplane shear (see, e.g., [10]), where the energy (1.1) reduces to the Mumford-Shah-type functional Ω |∇u| 2 dx + H d−1 (J u ), (1.2) for a scalar-valued displacement u ∈ SBV (Ω), the space of special functions of bounded variation. It is indeed well-known that a variational approximation of (1.2) by means of local integral functionals of the formˆΩ f ε (∇u) dx , defined on Sobolev spaces, is doomed to failure (see [12,Introduction]). Over the last years, this has motivated a great effort to provide suitable approximations of (1.2) by means of more manageable functionals, leading to the convergence of minimum points. A number of different approaches has been proposed, which in some cases have also been generalised to the (more challenging) setting of (1.1), both in a discrete and in a continuous, infinite-dimensional, setting. A very popular approach, originally proposed by Ambrosio and Tortorelli [4,5] and generalised to the linearly elastic setting in [28,17], provides an approximation of (1.1) and (1.2) by elliptic functionals (with parameter ε > 0) at the price of adding an auxiliary phase-field variable v ∈ H 1 (R d ; [0, 1]). The heuristics behind these functionals, taking the form is to approximate the discontinuity set K with the ε-layer {v ε ∼ 0}. Also discretizations of the above functionals by means of either finite-difference or finite-elements with mesh-size δ, independent of ε, have been considered. For a suitable fine mesh, with size δ = δ(ε) small enough, these numerical approximations Γ-converge, as ε → 0, to the Mumford-Shah functional (see [6,8], and [7] for the case of a stochastic lattice). A similar result for the energy (1.1) has been recently provided in [24]. For other discrete approaches based on finite differences or finite elements we may mention [15,20,38], in the context of the Mumford-Shah functional, and [35,1] for the Griffith model. Closer to the purpose of our paper are, however, variational approximations by means of nonlocal integral energies. Following a conjecture by De Giorgi, Gobbino proved for instance in [30] that the functionals 1 dx dy Γ-converge to (1.2) when ε → 0. A discretization of this model on graphs has been recently analysed in [14], and adaptions to the stochastic setting have also been provided ( [37]). Another method, introduced in [12], is based on non-local integral functionals whose density depends on the average of the gradient on small balls, in order to prevent that the effect of large gradients be concentrated on small regions. There, functionals of the form

4)
B ε (x) denotes the closed ball of radius ε centred at x ∈ Ω and ffl B v dx is the average of v on B. The functionals (1.3) Γ-converge, as ε → 0, to the functional Later on, in [13], it has been shown that more general energies of the form where |u + −u − | is the jump of u across J u , can be obtained by considering non-local approximating functionals as in (1.3) with varying densities f = f ε , and the function θ is computable from f ε . This analysis has been continued in [33,34] for functionals with bulk terms having linear growth in the gradient.
The very first non-local approximation of Griffith-type energies on the footsteps of [12], inspired by the subsequent generalization [23] of such model, has been provided in [36]. There, non-local convolution-type energies of the form are considered, where f ε is a suitable sequence of densities, ρ is a convolution kernel with support suppρ and ρ ε (z) is the usual sequence of convolution kernels ρ(z/ε)/ε d . The Γ-limit of (1.5) with respect to the L 1 convergence is shown to be the Griffith-type functional where the anisotropy φ ρ depends on the geometry and on the size of suppρ, and the function u belongs to the space SBD p of special functions of bounded deformation with Eu ∈ L p , which is a (proper) subspace of GSBD p . The argument in [36] introduces some novelties with respect to [12,23], in particular for the proof of the lower bound, which is obtained by means of a delicate construction based on a slicing technique. However, as it happens when dealing with the space SBD p , in order to obtain compactness of sequences of competitors with equibounded energy, an L ∞ bound has to be imposed, which is quite unnatural in Fracture Mechanics.
Our results: The purpose of our paper is to provide a variational approximation of the functional (1.1) in the spirit of [12]. We will namely show that for f complying with (1.4), the functionals Γ-converge to the functional (1.1) in the L 1 (Ω)-topology (Theorem 3.1). The proof strategy is based on the localization method for Γ-convergence (see, e.g., [25, for each u ∈ GSBD p (A), A ⊂ Ω, and ξ unit vector in R d . Above, the symbol ·, · denotes the scalar product in R d . As the two terms on the right-hand side are mutually singular, the Γ-liminf estimate can be recovered out of these two separate estimates by a standard technique (Lemma 2.9). While this general scheme has also been pursued in [12], getting to (1.7) is rather different in our paper than it was to obtain analogous estimates in theirs. Indeed, in the SBV -context of Mumford-Shah-type functionals, one has the possibility of lowering the energy by truncating competitors. Hence, the main estimates can be proved for functions in SBV ∩ L ∞ , as it is done for instance in [12, Proposition 4.1 and Proposition 5.1], where L ∞ -bounds are explicitly exploited. A similar tool is not available in the bounded deformation setting. Hence, we have to renounce to the semi-discrete approach of [12] and follow a different strategy, which is closely related to the the heuristics of the model (1.6).
The main idea for obtaining the first estimate in (1.7) is contained in the proof of Proposition 4.1. Using an energy estimate and the coarea formula, there we show that, for a given error parameter δ, the set where the averages ε ffl B (1−δ)ε (x) W (Eu(x)) dx exceed a given threshold can be included in a set K ′ ε with vanishing area and bounded perimeter. This allows one to show that the Γ-limit of the energies (1.6) is controlled from below by a functional of the type (1.1). The optimal constant in the bulk term can be recovered, as done in Proposition 5.1, by replacing a sequence of competitors u ε with their averages on balls of radius ε at points x ∈ Ω \ K ′ ε . Indeed, K ′ ε is the set where, intuitively, the energy does concentrate on lower dimensional manifolds and the bulk contribution can be neglected. It is worth mentioning that such an optimal estimate for the bulk term is not derived by means of any slicing procedure, which would not comply well with the general form of the bulk energy we are considering.
The second estimate in (1.7) is instead obtained by means of a slicing argument in the fixed direction ξ (see Proposition 5.2), first reconducting the problem to the analysis of the one-dimensional version of the functional (1.3) (which can be performed with elementary arguments, see [11,Theorem 3.30]) and then exploiting the slicing properties of GSBD functions recalled in Section 2.2. Finally, the Γ-limsup inequality (Proposition 6.1) can be obtained by a direct construction for a regular class of competitors having a "nice" jump set, and which are dense in energy according to recent approximation results, summarized in Theorem 2.3. Let us remark that our proof strategy can also be applied, with obvious modifications, for an alternative and, in our opinion, slightly simpler proof of the results in [12].
To end up this review of our results, we want to motivate our choice of the L 1 -topology and warn the reader of a related issue. Actually, while L 1 -convergence is a natural choice in the context of the Mumford-Shah functional, both for the possibility of using truncations and for the presence of L p -fidelity terms, when dealing with fracture models it would be preferable to deal with the convergence in measure. Indeed, Proposition 4.1 in principle only allows one for applying Theorem 2.4, which provides subsequences that are (essentially) converging in measure. 1 However, dealing with sequences converging in L 1 allows us to deduce the convergence of the averaged functions in Lemma 2.7 which are a useful tool in our proofs. Notice that compactness in L 1 can be easily enforced by adding a lower-order fidelity term, for which a completely satisfactory compactness and Γ-convergence result can be stated and proved (Theorem 3.2). It then seems to us that adding such a term, although not completely justified from the point of view of fracture mechanics, does not really affect our methods and results.
Outline of the paper: The paper is organized as follows. In Section 2 we fix the basic notation and collect some definitions and results on the function spaces we will deal with (Section 2.2), together with some technical lemmas (Section 2.3) which will be useful throughout the paper. In Section 3 we list the main assumptions, introduce our model (eq. (3.3)), and state the main results of the paper, given with Theorem 3.1 and Theorem 3.2. Section 4 contains the compactness result of Proposition 4.1. Section 5 is devoted to the Γ-liminf inequality: the separate estimates from below of the bulk term and the surface term of the energy are contained in Sections 5.1 and 5.2, respectively; the proof of the Γ-liminf inequality is the content of Section 5.3. The upper bound is provided in Section 6.

Notation and preliminary results
2.1. Notation. The symbol ·, · denotes the scalar product in R d , while | · | stands for the Euclidean norm in any dimension. The symbol Ω will always denote an open, bounded subset of R d . The Lebesgue measure in R d and the s-dimensional Hausdorff measure are written as L d and H s , respectively.
The symbol S d−1 will denote the (d−1)-dimensional unit sphere. The family of the open subsets of Ω will be denoted by A(Ω).

GBD and GSBD functions.
We recall here some basic definitions and results on generalized functions with bounded deformation, as introduced in [26]. Throughout the paper we will use standard notations for the spaces SBV and SBD, referring the reader to [3] and [2,9,39], respectively, for a detailed treatment on the topics.
Let ξ ∈ R d \{0} and Π ξ = {y ∈ R d : ξ, y = 0}. If y ∈ Π ξ and Ω ⊂ R d we set Ω ξ,y := {t ∈ R : y + tξ ∈ Ω} and Ω ξ := {y ∈ Π ξ : Ω ξ,y = ∅}. Given u : while if h : Ω → R, the symbol h ξ,y will denote the restriction of h to the set Ω ξ,y ; namely, The presence of the exceptional set A ∞ u in the statement of Theorem 2.4 is no real issue in the context of the Griffith model, as setting u = 0 there is optimal for the energy, see [19].
Let ξ ∈ S d−1 . For any y ∈ R d we denote by y ξ and y ξ ⊥ the projections onto the subspaces Ξ := {tξ : t ∈ R} and Π ξ , respectively. For σ ∈ (0, 1) and x ∈ R d we define the cylinders ) is a bounded Radon measure on Ω whose total variation satisfies for every Borel subset B of Ω.
If u ∈ GBD(Ω) and ξ ∈ R d \{0} then, in view of [26, Theorem 9.1, Theorem 8.1], the following properties hold: By [26,Remark 4.5] one has the inclusions BD(Ω) ⊂ GBD(Ω) and SBD(Ω) ⊂ GSBD(Ω), which are in general strict. Some relevant properties of functions with bounded deformation can be generalized to this weak setting: in particular, in [26, Theorem 6.2 and Theorem 9.1] it is shown that the jump set J u of a GBD-function is H d−1 -rectifiable and that GBD-functions have an approximate symmetric differential Eu(x) at L d -a.e. x ∈ Ω, respectively. The space GSBD p (Ω) is defined through: Every function in GSBD p (Ω) is approximated by bounded SBV functions with more regular jump set, as stated by the following result ([17, Theorem 1.1]).

Theorem 2.3.
Let Ω ⊂ R d be a bounded open Lipschitz set, and u ∈ GSBD p (Ω; R d ). Then there exists a sequence u n such that A further approximation result, by Cortesani and Toader [22,Theorem 3.9], allows us to approximate GSBD p (Ω) functions with the so-called "piecewise smooth" SBV -functions, denoted W(Ω; R d ), characterized by the three properties (2.8) We recall the following general GSBD p compactness result from [19], which generalizes [26,Theorem 11.3]. In the statement the symbol ∂ * denotes the essential boundary of a set with finite perimeter. We keep this general form of the statement. However, since we have to enforce L 1convergence of sequences with bounded energy, the situation which is relevant for our purposes is described in Remark 2.5 below.
2.3. Some lemmas. We recall here the following property of commutability of the integral averages with the gradient operator for a Sobolev vector-valued function. The proof is based on standard arguments by test functions, so we omit the details. (2.10) We will make use of the following convergence properties of averaged functions. These are probably well-known, a short proof is however added for the reader's convenience.
We will make also use of the following property of finite coverings of a bounded set.
Remark 2.8. Let R > 0 and E ⊂ R d be such that E ⊂ B R . Then, for every r > 0, there exists a finite subset and #(E ′ ) depends only on r, R and d. If, in addition, R r ≤ λ, then there exists a constant C = C(λ, d) such that #(E ′ ) ≤ C.
For this, we note that the family of balls 2 (z) ∩ E and so doing we construct the set Then, the desired covering of E is given by {B r (x) : x ∈ E ′ }. Note also that From (2.11) and a simple scaling argument, we infer that C(r, R, d) is uniformly bounded when the ratio R r is bounded. The following result, dealing with the supremum of a family of measures, will be useful for the derivation of the Γ-liminf inequality (see, e.g., [11,Proposition 1.16]). We also remark the following approximation property from below for lower semicontinuous increasing functions with truncated affine functions. A proof is given for the reader's convenience.
Then there exist two positive sequences (a i ) i∈N , Proof. For all (h, k) ∈ N 2 set Above the function f (t) t is extended by continuity with the value α for t = 0. We clearly have that a hk ≤ α for all (h, k) ∈ N 2 ; furthermore α is an accumulation point for the family a hk , hence α = sup{a hk : (h, k) ∈ N 2 }. With the monotonicity of f we have b hk ≤ β for all (h, k) ∈ N 2 ; furthermore β is an accumulation point for the family b hk , hence β = sup{b hk : (h, k) ∈ N 2 }.
By construction we have so that min{a hk t, b hk } ≤ f (t) for all (h, k) ∈ N 2 and t ∈ R. It then simply suffices to consider an enumeration of N 2 to conclude the proof.

2.4.
A one-dimensional Γ-convergence result. We recall here a one-dimensional Γ-convergence result which will be useful in the sequel. In the statement below, functions in L 1 (I) with I ⊂ R are extended by 0 outside of I, so that the functionals H ε are well-defined (actually, the result is not affected by the considered extension).

The non-local model and main results
In this section we list our assumptions and introduce the main results of the paper. Let Ω ⊂ R d be an open set with Lipschitz boundary, let 1 < p < +∞ and f : [0, +∞) → [0, +∞) a lower semicontinuous, increasing function satisfying Let W : R d×d → R be a convex positive function on the subspace M d×d sym of symmetric matrices, such that For every ε > 0 we consider the functional F ε : for every Borel set B ⊆ Ω and for every w ∈ L 1 (B). We will deal with a localized version of the energies (3.3). Namely, for every A ⊆ Ω open set, we will denote by F ε (u, A) the same functional as in (3.3) with the set A in place of Ω.
The following theorem is the first main result of this paper.
Notice that there is a mismatch between part (i) and (ii) of the previous statement. Indeed, the compactness property in (i) does not entail the L 1 -convergence of a subsequence of u ε . It only allows one to apply Theorem 2.4, which has a weaker statement. However, the L 1 -convergence on the whole Ω can be easily enforced with the addition of a lower order fidelity term, as we have discussed in Remark 2.5. This motivates the statement below.

set for every open set
+∞ , otherwise on L 1 (A; R d ). (3.6) When A = Ω, we simply write G ε (u) in place of G ε (u, Ω). Then we have the following result.
(ii) The energies (G ε ) Γ-converge, as ε → 0, to the functional with respect to the L 1 convergence in Ω.
Remark 3.3. Existence of minimizers for the functional G, and also for F if coupled with a Dirichlet datum, directly follows from Theorem 2.4 (see [19] for details). For fixed ε, the functionals F ε and G ε are lower semicontinuous in W 1,p (Ω; R d ), but clearly not coercive. However, as done in [12,Corollary 3.2], one can perturb f with a sequence f ε of functions having linear growth at infinity, and satisfying f (t) ≤ f ε (t) ≤ f (t) + a ε t for a sequence a ε = o(ε) as ε → 0 and still recover a Γ-convergence result. Their argument would also apply to the present situation: notice that only the Γ-limsup inequality has to be adapted, and this is straightforward in the space of regular approximating functions provided by Theorem 2.3. We omit the details of this generalization. If we now replace f with f ε , existence of minimizers for G ε in W 1,p can be obtained via the direct method. Then, Theorem 3.2 (ii) also gives convergence of the minimizers to a minimizer of G in GSBD p (Ω).

Compactness
With the following proposition, we prove the compactness statements in Theorem 3.1(i), and Theorem 3.2 (i), respectively.
Proof. Let ε > 0 and δ ∈ (0, 1) be fixed 2 . It suffices to consider here only the case f (t) = min{at, b} with a, b > 0. In the general case one can indeed find a, b > 0 with f (t) ≥ min{at, b} for all t, using Lemma 2.10, and deduce the result a fortiori. Hence, let us assume f (t) = min{at, b}. We define and the function Correspondingly, we introduce the compact set The set K ε is actually also depending on the fixed δ (as well as the sets K ′′ ε and K ′ ε used below) but we omit this dependence to ease notation. We first note that, setting Now, from the inclusion (4.3) and the fact that Then, applying the coarea formula to the 1-Lipschitz function g(x) := dist(x, K ε ) (see for instance [27,Theorem 3.14]) in the open set It follows that we can choose 0 < δ ′ ε < δε such that, for it holds For every ε > 0, we set Note that from (4.4) an the bound F ε (u ε , A) ≤ C it follows that We prove the following Claim: there exists a constant N > 0 depending only on d such that for every x ∈ A. For this, we first note that by definition of u ε , and since W (0) = 0 is the minimum value of W , one has W (Eu ε (x)) ≤ W (Eu ε (x)) for a.e. x. Now, when x ∈ A\K ′ ε , it holds x ∈ K ε so that by definition of K ε we have On the other hand, if x ∈ K ′ ε , then Remark 2.8 shows the existence of a finite subset of A\K ′ ε , say {x 1 , x 2 , . . . , x N }, where N only depends on the dimension d, such that We then have, with (4.7) and (4.9), where in the latter inequality we used the fact that the points x i ∈ K ′ ε . This concludes the proof of Claim.
Since W ≥ 0 and f is nondecreasing, we have the estimate W (Eu ε (y)) dy dx . (4.10) Moreover, if t ≤ N b a , one has the elementary inequality min{at, b} ≥ a N t. With this, recalling that W (Eu ε (x)) ≤ W (Eu ε (x)) for a.e. x, using the Claim, (4.10) and the monotonicity of f we obtain the estimate where we changed variables y = x + (1 − δ)εz and used Fubini's Theorem. Since W ≥ 0 with a further change of variables and using (3.2) we conclude Since by definition (4.7) we have that J uε = ∂K ′ ε , with (4.6), (4.11) and from the assumption F ε (u ε , A) ≤ C we deduce that u ε ∈ SBV p (A; R d ). Moreover, setting c 0 := 1 2 min{ a N , bδ 2 }, we infer the lower bound Combining with (4.8), this proves (i).

5.
Estimate from below of the Γ-limit 5.1. Estimate from below of the bulk term. We begin by giving a first estimate of the Γliminf of the functionals F ε . This estimate is optimal (up to a small error) for the bulk part of the energy, while it is not, for what it concerns the surface part. An optimal estimate for this term will be provided separately by means of a slicing argument (see Proposition 5.2 below). As the two parts of the energy are mutually singular, the localization method of Lemma 2.9 will eventually allow us to get the Γ-liminf inequality.
Proof. We divide the proof in two steps.
Step 1: we first consider the case f (t) = min{at, b}, with a, b > 0. Observe that in this case the value α given by (3.1) coincides exactly with a. We can clearly assume that otherwise the assertion is immediate. Corresponding to the fixed δ > 0 and for every ε > 0, we define the constant C δ , and the sets K ε and K ′ ε as in (4.1), (4.2) and (4.5), respectively. We define Then (iii) immediately follows from Lemma 2.7(i) and the fact that, by construction and (4.4), , so that with (4.6) we deduce (ii) for M δ = 1 δb . To prove (i), we observe that, since K ε ⊂ K ′ ε and A ⊂⊂ Ω, it holds for all x ∈ A \ K ′ ε . As C δ > 1 and f (t) = min{at, b}, we deduce the elementary inequality With (5.3), (5.4), the monotonicity of f , the convexity of W , (2.10) and (5.2) we get W (Eu ε (y)) dy dx which implies assertion (i). This concludes the proof of Step 1.
Step 2: for a general f complying with (3.1), use Lemma 2.10 to find a δ , b δ > 0 with a δ ≥ α(1−δ) and f (t) ≥ min{a δ t, b δ } for all t ∈ R, and perform the same construction as in the previous step. This gives (iii), (ii) (with M δ := 1 δb δ ) and that is (i).

5.2.
Estimate from below of the surface term. For any A ⊂ Ω open set, we denote by F ′ (u, A) the lower Γ-limit of F ε (u, A); that is, In the estimate of F ′ (u, A), it will be useful the following property, which is an immediate consequence of the definition: We note also that, since F ε (u, ·) is superadditive as a set function, the lower Γ-limit F ′ (u, ·) inherits an analogous property; namely, Proposition 5.2. Assume (3.1) and (3.2). Let δ ∈ (0, 1) be fixed, and consider a sequence ε j → 0.
Then u ∈ GSBD p (A) and for every ξ ∈ S d−1 .
Proof. It follows from Proposition 4.1 and Theorem 2.4 that u ∈ GSBD p (A). To prove (5.7), we first note that, by virtue of the growth assumption (3.2), we have for every ξ ∈ S d−1 . Thus, for every fixed ξ, since f is non-decreasing, it will be sufficient to provide a lower estimate for the energies We proceed by a slicing argument. If for each x ∈ A we denote by x ξ and y ξ the projections of x onto Ξ and Π ξ , respectively, we have , and denote (with a slight abuse of notation) still with z the (d − 1)-dimensional variable in B d−1 r δ εj (y ξ ). Set Since the function t → f (c(d, δ)t) still tends to β when t → +∞, applying Theorem 2.11 to the one-dimensional energies Consequently, from (5.10) and (5.11) we obtain that Taking into account (5.9), with Fatou's Lemma we then have and the proof of (5.7) concludes.

5.3.
Proof of the Γ-liminf inequality. We summarize the results of the previous sections in the following Proposition. The Γ-liminf G ′ of the sequence G ε is defined as in (5.5), with G ε in place of F ε . It trivially holds G ′ (u, A) ≥ F ′ (u, A) for each open subset A ⊂ Ω and u ∈ L 1 (A; R d ).
for every ξ ∈ S d−1 . If it additionally holds G ′ (u, A) < +∞, then one also has Proof. First we note that, by the definition of Γ-liminf, there exists a sequence (u j ) converging to u in L 1 (A; R d ) such that The first equality and Proposition 5.2 give that, if F ′ (u, A) < +∞, then u ∈ GSBD p (A). By the second one, the superadditivity of the liminf and Fatou's lemma, we have Hence, if (i) is proved, (iii) follows immediately. We only have to confirm (i) and (ii). To this aim, let δ ∈ (0, 1) be fixed. Then, by applying Proposition 5.1 to the sequence (u j ), there exists a sequence of functions (v δ j ) ⊂ SBV p (A; R d ), converging to u in L 1 (A) as ε j → 0, such that . Combining (a) and (b) with the equiboundedness of F εj (u j , A), one can apply the lower semicontinuity part of Theorem 2.4 to the sequence (v δ j ). Taking into account that A ∞ u = ∅ because u ∈ L 1 (A; R d ), by the convexity of W and (2.9), (ii), we have By letting δ → 0 above we then obtain (i) As for (ii), by Proposition 5.2, in particular from (5.7), we get for every ξ ∈ S d−1 , so that (ii) follows by taking the limit as δ → 0 again.

(6.4)
Proof. We only prove (6.3) by using the density result of Theorem 2.3, as (6.4) follows by an analogous construction with the additional property (2.7). In view of Theorem 2.3 and remarks below, by a diagonal argument it is not restrictive to assume that u ∈ W(Ω; R d ) and that J u is a closed subset of any of the coordinate hyperplanes, that we denote by K.
Proof of Theorems 3.1 and 3.2. The two results follow by combining Propositions 4.1, 5.4, and 6.1