Characterization of generalized Young measures generated by $\mathcal A$-free measures

We give two characterizations, one for the class of generalized Young measures generated by $\mathcal A$-free measures, and one for the class generated by $\mathcal B$-gradient measures $\mathcal Bu$. Here, $\mathcal A$ and $\mathcal B$ are linear homogeneous operators of arbitrary order, which we assume satisfy the constant rank property. The characterization places the class of generalized $\mathcal A$-free Young measures in duality with the class of $\mathcal A$-quasiconvex integrands by means of a well-known Hahn--Banach separation property. A similar statement holds for generalized $\mathcal B$-gradient Young measures. Concerning applications, we discuss several examples that showcase the rigidity or the failure of $\mathrm{L}^1$-compensated compactness when concentration of mass is allowed. These include the failure of $\mathrm{L}^1$-estimates for elliptic systems and the failure of $\mathrm{L}^1$-rigidity for the two-state problem. As a byproduct of our techniques we also show that, for any bounded open set $\Omega$, the inclusions \[ \mathrm{L}^1(\Omega) \cap \ker \mathcal A \hookrightarrow \mathcal M(\Omega) \cap \ker \mathcal A, \] \[ \{\mathcal B u\in \mathrm{C}^\infty(\Omega)\} \hookrightarrow \{\mathcal B u\in \mathcal M(\Omega)\}, \] are dense with respect to area-functional convergence of measures


Introduction
The last decades have witnessed an extensive development of the study of nonconvex variational energies related to equilibrium configurations of materials in a wide range of physical models (such as the study of crystalline solids and thermoelastic materials, linear elasticity, perfect plasticity, micro-magnetics, and ferromagnetics, among others [14,18,24,36]). Often, these models consist in a minimization principle for integrals of the form (1) w where Ω ⊂ R d is an open and bounded set, f : Ω × R N → R satisfies a uniform p-growth condition |f (z)| 1 + |z| p , and the configurations w : Ω → R N obey a set of physical laws determined by a system of linear PDEs, where, depending on the particular model, either Aw = 0 in the sense of distributions on Ω, or (2) w = Bu for some potential u : Ω → R M .
We shall refer to the first scenario as the A-free framework and to the latter as the potential or B-gradient framework. In order to keep the exposition as simple as possible, we shall henceforth adopt the A-free perspective.
In these circumstances, designs with near to minimal energy exhibit compatible equilibrium behavior at microscopical scales, while, at larger scales, configurations adapt by gluing together the low energy patterns allowed by the governing equations in (2)/(3). This interplay conveys the formation of finer and finer oscillations, often resulting in some form of L p -weak convergence w j ⇀ w when p > 1, or weak- * convergence (in the sense of measures) when p = 1 [5,6,11,16,29,30,42,43,58,59]. In general, such weak forms of convergence are incompatible with the lower semicontinuity of the energy, which is usually the starting point for minimization principles. Additionally, the case p = 1 may be ill-posed in the sense that, independently of In this paper we focus on the case p = 1, which requires a careful study of oscillations and concentrations occurring along weak- * convergent sequences of measures satisfying (2)/(3). In this regard, an equivalent approach towards the understanding of (1)- (2) consists of characterizing all generalized Young measures (see [25]) generated by sequences {µ j } ⊂ M(Ω; R N ) ∩ ker A. Let us recall that, formally, a generalized Young measure associated to a sequence {µ j } ⊂ M(Ω; R N ) is a triple ν = (ν x , λ, ν ∞ x ) x∈Ω conformed by a non-negative measure λ ∈ M + (Ω) and two families {ν x }, {ν ∞ x } of probability measures over the target space R N , satisfying the fundamental property that for all sufficiently regular integrands f : Ω × R N → R with linear growth at infinity. The main result of this paper is contained in Theorem 1.1 and states that a generalized Young measure ν, with zero boundary-values λ(∂Ω) = 0, is generated by a sequence A-free measures if and only if (see Definition 1.4) f ⟪id R N , ν⟫ ≤ ⟪f, ν⟫ for all A-quasiconvex integrands f : Ω × R N → R .
This separation result implies that the class of generalized A-free Young measures is a convex set characterized by duality in terms of all A-quasiconvex integrands. In addition to this duality characterization, we give a characterization in terms of the blow-up properties of generalized Young measures (see Theorem 1.2). More precisely, we prove that ν as above is generated by a sequence of A-free measures if and only if its tangent cones Tan(ν, x) almost always contain a generalized Young measure that is generated by A-free measures. Lastly, in Theorem 1.3, we establish the following approximation result: if µ ∈ M(Ω; R N ) is a bounded A-free measure, then there exists a sequence of A-free functions {w j } ⊂ L 1 (Ω; R N ) that converges to µ in the sense of the generalized area functional, i.e., w j L d * ⇀ µ as measures on Ω, and Ω 1 + |w j | 2 dx → Ω 1 + |µ ac | 2 dx + |µ s |(Ω).
We also prove analogous results in the B-potential setting (1)- (3), for generalized measures generated by sequences of the form {Bu j } ⊂ M(Ω; R N ). These are contained in Theorem 1.5, Theorem 1.4 and Theorem 1.6.
1.1. State of the art. The work of Young [70][71][72] and the use of (classical) Young measures plays a fundamental role in representing solutions of optimal control problems. The study of Young measures, from the point of view of partial differential equations, started with the work of Tartar & Murat, who, motivated by problems in continuum mechanics and electromagnetism, introduced the theory of compactness by compensation [6,51,53,66,67]. The first characterization of Young measures in the PDE-constrained context is due to Kinderlehrer & Pedregal [37,38] for the potential configuration w = ∇u, of a Sobolev function u ∈ W 1,p (Ω; R m ) with p > 1. This characterization of L p -gradient Young measures accounts for the validity of Jensen's inequality between gradient Young measures and (curl-)quasiconvex integrands. 1 More precisely, the authors showed that a (purely oscillatory) family of probability distributions {ν x } x∈Ω on the space of m × d matrices M m×d is a Young measure generated by a p-equi-integrable sequence of gradients ∇u j ⇀ ∇u if and only if for all quasiconvex integrands f : M m×d → R with p-growth at infinity. The characterization also covers the case p = 1, but only when the generating sequences are assumed to be equi-integrable. The extension of this result to generalized Young measures generated by gradients, which is instead associated to the space BV(Ω; R m ) of functions of bounded variation, is due to Kristensen & Rindler [42]. There, the authors show that a generalized Young measure ν = (ν x , λ, ν ∞ x ) x∈Ω is generated by a sequence of gradient measures if and only if a version of (6) holds for the absolutely continuous part of ν, that is, for all quasiconvex integrands f : M m×d → R with linear-growth at infinity, where Du = D ac uL d + D s u. Somewhat surprisingly, this conveys that the nonlinear moments of the purely concentration part (λ s , ν ∞ x ) of (ν, λ, ν ∞ ) x∈Ω are fully unconstrained. (This is a consequence of Alberti's rank one theorem [3] and a recent rigidity result for positively homogeneous rank-one convex functions established by Kirchheim & Kristensen [39].) The efforts to establish an A-free variational theory for Young measures initiated with the work of Dacorogna [20], who studied A-free functions w that are represented by potentials w = Bu where B is a suitable first-order operator. However, it was the seminal work of Fonseca & Müller that laid the foundations for an A-free setting under the more general assumption of A satisfying the constant rank property; see (8) below. 2 The authors generalized Morrey's notion of quasiconvexity to the A-free setting and showed that the necessary and sufficient condition for the lower semicontinuity of (1)- (2), under p-growth and p-equi-integrability assumptions, was precisely the A-quasiconvexity of the integrand. Fonseca & Müller 1 The importance of quasiconvexity in the calculus of variations was first observed by Morrey [46,47], who showed quasiconvexity is a sufficient and necessary condition for the lower semicontinuity of (1)-(3), when B = D is the gradient operator. 2 A recent result of Raita [57], crucial to this work, establishes that Dacorogna's assumption and the constant rank assumption are locally equivalent, up to considering B of higher-order (see Lemma 5.1).
also extended Kinderlehrer-Pedregal's characterization theorem to the A-free setting by showing that a family of probability distributions {ν x } x∈Ω ⊂ Prob(R N ) is a Young measure generated by a p-equi-integrable sequence of A-free maps {w j } if and only if the following two conditions hold: (i) there exists w ∈ L p (Ω; R N ) such that Aw = 0 and z dν x (z) as functions in L p (Ω; R N ), (ii) at L d -almost every x ∈ Ω, Jensen's inequality holds and all A-quasiconvex integrands h : R N → R with p-growth at infinity. The generalization of this result to generalized Young measures without the p-equiintegrability assumption in the range 1 < p < ∞ was later established by Fonseca & Kružík [27]. In the generalized Young measure framework for p = 1, the only characterization results are restricted to two well-known potential structures, gradients B = D ( [42]) and symmetrized gradients B = E ( [23]). 3 The well-established proofs for the case when B = D, E cited above rely on the strong rigidity properties that gradients and symmetric gradients possess. However, such properties are not known to hold for general higher-order operators. Up to now, the only A-free result in the generalized setting was a partial characterization due to Baía, Matias & Santos [12]. There, the authors characterize all generalized Young measures generated by A-free measures under the following somewhat restrictive assumptions: (a) The operator A is assumed to be of first-order. This implies that its associated principal symbol map ξ → A(ξ) is a linear map. In turn, this allows for rigidity and homogenization-type arguments which unfortunately fail for higher order operators. (b) The characterization is restricted to Young measures generated by sequences µ j * ⇀ µ, where the limiting measure µ satisfies the following Morrey-type bound sup r>0 |µ|(B r (x)) r 1+α < ∞ for some α > 0.
This upper-density bound on µ is in general too restrictive for applications as it rules out 1-rectifiable measures. For instance, every non-degenerate closed smooth curve γ : [0, 1] → Γ ⊂ R d defines a divergence-free measure by setting µ = (γ/|γ|) H 1 Γ. The purpose of this work is to give a full characterization of all generalized Young measures generated by A-free measures, as well as a characterization of generalized Young measures generated by B-gradients, for operators A and B satisfying the constant rank property. Therefore, we extend the aforementioned results into a unified general setting that allows for the appearance of mass concentrations in the case p = 1. Our strategy departs from previous ones (even in the case of gradients) in the sense that we do not work with averaged Young measure approximations nor rely on the rigidity of PDE-constrained measures. Instead, we work with Lebesguepoint continuity properties and the gluing of local generating sequences at the level of potentials; it is for this last point that the constant rank property is fundamental because it guarantees Sobolev-type regularity estimates when the kernel of A (or B) is removed. It is worth to mention that the characterization for A-free measures presented here does not deal with characterizations up to the boundary. The main assumption being that a generating sequence w j : Ω → W does not concentrate mass on the boundary ∂Ω. In this regard, the work of Kružík [13] addresses the characterization of generalized gradient Young measures up to the boundary; such results for general operators A are yet to be explored.

1.2.
Comments on the constant rank assumption. It is worthwhile to briefly discuss the role that the constant rank assumption plays for both the A-free setting and the B-potential setting. On the one hand, potentials allow for localizations of the form Bu → B(ϕu). In the case of gradients, these localizations are stable thanks to Poincaré's inequality u L p Du L p . For general B, this type of Poincaré estimates only holds after removing the kernel of B, i.e., u − π B u L p Bu L p , where π B is the (L 2 -)projection onto ker B. At the time the Fonseca-Müller characterization was given, one of the challenges was the lack of a potential structure for A-free fields. In this regard, Fonseca and Müller's strategy in the A-free setting departs from the one of Kinderlehrer and Pedregal, because localizations had to be carried out at the level of the A-free field w → ϕw. In order to handle this localization, the authors relied on the use of the equivalent estimate w − π A w L p Aw W −k,p , often referred as the Fonseca-Müller projection in the calculus of variations (it is, in fact, a Calderón-Zygmund-type bound). In either case, the constant rank property is a sufficient and necessary condition for the boundedness of both Poincaré's and Fonseca-Müller's estimates (see [32]).
While most of the physically relevant applications are modeled by operators that do satisfy the constant rank property (see Section 2), the variational theory in the setting (1)-(2)/(1)- (3) is not restricted to operators satisfying the constant rank property. Notably, Müller [49] characterized all (classical) Young measures generated by diagonal gradients. This setting is associated to the operator A(w 1 , w 2 ) = (∂ 2 w 1 , ∂ 1 w 2 ), which is one of the simplest examples of an operator that does not satisfy the constant rank property (see [66]). Other related work about the understanding of PDE-constraints where the constant rank property is not a main assumption include [9,23,68], and more recently [10].
1.3. Set-up and main results. We assume throughout the paper that U ⊂ R d is an open set, and that Ω ⊂ R d is an open and bounded set satisfying L d (∂Ω) = 0. Here, L d denotes the d-dimensional Lebesgue measure.
We work with a homogeneous partial differential operator A (or B), from W to X (or V to W ), of the form where W, X (and V ) are finite dimensional inner product euclidean spaces. Here α ∈ N d 0 is a multi-index with modulus |α| = α 1 + · · · + α d and ∂ α represents the distributional derivative ∂ α1 1 · · · ∂ α d d . Our main assumption on A (and B) is that it satisfies the following constant rank property: there exists a positive integer r such that where the tensor-valued k-homogeneous polynomial is the principal symbol associated to the operator A. Here, ξ α := ξ α1 1 · · · ξ α d d . We also recall the notion of wave cone associated to A, which plays a fundamental role for the study of A-free fields, as discussed in the work of Murat & Tartar [51,53,66,67]: The wave cone contains those Fourier amplitudes along which it is possible to construct highly oscillating A-free fields. More precisely, P ∈ Λ A if and only if there exists ξ ∈ R d \ {0} such that A(P ϕ(x · ξ)) = 0 for all ϕ ∈ C k (R). Let us begin our exposition by introducing a few concepts about the theory of generalized Young measures (as introduced in [25], and later extended in [4]): where S W is the unit sphere in W , and (iv) the map x → | q |, ν x belongs to L 1 loc (U ). If moreover, (iv) the map x → | q |, ν x belongs to L 1 (U ), and (v) λ is a finite measure, then we say that ν is a generalized Young measure. We write Y loc (U ; W ) to denote the set of locally bounded generalized Young measures, and Y(U ; W ) to denote the set of generalized Young measures.
Notation. In the following and when no confusion arises, we will often refer to generalized Young measures simply as Young measures. We will also write for all integrands f ∈ E(U, W ); see Section 4.2 for the precise definition of E(U ; W ). In this case we write µ j Y → ν on U .
Next, we incorporate the PDE constraint into the concept of generalized Young measure. Let us recall that W −k,p (U ) = (W k,p ′ 0 (U )) * for all 1 ≤ p < ∞. Here, p ′ is the dual exponent of p and W k,p ′ 0 (U ) is the closure of C ∞ c (U ) with respect to the W k,p ′ -norm.
We write Y A (U ) to denote the set of such Young measures. We will also require to define a weaker notion of recession function. For a Borel integrand h : W → R with linear growth at infinity, we define its upper recession function as Differently from h ∞ , the upper recession function always exists and defines an upper semicontinuous and positively 1-homogeneous function on W .
We are now in position to state our main characterization results. The first result extends the Hahn-Banach-type characterization [28,Theorem 4.1] to sequences of weak- * convergent measures: (ii) at L d -almost every x ∈ Ω, the Jensen-type inequality h(µ ac (x)) ≤ h, ν x + h # , ν ∞ x λ ac (x) holds for all A-quasiconvex upper-semicontinuous integrands h : W → R with linear growth at infinity, and then the purely singular part (λ s , ν ∞ ) of ν is unconstrained since then (iii) is equivalent to the trivial set inclusion In Section 2, we shall revise a few examples of operators that satisfy this property.
Remark 1.2. The condition at regular points, embodied by property (ii), conveys a similar constraint for the supports of ν x and ν ∞ x on a set of full L d -measure. The results contained in Corollary 4.1 imply that ν x is the µ ac (x) translation of a probability measure supported on W A , i.e., The same corollary also conveys that property (iii) holds λ-a.e., that is, On the other hand, the property at singular points (iii) is equivalent to the complementary Jensen's inequality This follows directly from the structure theorem for A-free measures [22, Theorem 1.1] and the rigidity results established in [39].
Our second result characterizes generalized A-free Young measures in terms of their tangent cone (in the spirit of [60]); definitions of tangent Young measures will be postponed to Section 4.2.
for all open and Lipschitz subsets U ⋐ R d with λ(∂U ) = 0.
We close the characterization of A-free Young measures with an application of the methods developed in this paper, which allows us to re-define A-free measures in terms of a pure A-free constraint: The following are equivalent: (i) ν is a generalized A-free Young measure, (ii) ν is generated by A-free measures.
Frequently, it has been accepted to impose a geometric assumption on U that guarantees the approximation of A-free measures by L 1 -integrable A-free fields in the strict sense of measures (see for instance [6,7,48]). More precisely, that U is a strictly star-shaped domain, i.e., there exists x ∈ U such that The approximation result contained in Theorem 1.3 below allows, in particular, to dispense with this assumption on the geometry of U . In order to state this result we need to introduce the following basic concept. The area functional of a measure is defined as In addition to the well-known weak- * convergence of measures, we say that a se- This notion of convergence turns out to be stronger than the conventional strict convergence of measures, which requires |µ j |(U ) → |µ|(U ). The usefulness of this form of convergence rests in the fact that the functional is area-continuous on M(U ; W ) for all integrands f ∈ C(U × W ) such that the strong recession function f ∞ exists on U × W (see [42,Theorem 5] and Area(w j L d , Ω) → Area(µ, Ω). Remark 1.3. Notice that we do not require Ω to be Lipschitz nor L d (∂Ω) = 0. The regularity of the recovery sequence {w j } can be lifted to be of class C k (Ω; W ) for any k ∈ N.

1.3.3.
Characterization of generalized B-gradient Young measures. In this section we state the characterization results that belong to the potential setting (1)-(3). Let B be a homogeneous linear operator of arbitrary order, from V to W , and assume that B satisfies the constant rank property (8). Let us first introduce the notion of B-gradient Young measure: We write BY(U ) to denote the set of these Young measures.
Remark 1.4. We do not require that the sequence of generating potential measures {u j } is uniformly bounded. However, of course, the sequence {Bu j } must be uniformly bounded since it generates ν.
Since B is satisfies the constant rank property, the results in [57] (see also Section 5) yield the existence of an annihilator operator for B. More precisely, there exists A from W to X as in (7) such that (11) Im A localization argument and an application of the Fourier transform imply that B-gradients are A-free fields and therefore, in this case, A more interesting question in this context is to understand how far is a generalized A-free measure from being a generalized B-gradient Young measure. The first step to answer this question is to notice that, by a slight modification of the proof of Theorem 1.2, we obtain the following local characterization: Now, before stating the analog of Theorem 1.1 for B-gradients, we will need to adapt some of the preliminary definitions of the A-framework into the B-framework. In the case of potentials, the role of the wave cone is replaced by the image cone which contains the set of B-gradients in Fourier space. The exactness property (11) has two direct consequences: Firstly, it implies (see [57,Corollary 1]) the equivalence between A-quasiconvexity and B-gradient quasiconvexity: and all z ∈ W .
Secondly, the wave cone of A coincides with the image cone of B (i.e., I B = Λ A ). These two observations and Theorem 1.2 imply that Y A (Ω) and BY(Ω) are structurally equivalent, except at their associated barycenter measures: and Area(Bu j L d , Ω) → Area(Bu, Ω).

Examples
In this section we review, with concise examples, a few of the most well-known Afree and B-gradient structures; most of which -with the exception of (e)-satisfy the spanning property Let us recall that, in this case, the point-wise relation (iii) of the singular part in Theorems 1.1 and 1.5 is superfluous (cf. Remark 1.1). In the following list of examples, the labels "A-free" or "potential" indicate the setting on which the operator is considered: (a) Gradients (potential). Let D be the gradient operator acting on functions u : Ω → R m . Clearly, D is first-order operator from R m to R m ⊗ R d . Moreover, the set of gradients in Fourier space is which is a generating set of R m ⊗ R d .
(b) Higher order gradients (potential). In the same context as the last point, the k-gradient operator is the space of k-th order symmetric tensors. The set of k-th order gradients in Fourier space is the set A standard polarization argument implies that (c) Symmetric gradients (potential). The symmetric gradient of a vector field u : R d → R d is defined as as Eu = sym(Du) = 1 2 (Du + Du T ). Clearly, E defines a first-order operator from R d to E 2 (R d ). The space of Fourier symmetric gradients is given by which, again by a polarization argument, can be seen to generate E 2 (R d ).
(d) Deviatoric operator (potential). The operator that considers only the shear part of the symmetric gradient is given by Tr(M ) = 0 }. The set of shear symmetric gradients in Fourier space is the set This set contains all the tensors of the form e i ⊗ e i − e j ⊗ e j and e i ⊗ e j + e j ⊗ e i for i = j, which conform a basis of the trace-free symmetric tensors.
(e) The Laplacian (A-free and potential). An interesting case is the Laplacian operator The Laplacian is a 2nd order operator from R to R. The A-free perspective of the Laplacian corresponds to the variational properties of the harmonic functions. The first statement of Lemma 3.1 gives Λ ∆ = {0} and This says that there are no concentration nor oscillation effects occurring along sequences of uniformly bounded harmonic maps. Of course, this is not surprising since harmonic functions satisfy local (W 1,∞ , L 1 )-estimates. The ∆-potential perspective is completely opposite (I ∆ = R). Indeed, since ∆ is a full-rank elliptic operator, then the second statement in Lemma 3.1 implies that ∆Y(Ω; R) ∩ Y 0 (Ω; R) = Y 0 (Ω; R), which says that being generated by the Laplacian of a sequence of functions represents no constraint. Heuristically, this is also not surprising due to existence of a fundamental solution for the Laplacian.
This defines a first-order operator from R d to R. It is straightforward to verify In particular, every div-quasiconvex function is convex (cf. Section 4.3). This, implies that both the constitutive relations of the absolutely continuous and singular parts of solenoidal Young measures are fully unconstrained: (g) Normal currents (A-free and potential). The following framework has recently received attention in light of the new ideas proposed to study certain dislocation models, which are related to functionals defined on normal 1-currents without boundary (boundaries of normal 2-currents). For a thorough understanding of these models, we refer the reader to [19,35] and references therein. Let 1 ≤ m ≤ d be an integer. The space of m-dimensional currents consists of all distributions T ∈ D ′ (Ω; m R d ). The boundary operator ∂ m acts (in the sense of distributions) on m-dimensional currents T as Therefore ∂ m is first-order operator from m R d to m−1 R d . De Rham's theorem implies that ∂ m is a constant rank operator. Indeed, Im ∂ m (ξ) = ker ∂ m−1 (ξ) for all ξ ∈ R d \ {0}. Hence rank ∂ m (ξ) is continuous on the sphere, and thus also constant. The space N m (Ω) of m-dimensional normal currents is defined as the space of m-currents T , such that both T and ∂ m T can be represented by measures: We say that T ∈ N m (Ω) is a current without boundary provided that ∂ m T = 0. In this context, we say that is an m-boundary Young measure. Notice that the symbol of ∂ m acts on m-vectors w ∈ m R d precisely as the interior multiplication ∂ m (ξ)w = w ξ. If e 1 , . . . , e d is a basis of R d , then ∂ m (e i )[e j ∧ v] = 0 for all i = j and all v ∈ m−1 R d . In particular If we consider ∂ m as a potential, then De Rham's theorem gives

Applications
In this section we discuss some applications of the dual characterizations. First, we give an explicit description of the Young measures, both from the A-free and potentials perspectives, associated to full-rank elliptic systems. The remaining sections are devoted to discuss, mostly via abstract constructions, the failure of classical compensated compactness results in the L 1 setting.
3.1. Young measures generated by full-rank elliptic operators. We show that, for A a full-rank elliptic operator, we can give a simple characterization of the A-constrained Young measures. Let us define first define ellipticity: Definition 3.1. We say that a homogeneous linear operator A of order k, from W to X, is elliptic if there exists c > 0 such that If moreover dim(W ) ≥ dim(X), then we say that A is a full-rank elliptic operator.
The following result says that the sets of A-free and A-gradient generalized Young measures are trivial for (full-rank) elliptic operators: Lemma 3.1. Assume that A is an elliptic operator from W to X. Then, If moreover A is a full-rank elliptic operator, then also Proof. Let us prove the first statement. Let us fix ν ∈ Y A (Ω) ∩ Y 0 (Ω; W ). The ellipticity of A implies that the only mean-value zero A-free smooth [0, 1] d -periodic map is the zero function. Indeed, if w ∈ C ∞ per ([0, 1] d ; W ) is A-free, then applying the Fourier transform (on the torus) to the equation gives If moreover w has mean-value zero, this shows that w(ξ) = 0 for all ξ ∈ Z d . Or equivalently, w = 0. Therefore, by definition, every integrand f ∈ E(W ) is A-quasiconvex. Since C(Ω) × E(W ) separates Y(Ω; W ) (see Lemma 4.1), then properties (i)-(iii) in Theorem 1.1 imply that ν must be an elementary Young measure, i.e., ν = δ µ := (δ µ ac , |µ s |, δ µ |µ| ) for some A-free µ ∈ M(Ω; W ). The ellipticity of A implies that Λ A = {0} and hence [22,Theorem 1.1] implies that µ = µ ac L d . This proves that ν = (δ w , 0, q ) for some A-free integrable map w.
We now show the statement for the potential perspective when A is a fullrank elliptic operator. If d = 1, then A is equivalent to the operator D k u acting on real-valued functions of one-variable. Therefore, in the case d = 1, the second statement follows directly from the compactness properties of BV(R)-functions and existence of primitives on open intervals of the real-line (quasiconvexity is equivalent to convexity in this case). We shall focus on the case d ≥ 2. Since A is a full-rank elliptic operator, the algebraic equation belongs to C ∞ per ([0, 1] d ; W ) and satisfies Au = w. Here q denotes the Fourier transform on the d-dimensional torus. This observation and a density argument convey that a function f : X → R is A-gradient quasiconvex if and only if f is constant. This, in turn, conveys that (ii)-(iii) Theorem 1.5 hold trivially for all ν ∈ Y(Ω; X). Now, we show that property (i) is also holds trivially. Since A(ξ) is onto for all non-zero frequencies, then T(ξ) := A(ξ) −1 exists and is homogeneous of degree here we are using that A is a tensor-valued homogeneous polynomial of degree k > k − d). Thus, inverting the Fourier transform, we find that Au = η in the sense of distributions on R d .
Moreover, since k − 1 > k − d, then up to a complex constant, for all multi-indexes α ∈ N d 0 such that |α| = k − 1. The multiplier m(ξ) = ξ α |ξ|A(ξ) is homogeneous of degree zero and smooth on S d−1 . Therefore, by an application of the Mihlin multiplier theorem, we deduce the bound Here, in passing to the last equality we have used that d ≥ 2 so that the Riesz potential norm F −1 (|ξ| −1 F ( q )) L q is an equivalent norm for W −1,q (R d ). Now, let µ ∈ M(Ω; X) be an arbitrary bounded measure and let η ∈ (E ′ ∩ M)(Ω; X) be its trivial extension by zero on R d . Define u := (K A ⋆ η) Ω and observe that the bound above and Lemma 4.2 imply that u ∈ W k−1,q (Ω) for all 1 ≤ q < d/(d − 1). Moreover, by construction, Au = µ in the sense of distributions on Ω.

3.2.
Failure of L 1 -compactness for elliptic systems. In this section we collect some results and examples that showcase the lack of rigidity occurring along sequences of A-free functions due to concentration effects. To account for this, let us recall that sequence of functions {u j } ⊂ L 1 (Ω) is said to converge weakly in L 1 (Ω) if and only if there exists u ∈ L 1 (Ω) such that for all g ∈ L ∞ (Ω).
We write u j ⇀ u in L 1 (Ω). Notice that in general u j L d * ⇀ uL d does not imply weak convergence in L 1 . This owes to concentrations that diffuse into an absolutely continuous part. In the generalized Young measure context, this corresponds to the analysis of ν ∞ x on points x where λ ac is non-zero and The Dunford-Pettis theorem gives the following criterion to rule out the appearance of diffuse concentrations: a sequence {u j } is sequentially weak pre-compact in L 1 (Ω) if and only if {u j } is equi-integrable, i.e., for every ε > 0 there exists some δ > 0 such that for any Borel set The examples given below are intended to exhibit how classical compensated compactness assumptions fail to prevent the lack of equi-integrability of PDE-constrained sequences. We begin with the following general result, which exploits the unconstrained behavior of the singular part of A-free Young measures: Au j = 0 on Ω, and In particular, Then, since Ω is an open set, the tripleν = (ν,λ, p) satisfies the weak- * measurability requirements to be Young measure in Y 0 (B R ; W ). We claim thatν is an A-free measure. According to Theorem 1.1 and Remark 1.1 it suffices to show that (a) id, δ A + L d + id, p λ = A, which follows by the assumption on p; and (b) that h, δ A + h # , p λ ac (x) ≥ h(0) for all A-quasiconvex integrands h : W → R with linear growth at infinity. The latter follows from Remark 1.2 and the fact that that the restriction of h # to W A is convex at zero, i.e., h # , τ ≥ h # (0) = 0 for all probability measures τ ∈ Prob(W A ) satisfying id W , τ = 0 (see [39]). Therefore, h, e. x ∈ Ω. Then, from Corollary 1.1 we deduce the existence of a sequence {µ j } ⊂ M(B R ; W ) of A-free measures that generatesν. By the theory discussed in Section 4.2 and a standard mollification argument we conclude that there exists a sequence δ j → 0 + (where δ j → 0 faster than j → ∞) such that The first two statements then follow by setting w j : Remark 3.1. The previous result also holds if SW z dp(z) ∈ Λ A and λ = cL d for some c ∈ R (see [39]).
A direct consequence of this lemma is the following failure of the L 1 -rigidity for elliptic systems (cf. [50, Section 2.6] and [10]) for A-free measures.
Corollary 3.1. Let L be a non-trivial subspace of W A and assume that L has no non-trivial Λ A -connections, i.e., {|w j |} is not locally equi-integrable on any sub-domain of Ω. Proof.
The assumption on L is equivalent to requiring that L ∩ Λ A = {0}. Since the class of probability measures satisfying p ∈ Prob(S W ), supp(p) ⊂ L ∩ W A and id, p = 0 is non-empty, we may chose at least one p with such properties. The previous lemma implies that the triple ν = (δ 0 , L d Ω, p) is a generalized A-free Young measure, generated by a sequence of A-free measures w j ∈ C ∞ (Ω; W ). The first and the last two statements of the corollary follow from this observation. To prove that dist(w j , L) → 0 in L 1 , let us consider the positively 1-homogeneous Here, we used that supp(p) ⊂ L and f ∞ | L ≡ 0.
The version of this result for constant rank potentials is the following: Then there exists a sequence {u j } ⊂ C ∞ (Ω; V ) such that {|Bu j |} is not locally equi-integrable on any sub-domain of Ω.
The proof of this corollary follows from a version of Lemma 3.2 for B-gradient Young measures, which we shall not prove, but that follows by similar arguments (to ones in the proof of Lemma 3.2) by using Theorem 1.5 instead.
Remark 3.2. In order to add some perspective to these results, let us recall a well-known result of Müller (see [50,Lemma 2.7]) that states the following: if L is a space of matrices containing no rank-one connections and then Du j → Du in measure. This may be understood as an L 1,w -rigidity for gradients. The previous corollary shows that even under L 1 -perturbations of elliptic systems, one cannot hope for sequential weak L 1 -compactness for gradients. (Here, one should not confuse weak L 1 -convergence with convergence in L 1,w .) 3.3. Failure of L 1 -compactness for the n-state problem. In the context of the rigidity properties for gradients,Šverák [65] showed that if K = {A 1 , A 2 , A 3 } is set of matrices such that rank(A i − A j ) = 1, then every sequence {u j } with uniformly bounded Lipschitz constant satisfies the following compensated compactness property: In particular, the restriction on K prevents the formation of any non-trivial microstructures.Šverák's proof also implies that if {Du j } is L 1 -uniformly bounded and dist(Du j , K) → 0 in L 1 (Ω), then, up to taking a subsequence, Notice however that neither of these compensated compactness results allows for concentrations. The first one assumes a uniform Lipschitz bound and the latter (implicitly) assumes equi-integrability since dist(Du j , K) → 0 in L 1 (Ω). For the two-state problem, Garroni & Nesi [31] have shown a similar result for divergence-free fields. More recently, De Philippis, Palmieri and Rindler [21] have extended this to general operators A. The precise statement is the following: then, up to extracting a subsequence, An interesting question to ask is what happens if we allow for concentrations, while still requiring the PDE constraint and requiring that (12) holds. The next two examples show that one cannot expect L 1 -compensated compactness if concentrations are allowed, even if the concentrations occur in the directions of {A 1 , A 2 }.
Av j = 0 in the sese of distributions on Ω.
Moreover, the sequence satisfies However, {|v j |} is not equi-integrable and, for any subsequence Similarly, we also have the following explicit example: Assume that d = 2 and consider the (2 × 2) matrices Then, there exists a uniformly bounded sequence And, for any subsequence {v j h }, Both examples follow directly from the following result (and its corollary below): Then, there exists a sequence of A-free functions An analogous statement holds for B-potentials.
Proof. By assumption we may find constants c 1 , . . . , c n ∈ [0, 1] as in the statement. The assertion then follows from Lemma 3.2.
Remark 3.3. The result of the corollary remains valid even if the vectors A 1 , . . . , A n are mutually Λ A -disconnected, i.e., In particular, {|w j |} is not equi-integrable on Ω.
Notice that, if A / ∈ Λ A , then

3.4.
Flexibility of divergence-free Young measures. So far we have seen how the lack of strong constraints for the concentration part of A-free measures is responsible for the lack of rigidity in a number of interesting scenarios. Now, we review the case of the scalar divergence operator in the A-free framework. As we have already seen, Λ div = R d and hence Since condition (ii) in Theorem 1.1 holds for all convex functions, and (iii) holds trivially in this case, divergence-free generalized Young measures are only constrained by the barycenter property (i). The following example exhibits how (13) yields the existence of rather ill-behaved weak- * convergent sequences of divergence-free fields: be an arbitrary compactly supported function of bounded variation and let p ∈ Prob(S d−1 ) be an arbitrary probability measure.
is the fundamental solution of the scalar divergence operator. Then, for every open and bounded Lipschitz set Proof. Since λ is a compactly supported and Dλ is a Radon measure, Young's inequality implies that w ∈ L 1 loc (R d ; R d ). Moreover, by construction we find that the barycenter of ν is the Radon measure µ = id, δ w L d + id, p λ = w L d + id, p λ and hence div(µ) = div(w) + id, p · Dλ = 0.
Thus, ν satisfies (i)-(iii) and hence it is an A-free measure on Ω. That ν can be generated on Ω by a sequence of smooth divergence-free fields follows from the theory discussed in Section 4.2.

Preliminaries
The d-dimensional torus is denoted by T d , and by Q we denote the closed ddimensional unit cube [−1/2, 1/2] d . We denote by Q r (x) the open cube with radius r > 0 and centered at x ∈ R d . 4.1. Geometric measure theory. Let X be a locally convex space. We denote by C c (X) the space of compactly supported and continuous functions on X, and by C 0 (X) we denote its completion with respect to the q ∞ norm. Here, C c (X) is the inductive limit of Banach spaces C 0 (K m ) where K m ⊂ X are compact and K m ր X. By the Riesz representation theorem, the space M b (X) of bounded signed Radon measures on X is the dual of C 0 (X); a local argument of the same theorem states that the space M(X) of signed Radon measures on X is the dual of C c (X). We denote by M + (X) the subset of non-negative measures. Since C 0 (X) is a Banach space, the Banach-Alaoglu theorem and its characterizations hold. In particular: 1. there exists a complete and separable metric d ⋆ : M(X)×M(X) → R. Moreover, convergence with respect to this metric coincides with the weak- * convergence of Radon measures (see Remark 14.15 in [45]), that is, 2. bounded sets of M(X) are d ⋆ -metrizable in the sense that d ⋆ induces the (relative) weak- * topology on the unit open ball of M(X).
In a similar manner, for a finite dimensional inner-product euclidean space W , M b (X; W ) and M(X; W ) will denote the spaces of W -valued bounded Radon measures and W -valued Radon measures respectively. The space M b (X; W ) is a normed space endowed with the total variation norm The set of all positive Radon measures on X with total variation equal to one is denoted by the set of probability measures on X. Here and all the follows we write to denote the open unit ball and the unit sphere on W respectively. Riesz' representation theorem states that every vector-valued measure µ ∈ M(Ω; W ) can be written as µ = g µ |µ| for some g µ ∈ L ∞ loc (Ω, |µ|; S W ). This decomposition is commonly referred as the polar decomposition of µ. The set is satisfied, is called the set of µ-Lebesgue points. This set conforms a full |µ|measure set of Ω, i.e, |µ|(Ω \ L µ ) = 0. In what follows, we shall always work with good representatives of µ-integrable maps. If g ∈ L 1 loc (Ω, µ; W ), then g satisfies If µ, λ are Radon measures over Ω, and λ ≥ 0, then the Besicovitch differentiation theorem states that there exists a set E ⊂ Ω of zero λ-measure such that where dµ dλ ∈ L 1 loc (Ω, λ; W ) is the Radon-Nykodým derivative of µ with respect to λ. Another resourceful representation of a measure is given by the Radon-Nykodým-Lebesgue decomposition which we shall frequently denote as where as usual µ ac : whenever the integrals above exist or if g is integrable.
is called a blow-up sequence. We write Tan(µ, x 0 ) to denote the set of all such tangent measures.
Using the canonical zero extension that maps the space M(Ω; W ) into the space M(R d ; W ) we may use most of the results contained in the general theory for tangent measures when dealing with tangent measures defined on smaller domains. The following theorem, due to Preiss, states that one may always find tangent measures.
Theorem 4.1 (Theorem 2.5 in [54]). If µ is a Radon measure over R d , then This property of Radon measures measures will play a silent, but fundamental role, in our results. We shall use it to "amend" the current lack of a Poincaré inequality for general domains; this, because (14) acts as an artificial extension operator for tangent measures restricted to the unit cube Q ⊂ R d . Returning to the properties of tangent measures, one can show (see Remark 14.4 in [45]) that, for a tangent measure τ ∈ Tan(µ, x 0 ), it is always possible to choose the scaling constants c m > 0 in the blow-up sequence to be for any open and bounded set U ⊂ R d containing the origin and with the property that σ(U ) > 0, for some positive constant c = c(U ); this process may involve passing to a subsequence. Then, from [54, Thm 2.6(1)] it follows that at µ-almost every x ∈ Ω we can find τ ∈ Tan(µ, x) as the weak- * limit a blow-up sequence of the form Yet another special property of tangent measures is that at, |µ|-almost every x ∈ R d , it holds that τ ∈ Tan(µ, x) if and only if |τ | ∈ Tan(|µ|, x), which in particular conveys that tangent measures are generated by strictly-converging blow-up sequences. If µ, λ ∈ M + (R d ) are two Radon measures with µ ≪ λ, i.e., that µ is absolutely continuous with respect to λ, then (see Lemma 14.6 of [45]) Then, a consequence of (15) and Lebesgue's differentiation theorem is that In fact, if f ∈ L 1 (Ω, W ), then it is an simple consequence from the Lebesgue Differentiation Theorem that

4.2.
Integrands and Young measures. Bounded generalized Young measures conform a set of dual objects to the integrands in E(U ; W ). We recall briefly some aspects of this theory, which was introduced by DiPerna and Majda in [25] and later extended in [4,42].
Notation. We remind the reader that U ⊂ R d denotes an open set and Ω ⊂ R d denotes an open an bounded set with L d (∂Ω) = 0.
Heuristically, E(W ) is isomorphic to the continuous functions on the compactification of W that adheres to it each direction at infinity. In particular, all f ∈ E(U ; W ) have linear growth at infinity, i.e., there exists a positive constant M such that |f (x, z)| ≤ M (1 + |z|) for all x ∈ U and all z ∈ W . With the norm the space E(Ω; W ) turns out to be a Banach space and S is an isometry with inverse exists and defines a positively 1-homogeneous function called the strong recession function of f . Moreover every f ∈ E(U ; W ) satisfies for all x ∈ U and z ∈ W .
In particular, there exists a modulus of continuity ω : [0, ∞) → [0, ∞), depending solely on the uniform continuity of Sf , such that for all x, y ∈ Ω and z ∈ W .
For an integrand f ∈ E(U ; W ) and a Young measure ν ∈ Y(U ; W ), we define a duality paring between f and ν by setting The barycenter of a Young measure ν ∈ Y loc (U ; R N ) is defined as the measure The generation of a Young measure is (cf. Definition 1.2) a local property in the sense that for all open Lipschitz sets U ′ ⋐ U with λ(∂U ′ ) = 0. In many cases it will be sufficient to work with functions f ∈ E(U ; W ) that are Lipschitz continuous and compactly supported on the x-variable. The following density lemma can be found in [42,Lemma 3].
Fundamental for all Young measure theory is the following compactness result, see [42, Section 3.1] for a proof.
Then, there exists a subsequence (not relabeled) and The Radon-Nykodým-Lebesgue decomposition induces a natural embedding via the identification µ → δ µ := (δ µ ac , |µ s |, δ gµ ). Notice that a sequence of measures {µ j } ⊂ M(U ; W ) generates the Young measure ν in U if and only if For a sequence {µ j } ⊂ M(U ; W ) that area-strictly converges to some limit µ ∈ M(U ; W ), it is relatively easy to characterize the (unique) Young measure it generates. Indeed, an immediate consequence of the Separation Lemma 4.1 and a version of Reshetnyak's continuity theorem (see [42,Theorem 5]) is that Since tangent Young measures are only locally bounded, it will also be convenient to introduce a concept of locally bounded A-free Young measure: The proof of the following result follows the same principles used in the proof of [6, Lem. 2.15] with A ≡ 0.
The following notion of translation or shift of a Young measure will be used to deal with the fact that W might be in fact larger than W A in the proof of Theorem 1.1.

Tangent Young measures.
Similarly to the case of measures, we can define the push-forward of Young measures.
Suppose that x ∈ Ω. A non-zero Young measure σ ∈ Y loc (R d ; W ) is said to be a tangent Young measure of ν at x if there exist sequences r m ց 0 and c m > 0 such that The set of tangent Young measures of ν at x ∈ Ω will be denoted as Tan(ν, x). Since Young measures can be seen, via disintegration, as Radon measures over U × W , the property of tangent measures contained in Theorem 4.1 lifts to a similar principle for tangent Young measures: Young measures also enjoy a Lebesgue-point property in the sense that a tangent Young measure σ ∈ Tan(ν, x) truly represents the values of ν around x. More precisely, we have the following localization principle for (L d + λ s )-almost every x 0 ∈ Ω: every tangent measure σ ∈ Tan(ν, x 0 ) is a homogeneous Young measure of the form We state two general localization principles for Young measures, one at regular points and another one at singular points. These are well-established results, for a proof we refer the reader to [58,59]; see also the Appendix in [6].
Then there exists a set S ⊂ U with λ s (U \ S) = 0 such that for all x 0 ∈ S there exists a singular tangent Young measure σ = (σ, γ, σ ∞ ) ∈ Tan(ν, x 0 ), that is, e. This properties tell us that certain aspects of the weak- * measurable maps ν and ν ∞ belonging to ν can be effectively studied by looking at tangent measures of ν itself. In a similar fashion to (14), at every x 0 where Proposition 4.3 holds, we may find a tangent Young measure σ ∈ Tan(ν, x 0 ) as in (21) with (22) τ (∂Q) = 0, and σ is generated by a blow-up sequence as in (20) where in any case c m can be taken to be (⟪| q |, ν Q r (x)⟫) −1 . At singular points we may assume without loss of generality that In all that follows we shall write C ∞ ♯ (T d ; W ) to denote the subspace of smooth, W -valued periodic functions with mean-value zero. We recall, from the theory discussed in [6, Section 2.5], that maps . This set contention will be crucial for the proof of Theorem 1.1.
Below we recall some well-known convexity and Lipschitz properties of A-quasiconvex functions.
Let D be a balanced cone of directions in W , that is, we assume that tA ∈ D for all A ∈ D and every t ∈ R. A real-valued function h : W → R is said to be D-convex provided its restrictions to all line segments in W with directions in D are convex. We recall the following Λ A -convexity property of A-quasiconvex functions contained in lemma from [28,Proposition 3.4] for first-order operators and in [6,Lemma 2.19] for the general case: In Section 9, we will require to work around the fact that W A may not necessarily be equal to W . The following definition and propositions will play an important role in this regard.
for some constant C = C(M, A).
Proof. Property (a) is a direct consequence of (25) and the definition of ( q )˜. Property (b) follows directly from (a). Property (c) follows from Lemma 4.3, property (b) and the fact thatf is invariant on W A -directions. Finally, we prove (d). Up to a linear isomorphism we may assume that D = Λ A ∪ W ⊥ A contains an orthonormal basis {w 1 , . . . , w s } basis of W . Of course, the change of variables carries a constant in the desired Lipschitz bound, but that constant depends solely on A. The difference between two points z 1 , z 2 ∈ W can be written as where the constant of the last estimate depends solely on s = s (A). Property (c) implies thatf is D-convex. Since moreover D is a spanning set of directions of W , then [39,Lemma 2.5] implies that for all x ∈ D and y ∈ W .
An iteration of this identity yields the upper bound Since Reversing the roles of z 1 and z 2 gives the desired Lipschitz bounds.
Corollary 4.1. Let α ≥ 0 and let p 1 ∈ Prob(W ), p 2 ∈ Prob(S W ) be two probability measures satisfying Proof. If W A = W , then we have nothing to show. Else, let X := W ⊥ A and let g ∈ E(X) be an arbitrary integrand. We also define h := 1 WA ⊗ g ∈ E(W ). It follows from Proposition 4.5(a) that Q A h = 1 WA ⊗ g. If we choose g ∈ C c (X), then h ∞ ≡ 0 and the assumption on p 1 , p 2 yields where (id W −p)[p 1 ] is the push-forward of p 1 with respect to (id W −p). Since C c (X) separates Prob(X), this implies that (id W −p)[p 1 ] = δ (idW −p)P0 . In particular This proves that supp(δ −(idW −p)P0 ⋆ p 1 ) ⊂ W A , and therefore also A similar argument and the previous identity further imply α g, (id W −p)[p 2 ] = 0 for all positively 1-homogeneous g : X → R.
The next two propositions will be used to address some technical details involving the proof of Theorem 1.1 and Remark A.1: Proposition 4.6. Let f ∈ E(Ω; W ) and assume that there exists a dense set D ⊂ Ω such that Q Af (x, 0) > −∞ for all x ∈ D. Then |Q Af (x, z)| ≤ C(1 + |z|) for all (x, z) ∈ Ω × W for some constant C depending on A and g E(Ω;WA) .
Proof. Since f ∈ E(Ω; W ), there exists a constant M = f E(Ω;W ) such that |f (z)| ≤ M (1 + |z|). It follows from the definition of A-quasiconvexity (testing with the field w = 0) that It follows from Proposition 4.5(c) and a suitable version of [40, Lemma 2.5] that, if x ∈ D, then where C = C(M, A). The cited result and its proof are originally stated for quasiconvex functions. However, a similar argument can be given for D-convex functions where D is a spanning balanced cone. This shows that the restriction of Q Af on (D × W ) has linear growth at infinity. We shall prove now that Q Af is finite for all x ∈ Ω. Let us assume that there exists x ∈ Ω \ D with Q Af (x, 0) = −∞. Let us fix L > 0 be a large real number. By our assumption on x, we may find a smooth The density of D allows us to find a sequence {x h } ⊂ D satisfying x h → x. We use once again the fact that f ∈ E(Ω; W ) to deduce that f is uniformly continuous on Ω × RB W where R > w ∞ . Hence, we may use a standard modulus of continuity argument to conclude that Letting L > 2C we conclude that |Q Af (x h , 0)| > C for h sufficiently large. This poses a contradiction to the bound (26). Repeating the first step with D = Ω, we find that This finishes the proof.
Proposition 4.7. Let f : W → R be locally bounded and assume that Q Af is locally finite. Fix ε ∈ (0, 1), δ > 0 and let w ∈ C ∞ ♯ (T d ; W ) be an A-free field satisfying wheref ε :=f + ε| q |. Then, for some constant C > 0 depending on A and the linear growth constant of f .
Proof. Since Q Af is finite, then it has linear growth with a constant C that depends solely on A and the linear growth constant of f (cf. Lemma 4.6). The same holds forf ε since Q Af ε ≥ Q Af up to taking C + 1 instead. Using the assumption we get The conclusion follows directly from this estimate.
Fix ε ∈ (0, 1). Then, there exists a modulus of continuity ω : for all x, y ∈ Ω and A ∈ W . The modulus of continuity depends on ε, A, the linear growth constant of f and the modulus of continuity of f .
Proof. We begin with two observations. First, that Q Af (x, q ) is finite implies that it has linear growth and that is globally Lipschitz with constants that depend solely on A and the linear growth constant of f (cf. Propositions 4.5 and 4.6). Now, let δ > 0 and let The previous proposition yields that ξ +w L 1 (Q) ≤ ε −1 C(1+|ξ|+δ). By definition, we get The desired bound follows by letting δ → 0 + and exchanging the roles of x, y.

Sobolev spaces.
In order to continue our discussion, we need to recall some facts of the theory of general Sobolev spaces. The following definitions and background results about function spaces and the Fourier transform can be found in the monographs of Adams [2, Section 1] and Stein [63, Section VI.5], as well as the full compendium of definitions and results contained in the book of Triebel [69].
Recall that T d ∼ = R d /Z d denotes the d-dimensional flat torus. Let ℓ ∈ N 0 and let 1 < p < ∞. The Sobolev space W ℓ,p (T d ) is the collection of Z d -periodic functions f all of whose distributional derivatives ∂ α f with 0 ≤ |α| ≤ ℓ belong to L p loc (R d ). The norm of W ℓ,p (T d ) is Since the torus is a compact manifold, we also have W ℓ,p (T d ) = W ℓ,p 0 (T d ). Calderón showed (see [2, Thm 1.2.3]) the equivalence W ℓ,p (T d ) = L ℓ,p (T d ) between the classical Sobolev spaces and the Bessel potential spaces, which are defined as where F and ( q ) denote the Fourier transform on periodic maps (see the next section). We shall henceforth make indistinguishable use of q W −ℓ,p and q L −ℓ,p as norms of W −ℓ,p . A standard Hahn-Banach argument (see for instance [17,Prop. 9.20] for the case ℓ = 1) shows that u ∈ W −ℓ,p (T d ) if and only if there exists a family {f α } 0≤|α|≤k ⊂ L p ′ (T d ) such that Here p ′ = p/(p − 1). If {ρ ε } ε>0 is a family of standard mollifiers at scale ε > 0, then the representation above implies that Remark 4.4. This shows that under standard mollification.
Crucial to our theory are the following direct consequences of Morrey's embedding theorem (see Corollary 9.14 in Sec. 9.3 and Remark 20 in Section 9.4 of [17]): Proof. Notice that q ′ > d. Then, Morrey's embedding and Ascoli-Arzelà's theorem convey the compact embedding W 1,q ′ 0 (U ) c ֒→ C 0 (U ). Since these are Banach spaces, the assertion follows directly from [17,Theorem 6.4].
Remark 4.5. Notice that it is not necessary to require that U is Lipschitz or a domain with any type of regularity.

Analysis of constant rank operators
Let us recall that our main assumption is that A is a linear operator of integer order k, from W to X, that satisfies the constant rank condition (27) ∀ξ ∈ R d − {0}, rank A(ξ) = const.
In this section we shall assume that A is a non-trivial operator, i.e., k ≥ 1 for otherwise all the results are trivially satisfied. The aim of this section is to give a simple extension of the well-known L p -multiplier projections for constant rank operators established by Fonseca & Müller in [28]. The Fourier transform acts on periodic measures by the formula Smooth periodic functions are represented by F −1 through the trigonometric sum The choice to primarily work with Fourier series lies in the following characterization for constant rank operators due to Raita [57, Thm. 1] and its direct implication on periodic maps (see Lemma 5.1 below): Let A be an operator from W to X as in (7). Then A satisfies the constant rank condition if and only if then there exists a constant rank operator B from V to W such that For the reminder of this section A and B will be assumed to satisfy the exactness relation (28), we call B an associated potential to A (we call A an associated annihilator of B). 4 Raita showed that every A-free periodic field is the B-gradient of a suitable potential. The following is a version for measures of the original statement [57, Lemma 5]:   (8) is verified, then the map ξ → π(ξ) is an analytic map on R d − {0}, homogeneous of degree 0. The Mihlin multiplier theorem implies that π defines an (L p , L p ) multiplier on R d for all 1 < p < ∞, and standard multiplier transference methods imply that if we setπ(ξ) = π(ξ), for ξ = 0, and π(0) = 0, then {π(ξ)} ξ∈Z d defines an (L p , L p )-multiplier on T d via the assignment (see Theorem 3.8, Corollary 3.16 and its remark below in [64] for further details):

By construction
5.1.1. Sobolev estimates. It is well-known that (8) implies the map ξ → A(ξ) † belongs to C ∞ (R d \ {0}; Lin(X * , W )) and is homogeneous of degree −k. Here, M † denotes the Moore-Penrose inverse of M , which satisfies the fundamental algebraic identity M † M = Proj (ker M) ⊥ . Partying from this identity and using that Au(0) = A(ξ) u(0) = 0 for all u ∈ C ∞ (T d ; W ), one finds that The advantage of this perspective, is that it allows one to define u A in terms of Au rather than u itself. 5 Recalling the seminal ideas of Fonseca and Müller [28], we can exploit the representation in (31) to deduce Sobolev estimates on u A directly from the regularity of Au, as one would do for elliptic operators (see also the exposition in [56]). In order to proceed with this task let us define the auxiliary spaces where ℓ ∈ [0, k] is a positive integer and 1 < p < ∞. These are Banach spaces of distributions when endowed with the natural norm u W −1,p + Au W −k+ℓ,p . Next, we show that the A-representative operator can be extended to an operator with Sobolev-type properties on W ℓ,p (T d ): 4 The class of operators B satisfying (28) may have more than element. 5 The fact that u A can be expressed as Lemma 5.2. Let 1 < p < ∞ and let ℓ ∈ [0, k] be a positive integer. There exists a continuous linear map T : W ℓ,p (T d ) → W ℓ,p (T d ) with the following properties:

A(T [u]) =
in the sense that T = T (ℓ ′ , p ′ ) is an extension of T = T (ℓ, p).
Proof. Let us define T on smooth maps as: so that property (1) follows from (31). In light of Remark 4.4, in order to prove that T extends to W ℓ,p (T d ) with linear bound as in (2), it suffices to prove (2) for u ∈ C ∞ (T d ; W ). Notice that once (2) has been established, it also suffices to verify that (3)-(4) hold for smooth maps; properties (3)-(4) for smooth maps follow from (29)- (30). We shall therefore focus in proving (2) for smooth maps. Fix an integer ℓ ∈ [0, k] and consider the multiplier where α ∈ N d 0 be a multi-index with |α| = ℓ. Consider the family {m(ξ)} ξ∈Z d defined by the rulem(ξ) = m(ξ) for all ξ ∈ Z d − {0} andm(0) = 0. Partial differentiation and the properties of the Fourier transform yield for all u ∈ C ∞ (T d ; W ). Hence, inverting the Fourier transform at both sides of the equation gives We readily verify that m is homogeneous of degree zero, analytic on R d \ {0}. Then, in light of the transference of multipliers discussed above, the Mihlin multiplier theorem implies that the assignment f → F −1 (m f ) extends to an (L p , L p )-multiplier on T d . In particular, Here, in passing to the last equality we have used that the Mihlin multiplier theorem implies that the norms Running through all multiindexes |α| = ℓ and using Poincaré's inequality for periodic mean-zero functions yields the sought assertion. Definition 5.1. Let ℓ ∈ [0, k] be an integer and let 1 < q < d/(d − 1). If µ ∈ M(T d ; W ) is a measure with Aµ ∈ W −k+ℓ,q (T d ), then by Corollary 4.2 we may define the A-representative of µ as where T is the linear map from Lemma 5.2.
Notice that µ A is well-defined regardless of the choice of q, it has mean-value zero, it satisfies Aµ A = Aµ in D ′ (T; X) and for some constant C = C(q, ℓ, A).

Localization estimates.
We close this section with a useful observation for estimates concerning the localization with cut-off functions. Let ϕ ∈ C ∞ c (R d ). The commutator of A on ϕ is the linear partial differential operator where ϕ acts as a multiplication operator. It acts on distributions η ∈ D ′ (R d ; W ) as [A, ϕ](η) = A(ϕη) − ϕAη. Due to the Leibniz differentiation rule, [A, ϕ] is a partial differential operator of order (k −1) from W to X, with smooth coefficients depending solely on the coefficients of the principal symbol A and the first k derivatives of ϕ. In particular, if µ ∈ M(R d ; W ) satisfies Aµ ∈ W −k,p loc (R d ), then by Corollary 4.2 we get 6. Proof of the approximation theorems 6.1. Proof of Theorem 1.3. Let us recall that we are given an A-free measure µ ∈ M(Ω; W ), and we aim to find a sequence of A-free measures that area-strict converges to µ. We may, without loss of generality assume that Ω ⊂ Q.
We give a variant of the construction given in Step 2 of [6, Sec. 5.1]: Let {ϕ i } i∈N ⊂ C ∞ c (Ω) be a locally finite partition of unity of Ω. For a measure or function σ on Ω, we set where ρ ε = ε −d ρ( q /ε) is a standard radial mollifier at scale ε > 0. Let us begin with a few observations: (a) Every µ i,ε is a compactly supported on supp(ϕ i ) ⋐ Ω ⊂ Q. Therefore we may naturally consider each µ i,ε as an element of C ∞ (T d ; W ). (b) The A-free constraint on µ implies that Aµ i,ε = [A, ϕ i ](µ ε ) for all i ∈ N 0 , where we recall that [A, ϕ i ] is a linear operator of order (k − 1). (c) As {ϕ i } i∈N is a locally finite partition, we can take linear operators inside and outside arbitrary sums subjected to it. In particular, By standard measure theoretic arguments we get that where in establishing (39) we have used that (36) Then, In light of (36)-(39), for every i ∈ N we may choose 0 < ε j (i) < dist(supp ϕ i , ∂Ω) such that (writingσ i,j := σ i,εj (i) ) Now, for a measure or function σ on Ω let us definẽ By construction we have which shows the first condition, that indeedμ j * ⇀ µ on Ω. In particular, the sequential lower semicontinuity of the area functional gives the lower bound Area(µ, Ω) ≤ lim inf j→∞ Area(μ j , Ω).
Next, we prove the upper bound: We use that 1 + |w| + |z| ≥ 1 + |w + z| for all w, z ∈ W to deduce that (recall that Ω ⊂ Q) where the last inequality follows from the intermediate step Here, the passing to the second inequality follows from Jensen's inequality and the radial symmetry of ρ ε . Hence, we conclude that lim sup j→∞ Area(μ j , Ω) ≤ Area(µ, Ω).
This, together with the lower bound and the convergenceμ j * ⇀ µ imply thatμ j converges area-strictly to µ on Ω. Lastly, we use the triangle inequality and the embedding W k,q ′ 0 (Ω) ֒→ W k,q ′ (T d ) to find that which shows that indeed Aμ j → Aµ strongly in W −k,q (Ω).
Step 2. Construction of the A-free sequence. Let us fix i ∈ N. We write v i,j := (μ i,j ) A and v i := (µ i ) A . Then, in light of the estimates from Lemma 5.2 and the triangle inequality we get This proves that v i,j → v i in L q (Ω). Now, let us look at the translations (μ i,j − v i,j )L d ∈ M(T d ; W ). These are A-free by construction, which lead us to define the following candidate for an A-free recovery sequence: A(ϕ i µ) = Aµ = 0 in the sense of distributions on Ω.
Claim 2. The sequence {u j L d } area-strict converges to µ. Since {μ j L d } already area-converges to µ, it suffices to show thatμ j and u j are asymptotically L 1 -close to each other (this is sufficient to ensure the asymptotic closeness of the area functional, which has a uniformly Lipschitz integrand). This is easily verified since This proves the second claim, which finishes the proof.
6.2. Proof of Theorem 1.6. Let us recall that we are given u ∈ M(Ω; V ) with Bu ∈ M(Ω; W ). We want to show there exists a uniformly bounded sequence Bu j L d area-strictly converges to Bu on Ω.
Proof. First we need to establish Sobolev estimates for the B-representatives of localizations of an arbitrary potential w ∈ M(Ω; V ).
In all that follows we may assume that Ω ⋐ T d . As in previous arguments, we shall indistinguishably identify compactly supported functions on Ω with their periodic extensions on T d . Let Ω ′ ⊂ Ω k B −1 ⋐ · · · ⋐ Ω 1 ⋐ Ω 0 be a nested family of equidistant Lipschitz open sets. By standard methods, we may find cut-off functions ψ 1 , . . . , ψ k ∈ C ∞ c (Ω; [0, 1]) satisfying 1 Ωr+1 ≤ ψ r ≤ 1 Ωr , Let w ∈ M(Ω; V ) be such that Bw ∈ M(Ω; W ). The main advantage of the potential framework is that we may localize inside the PDE: we define a sequence of smooth functions by setting σ r := ψ r w. Computing the B-gradient we find that, for 1 ≤ r ≤ k B − 1 it holds The idea now is to deduce Sobolev estimates for their B-representatives following a standard bootstrapping argument: Since B satisfies the constant rank condition, we may define w r := (σ r ) B . Then, by Lemma 5.2, we deduce the a priori estimates where the constant C may change from line to line and depends solely on dist(Ω ′ , Ω), B and q. Iteration of this bounds until the (k B − 1) th step yields for yet another constant C = C(q, B, dist(Ω ′ , ∂Ω)). With these estimates in hand, the approximation argument follows almost the same lines as the one used in the proof of Theorem 1.3. Therefore, we shall only give a sketch of the proof: Let {ϕ i } ⊂ C ∞ c (Ω) be a locally finite partition of unity, and assume that ϕ i = (ψ i ) k B −1 so that we may apply the previous estimates on localizations of the form ϕ i w. 1. Set w i,j := (ϕ i (u ⋆ ρ εj )) B and w i := (ϕ i u) B . Then, first applying the bootstrapping argument above with w = u, and subsequently with w = u⋆ρ εj −u, conveys the estimate Here C = C(q, ϕ i , B). 2. Similarly to the previous proof, we define where, for each i ∈ N, ε j(i) > 0 is chosen so that This proves the convergence Bw j * 4. We decompose Bu j = I ac j + I s j + II j , where Young's inequality implies |I s j |(Ω) ≤ |B s u|(Ω), and from the estimates of Step 2 we deduce that Thus, the inequality 1 + |z + z ′ | ≤ 1 + |z| + |z ′ | implies the upper bound Area(Bu j , Ω) = Area(I ac j + I s j + II j , Ω) q Area(B ac u, Ω) + |I s j |(Ω) This proves that lim sup j→∞ Area(Bu j , Ω) ≤ Area(Bu, Ω). We thus conclude that lim j→∞ Area(Bu j , Ω) = Area(Bu, Ω) as desired. This finishes the proof.

Helmholtz decomposition of generating sequences
In this section A and B are constant rank operators from W to X and V to W , of respective orders k and k B . When we work in the A-free context, B will denote an associated potential of A, which was discussed in the previous section (cf. (28)) and for which Lemma 5.1 holds. In all that follows U ⊂ R d is an open set and we assume that The following lemma establishes that oscillations and concentrations generated along A-free sequences are, in fact, only carried by B-gradients: can be decomposed into the B-gradient of a potential u ∈ W k B −1,q (Ω ′ ) an A-free field v ∈ L q (Ω ′ ), i.e., Lastly, if there exists u 0 ∈ W k B −1,q (Ω ′ ) such that [ν] Ω ′ = Bu 0 , then v may be chosen to be the zero function.
Proof. Let ϕ ∈ C ∞ c (Ω; [0, 1]) be a cut-off function satisfying Ω ′ ⊂ {ϕ ≡ 1}. Without loss of generality we may assume that supp(ϕ) ⊂ Q. Let {µ j } be a sequence of measures generating ν and satisfying Aµ j → 0 in W −k,q (supp ϕ). We define a sequence of compactly supported measures on U by setting σ j := ϕµ j and σ 0 := ϕµ. Using the trivial extension by zero, we may regard each measure σ j as an element of M(T d ; W ). For a function τ on the torus we write τ := T d τ (if τ is a measure, we set τ := τ (T d )) to denote its mean-value. Next, define the sequence {w j } ⊂ L q (T d ; W ) of mean-value zero maps as

Indeed, thanks to Lemma 5.2 and Corollary 4.2 we obtain
Since µ j * ⇀ µ in M(U ; W ), it holds Indeed, Aµ j → 0 in W −k,q (supp ϕ), while the convergence involving the commutator follows from the fact that (cf. Corollary 4.2) µ j → µ W −1,q (U ) and that [A, ϕ] is an operator of order at most (k − 1). Hence, it follows from the estimates in Lemma 5.2 that This allows us to define an asymptotically L q -close sequence to σ j by setting Next, we exploit the potential property of mean-value zero A-free functions on the torus. Proposition 5.1 yields potentialsũ j ⊂ L q (T d ; V ) satisfying Bũ j = z j for all j ∈ N 0 , each of which we may assume to be given by its own B-representative, i.e.,ũ j = (ũ j ) B . Applying once more the estimates of Lemma 5.2 (for B instead of A), we find that Since supp(f 0 ) ⊂ Ω \ Ω ′ , then w 0 is an A-free measure on Ω ′ . We readily check, settingŨ This proves the first assertion on the decomposition of the barycenter µ on Ω ′ . Moreover, since λ(∂Ω ′ ) = 0, we obtain Therefore, using that w j → w 0 strongly in L q (T d ), it follows from Proposition 4.1 that We are left to see that we can adjust the boundary of {Ũ j } j∈N to match the values of u :=Ũ 0 near ∂Ω ′ . For a positive real t > 0 we define to be a cut-off of Ω ′ 2t with ϕ ≡ 0 on Ω ′ t , and such that ϕ t k B ,∞ t −k B . Let δ h ց 0 be an infinitesimal sequence of positive reals. We define a sequence with u-boundary values by setting (43) u h,j : In particular, setting u h := u h,j(h) , we can estimate the total variation of Bu h as Notice that this not only implies that {Bu h } is uniformly bounded, but also that the sequence does not concentrate mass on the boundary ∂Ω ′ . Therefore, up to extracting a subsequence (which we will not relabel), the sequence generates a Young measure on Ω ′ which does not carry mass into the boundary, i.e., On the other hand, our construction gives the equivalence of measures Bu h = BŨ j(h) when these are restricted to the set Ω ′ 2δ h . Since δ h ց 0, we deduce from (42)-(43) that σ ≡ ν on Ω ′ , and therefore with u h ≡ u on a neighborhood of ∂Ω ′ . The last statement follows by noticing that v = B(u 0 − u) and hence we may simply re-define the sequence of potentials as U h := u h + u 0 − u. This finishes the proof of the lemma.
The proof of following lemma follows by verbatim from the first step of the proof of the lemma above: Then, for a bounded open subset Ω ′ ⊂ U , there exist u ∈ W k B −1,q (Ω ′ ) and v ∈ L q (Ω ′ ) such that Moreover, there exist sequences {u j } ⊂ W k B −1,q (Ω ′ ) and {v j } ⊂ L q (Ω ′ ) such that The following two results show that tangent A-free Young measures and Bgradient Young measures differ only by a constant shift: be an A-free measure and let σ ∈ Tan(ν, x) be a tangent Young measure. Then, for every Lipschitz domain ω ⋐ R d with λ(∂ω) = 0, there exist a potential u ∈ W k B −1,q (ω) and a vector z ∈ W such that Bu + z L d = [σ] as measures on ω.

Moreover, there exists a sequence {u
u h ≡ u on a neighborhood of ∂ω ′ , Furthermore, if x ∈ Ω is a singular point of ν or if there exists u 0 ∈ M(Ω; V ) such that [ν] = Bu 0 , then z = 0 ∈ W .
Proof. The locality property (19) of Young measures and the local decomposition of generating sequences given in Lemma 7.1 imply that it is enough to show the assertion when We consider two cases: when x ∈ Ω is a regular or a singular point of λ. Regular points: Every tangent Young measure σ ∈ Tan(ν, x) is generated by a sequence of the form Recall from (17) that Hence, from the linearity of the push-forward and the compactness of Young measures, it follows that (here we use that λ(∂ω) = 0).
The assertion follows by taking z = v(x). If [ν] = Bu 0 , then a localization argument and (25) imply that In particular, there exist ξ i , . . . , ξ r ∈ S d−1 and a 1 , . . . , a r ∈ V such that This allows us to construct a smooth primitive of z as follows: Let η(t) = t k B /k B ! and define u z (x) := a 1 η(x · ξ 1 ) + · · · + a r η(x · ξ r ) ∈ C ∞ (R d ; V ). By construction satisfies we have which implies that Bu z = z. Therefore, by Lemma 7.1, z can be taken to be the zero constant.
Singular points: This proof is easier since instead of (44)- (45) we have Recall however from (24) that Therefore, using the same arguments as before (with different normalization constants) yields z = 0. This completes the proof. It follows from the main assumption, that there exists a full Λ-measure set B ⊂ Ω with the following property: at every x ∈ B there exists a tangent Young measure σ = (ν x , κ, ν ∞ x ) ∈ Tan(ν, x) satisfying (22) and (without carrying the x-dependence on several of the following elements) In what follows we shall simply write c rj = c rj (x) when no possible confusion arises. Particular consequences of the convergence above are the following: at every x ∈ B we can find a blow-up sequence (50) c and (composing with the identity map id W ) also (51) Applying Lemma 7.2 on the sequence γ j and the sets U = R d and Ω ′ = Q, which yields (cf. Corollary 7.2) Step 1. Construction of a disjoint cover of B. Fix m ∈ N and let ϕ ∈ C(Q). At every x ∈ Ω we define ρ m (x) as the supremum over all radii 0 < r j (x) ≤ 1 m (where r j (x) ց 0 is the sequence from the previous step at a given x ∈ B) such that Next, define the cover (of open cubes) with centers in B given by Notice that, since ρ m (x) > 0 exists for all x ∈ B, then Q m is a fine cover of B and hence we may apply Besicovitch's Covering Theorem, with the measure Λ, to find a disjoint sub-cover O m = {Q x,m }, where each Q x,m is of the form Q Rm (x) for some Step 2. An adjusted generating sequence of σ. Let x ∈ B be fixed and let σ = σ(x) be the A-free tangent Young measure from the beginning of the proof. Now, we apply Corollary 7.1 to find a sequence {w h } ⊂ W k B −1,q (Q) satisfying Since it will be of use later, let H(m) ∈ N be sufficiently large so that We also consider η m , ϕ m ∈ C ∞ (Q; [0, 1]) be two cut-off functions (with disjoint support) that satisfy the following properties: Step 3. Boundary adjustment for generating sequences of σ(x). The next step is to define an A-free sequence generating σ = σ(x) on Q, which also has a blow-up of µ as boundary values. This should allow us to freely glue each of this approximations together while keeping the A-free constraint Fix m ∈ N and let Q x,m ∈ O m . We begin by constructing a sequence on Q, which we shall later translate to Q x,m ∈ O m . Bearing in mind all the x-dependencies that we have omitted in the previous steps, define the A-free sequence Here, let us recall that the commutator [B, χ] := B(χ q )−χB is a differential operator of order at most k B − 1 (with coefficients involving the coefficients of B and the derivatives of χ of order less or equal than k B ). By this token, if h ≥ H(m), we may estimate the total variation of q h,m as whence it is established that (q h,m ) h≥h(m) is uniformly bounded in M(Q; W ). In fact, we get that lim sup m→∞ |q m |(Q \ Q 1− 1 m ) = 0; this follows from the property κ(∂Q) = 0. Therefore, passing to further subsequence of the h's if necessary (not relabeled), we may assume that On the other hand, observe that q h,m (Q 1− 4 m ) =γ h(m) + (v Rm − z) and hence, by Lemma 4.1 and the locality of Young measures, it must holdσ ≡ σ in Y(Q 1− 4 m ; W ). Since this holds for all h ∈ N and neitherσ or σ charge the boundary ∂Q, it follows that In particular, the uniform bound above and (50) ensure that we may find another subsequence h(m) ≥ H(m) satisfying Step 4. Gluing together and generating ν. So far, we have constructed generating sequences for specific tangent Young measures of ν on every x where there is a cube Q x,m ⊂ O m . The rest of the proof can be summarized in the following two steps: First, we construct an A-free sequences by gluing together the Q → Q x,m pushforwards of each q h(m),m . Second, we show the new global sequence is uniformly bounded.
Step 5a. Gluing the generating sequences. For x ∈ R d and r > 0 we define the map G x,r (y) = (T x,r ) −1 = x + ry, which is defined for all y ∈ R d . Fix a cube Q x,m in O m and define an A-free measure there by setting Notice that where Q ′ x,m is the concentric sub-cube of Q x,m given by z + (1 − 1 m )(Q x,m − x). Therefore, the measure defined as is well-defined in Ω. Moreover, it is also A-free on Ω (cf. (63)) and its total variation in Ω can be controlled as follows (recall that the push-forward is mass preserving) Step 4b. The new A-free sequence generates ν. This last step consists of checking that ν is indeed an A-free Young measure in Ω. In light of the previous steps, it suffices to check that τ m generates ν in Ω. First, we estimate how close U m is from generating ν on Q x,m . Fix ϕ ∈ C(Ω). Every cube Q x,m ∈ O m has diameter at most √ dm −1 and therefore there exists a modulus of continuity (depending solely on ϕ) such that ϕ(x) − ϕ ∞ (Q x,m ) ≤ ω(m −1 ) for all Q x,m ∈ O m ; the same bound holds for any dilation of ϕ on the corresponding dilation of Q x,m .
Let p ∈ N and let M p to be the linear growth constant of h p . We define Let m ≥ p. Regarding U m as an element of M(Ω; W ) through the trivial extension by zero, we obtain the estimate Therefore, adding up these estimates for each cube Q x,m on Q m yields ≤ δ(m).
Conclusion. Since the family {ψ p ⊗ h p } p∈N separates E(Ω; W ), we conclude that the (uniformly bounded) sequence of A-free measures {τ m } generates ν, i.e., This finishes the proof.

8.2.
Proof of Corollary 1.1. If ν ∈ Y A (Ω) is such that λ(∂Ω), then from Theorem 1.2 we may assume that there exist tangent A-free measures of ν at (L d + λ)a.e. in Ω. The proof follows from the sufficiency part of the proof above, in particular from Step 4. The recovery sequence τ m constructed there is A-free and it also generates ν. Sufficiency. Due to a small clash of notation, we re-write assumption (i) as We will show that there exists a sequence {V m } ⊂ M(Ω; V ) with {BV m } ⊂ M(Ω; W ) that generates ν. This will be deduced from the constructions contained in the proof of the sufficiency of Theorem 1.2, but first we need to recall the following fact of blow-downs: if w ∈ M(Q; V ) and Bw ∈ M(Q; W ), then the blow-down of Bw, centered at x and at scale r, is given by Notice that w r,x ∈ M(Q r ; V ) and Bw r,x ∈ M(Q; W ). Moreover, the mass preserving property of push-forwards gives (66) |Bw r,x |(Q r,x ) = |Bw|(Q) We are now ready to prove the assertion. Let us recall from (b) that ν satisfies the sufficiency assumptions of Theorem 1.2 and hence we may apply the elements contained in its proof to ν. In particular, the sequence {γ j } (introduced in (52)) has elements of the form Bu (rj) + v (rj ) . However, since by assumption [ν] = Bu 0 , Corollary 7.2 says that we may assume z = 0 in (52). In particular, keeping the notation of the previous proof, the sequence {q h,m } (defined in Step 3) is a sequence of B-gradients. Indeed, since z = 0 it follows that v (Rm) = C m T x,Rm [Bu 0 Q x,m ] − Bu and by the (inverse) property of blow-downs we get It follows that the sequence {τ m } ⊂ M(Ω; W ) (defined in in p. 44), which generates our Young measure ν, has the form where, according to (65) and ignoring the x-dependence, the U m are defined as In particular, the identity for blow-downs above implies that U m = BW m , where W m is the Radon measure given by By construction we have W m ≡ u 0 (as measures) on a neighborhood of ∂Q x,m . This compatibility across the partition ensures that The proof of the following proposition is contained in Lemma 5.3 of [12]. There the authors state their main results under additional assumptions. However, the proof of this specific proposition makes not use of such assumptions and can be worked out by verbatim in our context. Proof. Fix 0 < θ < 1 and let . We also write ν θ := θν 1 + (1 − θ)ν 2 ∈ E(Ω; W ) * . Our goal is to show that ν θ is an A-free Young measure on Ω. To show this we will construct a sequence of A-free measures om Ω which generate the functional ν θ .
Since this will be a fairly long and technical proof we will begin by describing a brief program of the proof. The foundation of our proof lies in a careful inspection of the infinitesimal qualitative behavior of points x ∈ Ω with respect to our Young measures ν 1 , ν 2 . The qualitative understanding of the set of tangent Young measures of ν i (i = 1, 2) at a given x ∈ Ω will be decisive in the choice of construction of an A-free recovery sequence for ν θ about that point. Once every point and their local constructions are established, the idea is to use Besicovitch's covering theorem to build a partition of Ω into disjoint tiles, each of which retrieves the infinitesimal properties of ν i and hence the recovery sequences of ν θ about their center points. The one but last step is to glue the aforementioned A-free recovery sequences from each tile into a globally A-free sequence which generates an arbitrarily close a piecewise constant approximation of ν θ . The conclusion of the argument then follows from a diagonalization argument between the larger scale of piece-wise constant approximations of ν θ where we glue the recovery sequences, and the smaller scale where the corresponding recovery sequences are effectively constructed.
Step 1. Qualitative analysis of points.
Since we are trying to capture the fine properties of ν 1 and ν 2 simultaneously, it will be convenient to define the measure Λ := λ s 1 + λ s 2 , which is a suitable substitute candidate to keep track of the interactions between singular points of λ 1 and λ 2 . We start by distinguishing regular points and singular points. It follows from the Radon-Nykodým theorem that at (L d +Λ)-almost every x ∈ Ω one of the following properties hold: either (67) x ∈ reg(Ω) := x ∈ Ω : is a regular point, or (68) x ∈ sing(Ω) := x ∈ Ω : is a singular point. Throughout this proof we shall call points with the first property (which holds L d -almost everywhere) regular points, and points satisfying the second property (which holds Λ-almost everywhere) will be called singular points; we shall only consider points x ∈ Ω that are either regular or singular points. In addition, we may assume without any loss of generality that the limits exist at every singular point x ∈ Ω. Next, we further partition sing(Ω) into sets which render precise information about the size relation between λ 1 and λ 2 . More precisely, we split sing(Ω) into sets and Λ(N ) = 0. If we set then, up to modifying N , we may assume that g 1 , g 2 are Λ-measurably continuous and Step 1a. Tangential properties of singular points. So far we have separated regular and singular points, and the latter by their weights with respect to λ 1 and λ 2 . The next step is to separate points in sing(Ω) with respect to the qualitative behavior of Tan(Λ, x). In particular, if x ∈ G 1 , then (2) If otherwise (1) does not hold for any tangent measure of Λ at x, we write x ∈ S. It follows from Lemma B.3 and the fact that blow-ups of blow-ups are blow-ups (see Theorem 2.12 in [54]) that x ∈ S =⇒ δ 0 ∈ Tan(Λ, x).
Step 1b. Selection of points with Lebesgue-type properties. We now turn to the selection of points which later shall be the centers of the tile partitions. As usual let {f p,q } p,q∈N ⊂ E(Ω; W ) be the family from Lemma 4.1 which separates points in E(Ω; W ) * .
Up to removing a set of L d -measure zero, we may assume that every x ∈ reg(Ω) is a Lebesgue point of the maps About singular points x ∈ sing(Ω), we shall be more careful and set B ∞ i ⊂ sing(Ω) to be the set of λ s i -Lebesgue points of the family of maps Each B ∞ i has full λ s i -full measure on Ω and hence B ∞ 1 ∪ B ∞ 2 has full Λ-measure on Ω. Therefore, in what follows there will be no loss of generality in assuming that sing(Ω) = B ∞ 1 ∪ B ∞ 2 ; this union may not be disjoint.
Step 2. Building a partition of cubes with good fine properties. Let m ∈ N, in this step we will address the construction of a full Λ-measure partition of Ω with O(m −1 )-asymptotic approximation Lebesgue-type properties. To begin, let us define a fine cover of L := reg(Ω) ∪ sing(Ω). At every x ∈ L we define ρ m (x) := sup 0 ≤ r ≤ 1 m : r satisfies the (P m (x)) property .
A radius r is said to satisfy (P m (x)) provided the following continuity properties hold for i = 1, 2 and all indexes p, q ≤ m: Qr (x) If x ∈ sing(Ω), then If x ∈ G 0 or x ∈ G ∞ , then we can only find D r satisfying (76) for λ 2 and λ 1 respectively. Lastly, if x ∈ S ∩ G 1 , then Moreover, s r can be chosen sufficiently small so that where µ Q r (x) = Bu + vL d is the decomposition provided by Lemma 7.1 for µ on Q r (x). Here we have used the short notation for the translations of a function w. Now, this is indeed a large amount of smallness conditions to keep track, but they are all fundamental if one wishes to avoid (trivial) partitions which do not reflect the behavior of ν 1 , ν 2 appropriately.  (1). We focus in showing (77)-(78) which will follow from the fact that δ 0 ∈ Tan(Λ, x). Indeed, in this case we may a sequence of infinitesimal radii r j ↓ 0 such that Then, by the strict convergence of the blow-up sequence we deduce that In particular, since x ∈ G 1 , we conclude that Choosing s ≤ 1 m in a way that s rj := sr j satisfies the required properties for A rj and Q rj (x) (this can be done by slightly modifying each r j in the blow-up sequence), we exhibit an infinitesimal sequence r j (and their associated s rj ) satisfying (77)-(78).
This proves the claim.
In particular, the cover Here, we have set O m := ∪ Qx∈Om Q x .
Step 3. Piece-wise homogeneous approximations of ν i . The idea behind defining O m is to construct a piece-wise homogeneous approximation of ν 1 , ν 2 of order 1 m as follows: Fix i ∈ {1, 2} and define, through duality, a sequence of functionals {ν The fact that these functionals are in fact Young measures follows directly from (79), the weak- * measurability properties of ν 1 and ν 2 , and the fact that simple Borel maps are measurable with respect to any Radon measure.
Proof of Claim 2. Let p, q ∈ N (we shall simply write f = f p,q ). First, we show that We consider p, q ≤ m ∈ N. We may estimate (cf. (79)) the difference of the integrals above by the sum of the two non-negative quantities and Using (72) and the linear growth of |f | ≤ M f (1 + | q |) we obtain It follows that lim m→∞ I m = 0.
On the other hand, we use (70)- (71) to bound II m as This shows that lim m→∞ II m = 0, whence (80) follows.
To prove the claim we are left to show that We may estimate the integrand above, for fixed m ∈ N, by Since {f p,q } separates E(Ω; W ) * , this proves Claim 2.
Step 4. Construction of a global A-free recovery sequence. Let us fix m ∈ N. Next, we define candidate recovery sequences for ν θ on Q x ∈ O m . This will be done depending on whether x belongs to R or S where these sets are the ones defined in Step 1a.
Step 4a. Cubes Q x ∈ O m centered at x ∈ R ∪ reg(Ω). We recall from step 1a and (76) that, if x ∈ R, then there are open Lipschitz sets D x ⊂ Q x ⋐ Ω satisfying On the other hand, since ν 1 , ν 2 are A-free Young measures on Ω, we may apply Lemma 7.1 to find sequences (to avoid adding unnecessary notation, we will omit the x-dependence of these sequences) of A-free measures {u j } ⊂ M(D x ; W ) and The same construction applies with for x ∈ reg(Ω) with the exception that we require D x ⊂ Q x to satisfy  Notice that by construction the measures x v j are A-free on Q x for all j ∈ N. Moreover, the w j 's can be extended by µ outside Q x and particular this extension preserves the A-free constraint. Moreover, in virtue of (85)-(86) and the locality of the weak- * convergence of Young measures it holds that Therefore, upon re-adjusting the sequence {w j } we may assume that Step 4b. Cubes Q x ∈ O m centered at x ∈ S. The constructions in these cubes will be completely different and it will consist of separating the generating sequences of ν 1 , ν 2 locally. Once again, by Lemma 7.1, we may find sequences of potentials and Now, let ϕ be a cut-off function satisfying (here Q x = Q r (x)) Due to the L p -continuity of the translations, we may choose n 1 = n 1 (m) ∈ N to be sufficiently large so that for all j ≥ n 1 .
We are now in position to define our recovery sequence candidate for ν θ on Q x by setting The purpose of this sequence is to shift u j and w j apart from each other, while preserving the µ-boundary conditions near ∂Q x (see Figure 2 below). Clearly, {q j } is a sequence of A-free measures on Q x with q j ≡ µ on a neighborhood of ∂Q x and q j ≈ θ 1 u j + θ 2 v j on Q r/2 (x). Notice that this construction differs from the previous one (when x ∈ R) in the sense that the θ i -weights are incorporated by simple multiplication. In general, this construction is too naive to work. However, in this case, it works because we have Λ ≈ δ 0 in Q x . Let us fix j ∈ N. Writing u = θ 1 u + θ 2 u and adding a zero, we may express q j as where as usual the commutator [B, χ] = B(ϕ q ) − ϕB is a linear operator of order at most (k B − 1) and whose coefficients depend solely on ϕ k,∞ and the principal symbol B. We obtain the following estimate for the total variation of q j : In particular, upon re-adjusting the sequence {q j } we may assume that . Observe that if f = f p,q with p, q ≤ m, then Hence, there exists n 1 ≤ n 2 = n 2 (m) ∈ N such that (with γ 1 for all j ≥ n 2 . An analogous estimate holds for ν 2 , w j , and θ 2 . Let us set . Then, by the definition of q j , a similar argument combined with the right translations ±s x e 1 yield (for j ≥ n 2 ) . Combining these estimates we obtain upon re-adjusting the sequence of j's (recall that we had written f = f p,q ) whenever p, q ≤ m.
Step 4c. Gluing the local recovery sequences. Every cube Q x ∈ O m is centered at some x ∈ L and since reg(Ω) ∪ R ∪ S = reg(Ω) ∪ sing(Ω) = L, the constructions in Steps 4a and 4b indeed cover all possible scenarios which can present. The next task is to glue the recovery sequences together to obtain an Afree global recovery sequence of the O(m −1 )-approximation of ν θ . For each m ∈ N, let us define the sequence Notice that by construction each w (m) j is A-free since each w x j and q x j is A-free on Q x and has µ-boundary values in an open neighborhood of ∂Q x .
Step 4d. Generation of the (m −1 )-approximations of ν θ . Appealing to the locality of weak- * convergence of Young measures, we show next that if p, q ≤ m, then (as j → ∞) where ν (m) θ is the Young measure which acts on f ∈ E(Ω, W ) by the representation formula ⟪f,ν Later, in the next step, we will show these Young measures are indeed O(m −1 )approximations of ν θ . This, together with a diagonalization argument with (93) will imply that ν θ ∈ Y 0,µ A (Ω). First, we show that the sequence {w  Since (w m j ) j is uniformly bounded on Ω, the desired limit approximation in (93) follows from 1) the locality of weak- * convergence of Young measures, 2) the generation properties (85)-(86) for points in reg(Ω) ∪ R, 3) the generation property at singular points in S the (92), and 4) the fact that O m is a full (L d + Λ)-partition of Ω.
Step 5. The sequence ν (m) θ approximates ν θ . Next we show that Accordingly, fix p, q ∈ N and choose m ≥ p, q. Let us, for the sake of simplicity, write f = f p,q and f ∞ = f ∞ p,q . Due to the high amount of terms and estimates involving this argument, let us write where each term contains partial sums subjected to a decomposition of the mesh O m in the following way: (a) the cubes Q x around regular points x ∈ reg(Ω). For i ∈ {1, 2}, the corresponding partial sum is given by (b) and now we cover the singular set sing(Ω), starting with the cubes around singular points x ∈ R ∩ G 0 In the first equality we have strongly used the Λ-Lebesgue property for the sets Q i x which is justified in Step 1a(1); the precise statement is contained in Corollary B.2. The same argument will be used in the estimates (c) and (d) below; (c) passing to points x ∈ R ∩ G 1 (in this case x ∈ B ∞ 1 ∩ B ∞ 2 ). For i = 1, 2 the partial sum reads (d) and to finally cover R, the singular points x ∈ R ∩ G ∞ : an analogous estimate to the one derived in (b) gives (e) Lastly, the cubes with centers x ∈ S which by definition are simply given by Since the singular set can be split into the disjoint union and since every possible cube Q x ∈ O m is centered at one (and only one) of the previous four sets, we deduce from inspecting the terms I i , II, III i , IV, V i that Moreover, it follows from the compactness of Young measures and the separation Lemma 4.1 that the convergence above implies (this may involve passing to a subsequence) This finishes the proof.

Characterization of singular Young measures.
In this section we establish a criterion for a family Y sing (Q) ⊂ Y 0 (Q; W ) to belong to Y A (Q). This family mimics the properties of singular tangent Young measures, and therefore this criterion will serve as a preparation for the proof of Theorem 1.1.
Remark 9.1. It is worthwhile to mention that our construction departs from the approach presented in [22,42]. There, the authors are able to work with a family Y sing (P 0 ) of Young measures that are one-directional. This, however, relies on the rigidity that gradients and symmetric gradients possess. In turn, this simplifies enormously the proof of the convexity of A-free young measures at the level of tangent Young measures; it also allows for the creation of artificial concentrations (cf. [23, Lemma 3.5]), which is crucial for the separation argument. It is precisely for this reason that the convexity of Y A,0 (µ, Ω) had to be conceived globally rather than at the level of tangent A-free Young measures.
Let us turn to the heart of the matter. Let P 0 ∈ W and let Y sing (P 0 ) := (δ 0 , λ, σ ∞ ) ∈ Y 0 (Q; W ) : σ ∞ y = σ ∞ 0 λ-a.e., id W , σ 0 = P 0 . We shall prove the following: Proposition 9.2. Let ν = (δ 0 , λ, ν ∞ 0 ) ∈ Y sing (P 0 ) for some P 0 ∈ Λ A . Assume that A[ν] = 0 in the sense of distributions on Q and further assume that Proof. Let us write µ := [ν] = P 0 λ. The idea is to find a suitable convex subset is of course not unique, and is part of the problem in turn. We shall work with the following set: Definition 9.1. Let Y sing A (µ) be the set of A-free Young measures σ ∈ Y 0 (Q; W ) satisfying the following properties: (a) [σ] = µ, (b) supp(σ y ) ⊂ W A for L d -almost every y ∈ Q, (c) supp(σ ∞ y ) ⊂ W A for λ σ -almost every y ∈ Q. Notice that Y sing A (µ) is non-empty since it contains δ µ .
Remark 9.2. Since properties (a)-(c) are convex properties, Proposition 9.1 and Theorem 9.1 imply that Y sing A (µ) is a non-empty weak- * closed convex subset of E(Q; W ) * .
Step 1. The separation property. Let us recall that, for every weak- * closed affine half-space H ⊂ E(Q; W ) * , there exists an integrand f H ∈ E(Q; W ) such that From the geometric version of Hahn-Banach's it follows that, either ν ∈ Y sing A (µ), or there exists f H ∈ E(Q; W A ) such that Since the former case is precisely what we want to prove, let us henceforth assume that the f H strictly separates Y sing A (µ) and ν. By assuming this, we shall reach a contradiction.
Step 2. Separation withf H . Letf H ∈ E(Q; W ) be the integrand defined as f (x, z) = f (x, pz), where p : W → W is the canonical linear projection onto W A . Notice that properties (b)-(c) and the properties of Y sing (Q) we get Step 2. Boundedness of Q AfH . The following version of [12, Lemma 5.5] states that the separation property with Y sing A (µ) conveys the finiteness of the A-quasiconvex envelope off H : . Then, there exists a dense set D ⊂ Q such that, for every y ∈ D, it holds Q Af (y, 0) > −∞.
The proof follows by verbatim from the proof of [12,Lemma 5.5]. The only difference is that, there, it is assumed that W = W A . However, this is of little importance in our setting due to properties (b) and (c), the properties off (cf. Section 4.3) and property (25).
Step 3. The contradiction: ν ∈ H. For an integrand f and ε > 0, we define f ε := f + ε| q |. By construction, property (A) from Remark A.1 is trivially satisfied for the integrand f = (f H ) ε and the domain Ω = Q. The finiteness of Q Af of D and Propositions 4.5, 4.6 and 4.8 imply that (f H ) ε also satisfies property (B) from Remark A.1. Lastly, we recall that |µ|(∂Q) = P 0 |λ|(∂Q) = 0, which implies that (C) in Remark A.1 is also satisfied. With these considerations in mind, we may apply Theorem A.1 to find a recovery sequence {u j } ⊂ L 1 (Q; W ) satisfying Passing to a further subsequence if necessary, we may assume that Claim. σ ∈ Y sing A (µ).
By construction, [θ] = w*-lim u j = µ, and hence property (a) is satisfied. The theory developed in Section 7 implies that, on strictly contained sub-cubes Q t ⊂ Q, we may assume it holds where the elements of the sequence {z j } N0 are mean-value zero A-free measures in M(T d ; W ) and v ∈ L q (T d ; W ). Property (25) and a standard mollification argument gives {z j } N0 ∈ M(T d ; W A ). In particular, the expression for (P 0 λ) Q t and the assumption P 0 ∈ Λ A gives vL d Q t ∈ L q (Q t ; W A ), whence u j Q t ∈ M(Q t ; W A ) ∀j ∈ N.
Since Q t ⊂ Q was arbitrarily chosen, this proves that θ ∈ Y 0 (Q; W A ) and therefore properties (b)-(c) hold. This proves that θ ∈ Y sing A (µ) and the claim is proved. For the ease of notation, let us write f := (f H ) ε for the next calculation. We use the main assumptions P 0 ∈ Λ A and supp(ν ∞ 0 ) ⊂ W A , together with Proposition 4.5 (recall that Q A f (x, q ) is (Λ A ∪ W ⊥ A )-convex) and Remark 1.2 to find that ≥ Q A f (y, µ ac (u))) dy + Here, in the one but last inequality we have dealt with the absolutely continuous part with the aid of the following property: if D ⊂ W is a spanning cone of directions, then every D-convex function g : W → R with linear growth at infinity satisfies (see for instance Lemma 2.5 in [39]) g(z + P ) ≤ g(z) + g ∞ (P ) for all z ∈ W and P ∈ D.
Hence, by combining (97)-(98) we conclude that Letting ε → 0 + , we conclude from (96) that ⟪f H , ν⟫ = ⟪f H , ν⟫ ≥ s H , which contradicts (95). This proves that ν must indeed belong to Y sing A (µ). This completes the proof of the proposition. 9.3. Characterization of regular Young measures. Now, we prove the analog of Proposition 9.2 for regular tangent Young measures. The proof follows the ideas of the original proof in [28], but it requires some minor changes to deal with the fact that W A and W may not coincide.
Proof. The proof is considerably simpler than the one at singular points since, for the separation argument, it will suffice to consider the following family of homogeneous Young measures: . The first step is to verify that Y reg A (P 0 ) is non-empty. Indeed, A(P 0 L d ) = 0 and therefore the elementary (purely oscillatory) homogeneous Young measure (δ P0 , 0, q ) belongs to Y reg A (P 0 ). Remark 9.3. The properties that define Y reg A (P 0 ) are convex properties. Therefore, Proposition 9.1 and Theorem 9.1 imply that Y reg A (P 0 ) is a non-empty weak- * closed convex subset of E(Q; W ) * .
Step 1. The separation property. By the geometric version of Hahn-Banach's theorem it holds that, either ν ∈ Y reg A (P 0 ), or there exists an affine weak- * closed halfplane H that strictly separates ν from Y reg A (P 0 ), i.e., there exists f H ∈ E(Q; W ) such that (100) ⟪f H , ν⟫ < s H ≤ ⟪f H , σ⟫ for all σ ∈ Y reg A (P 0 ).
We shall henceforth assume that f H strictly separates ν and Y sing A (P 0 ). For an integrand f ∈ E(Q; W ), let us consider the homogenized integrand f hom ∈ E reg (W ) defined as f hom (y, z) := Q f (ξ, z) dξ, ∀(y, z) ∈ Q × W.
In particular, Fubini's theorem gives This tells us there is no loss of generality in assuming that f H ∈ E reg (W ).
Step 3. Reduction to the case P 0 = 0 and separation withf H . At regular points, we may no longer use the property P 0 ∈ Λ A . This prevents us to argue in the same way as for singular points, that it indeed suffices to work withf H . There is turnaround to this issue by using the shifts introduced in Definition 4.2: Remark 9.4. By Corollary 7.2, the shifted Young measure Γ P0 [σ] is an homogeneous A-free Young measure with barycenter zero if and only if σ is an homogeneous A-free Young measure with barycenter P 0 .
This remark implies that Shifts P0 {Y reg A (P 0 )} = Y reg A (0). A Young measure (σ 0 , βL d , σ ∞ 0 ) in the latter set satisfies the crucial property that supp(σ 0 ) (and supp(σ ∞ 0 )) are contained in W A for L d -a.e. (βL d -a.e.) y ∈ Q; this follows from the same argument used to prove the claim that θ ∈ Y(Q; W A ) in the singular points' case (here one uses that 0 ∈ Λ A ). On the other hand, Corollary 4.1 implies that Γ P0 [ν] ⊂ W A and also that, either α = 0, or supp(ν ∞ 0 ) ⊂ W A . In particular, we get Therefore, there is no loss of generality in assuming that P 0 = 0 and f H =f H .
Step 3. The contradiction: ν ∈ H. The argument is essentially the same as the one for singular points, which relies on the upper bound from the relaxation argument. The relaxation argument, however, is nowhere near as involved as the one contained in Theorem A.1. First, we show that Q AfH is finite with lower bound s H . Let w ∈ C ∞ ♯ (T d ; W A ) ∩ ker A. By a well-known homogenization argument (see for instance [28,Proposition 2.8]) the sequence w j (y) := w(jy) generates the homogeneous purely oscillatory Young measure θ = (δ w , 0, q ), where δ w , h := Q h(w(y)) dy for all h ∈ C(W ), w j * ⇀ 0 in L ∞ .
Taking the infimum over all w, we conclude that Q AfH (0) ≥ s H , as desired. Now, we are in position to use (99) with h = Q AfH to get (recall that Q AfH is globally Lipschitz, see Propositions 4.5 and 4.6): This poses a contradiction to inequality (100). This proves that ν ∈ Y reg A (0), as desired.
9.4. Proof of Theorem 1.1. Necessity. Property (i) is obvious since, by definition, an A-free measure is generated by a uniformly bounded sequence of (asymptotically) A-free measures, therefore its barycenter is an A-free measure. Properties (ii)-(iii) have been established in [6]: condition (ii) is contained in Proposition 3.1, and condition (iii) in Lemma 3.2; condition (iii') is contained in Proposition 3.3.
Sufficiency. The theory discussed in Section 4.2.1 implies that at (L d + λ s )almost every x 0 ∈ Ω there exists a regular or a singular tangent measure ν 0 ∈ Tan(ν, x 0 ). The idea is to show that each of these tangent measures are in fact locally A-free Young measures and argue by the local characterization from Theorem 1.2. By the dilation properties of tangent Young measures, it suffices to show the following: However, in general one cannot expect this as for instance there might be mass sitting on λ(∂Q r ). We may then assume that θ 1 := λ(Q r ) = ϕ(r − ) ≤ θ and (103) 0 ≤ θ − θ 1 ≤ ϕ(r + ) − ϕ(r − ).
This will be our first approximation. The second step to carry the same approximation now on {q 1 r , . . . , q 2d r }, the (2d) open (d − 1)-dimensional faces of ∂Q r , which are axis-directional translated (d − 1)-dimensional cubes (centered at the origin) in R d−1 ; the number (2d) of faces will not be relevant for our construction. Define the maps Repeating the same procedure as in the first step on each of the faces simultaneously, we update the error of our estimate to where r 1 := sup{ s ∈ (0, r) : where C 2 := Q r ∪ q 1 r1 ∪ · · · ∪ q 2d r1 . Moreover, by construction (since each one of the added faces are concentric to each face of Q r ), C 2 is a (semi-open/semi-closed) convex set satisfying Q r ⊂ C 2 ⊂ Q r . There are two cases, either θ = θ 2 and then λ(C 1 ) = θ, or θ − θ 2 > 0 and we must keep adding bits of ∂Q r , which may require to perform a similar argument on the (d − 2)-dimensional concentric faces of ∂Q r and the (d − 2)-dimensional faces of each q i s , 1 ≤ i ≤ 2d. In general, the (j + 1)th step is to iterate this argument (when θ − θ j > 0) on all the possible (d − j)-dimensional faces of ∂Q r and all the other (d − j)-dimensional faces resulting of adding the previous (d − j + 1)-dimensional concentric cubical caps. The key part of the construction is that, at the end of the (j + 1)th step, one obtains a convex set C j ⊂ C j+1 satisfying Q r ⊂ C j+1 ⊂ Q r and 0 ≤ θ − λ(C j+1 ) =: θ − θ j+1 for some θ j+1 ≥ θ j .
We now argue why there exists 1 ≤ j ≤ d such that θ j = θ. If d = 2, then by the same argument that in the one-dimensional case we get that the maps ϕ i,2 are continuous, and hence it must be that θ 2 reaches θ. If d = 3, then the maps ϕ i,3 of the third step are continuous and hence at most θ 3 reaches θ. In general, the description of this procedure is tedious, but it is inductively natural and always reaches and endpoint (at most after d-steps) where we find a convex set C containing the origin and satisfying Q r ⊂ C ⊂ Q r and λ(C) = θ.
This finishes the proof. Proof. Let τ ∈ Tan(λ, x) and set 0 < α = λ({x}). Since Tan(λ, x) is a d-cone, it is enough to show that τ = δ 0 when τ is a probability measure on Q. Moreover, we may also assume the blow-up sequence converging to τ has the form It follows from the strict-convergence γ → τ on Q that τ (Q s ) = lim j→∞ γ j (Q s ) = 1 α lim j→∞ λ(Q srj (x)) = 1 for all s > 0.
Since τ is a probability measure on Q, this shows that τ Q = δ 0 as desired.