Characterization of Generalized Young Measures Generated by Symmetric Gradients

This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer–Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The “local” proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti’s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences.


Introduction
Young measures quantitatively describe the asymptotic oscillations in L p -weakly converging sequences. They were introduced in [48][49][50] and later developed into an important tool in modern PDE theory and the calculus of variations in [8,9,44,45] and many other works. In order to deal with concentration effects as well, DiPerna and Majda extended the framework to so-called "generalized" Young measures, see [2,17,21,28,30,43]. In the following we will refer also to these objects simply as "Young measures".
When considering generating sequences that satisfy a differential constraint like curl-freeness (i.e. the generating sequence is a sequence of gradients), the problem immediately arises to characterize the resulting class of Young measures. In applications, these results provide very valuable information on the allowed oscillations and concentrations that are possible under this differential constraint, which usually constitutes a strong restriction.
The first general classification results are due to Kinderlehrer and Pedregal [23,24], who characterized classical gradient Young measures, i.e. those generated by gradients of W 1, p -bounded sequences, 1 < p ≤ ∞. Their theorems put such gradient Young measures in duality with quasiconvex functions as introduced by Morrey [34]. For generalized Young measures the corresponding result was proved in [21] (also see [22]) and numerous other characterization results in the spirit of the Kinderlehrer-Pedregal theorems have since appeared, see for instance [11,19,20,29].
Characterization theorems are of particular use in the relaxation of minimization problems for non-convex integral functionals, where one passes from a functional defined on functions to one defined on Young measures. A Kinderlehrer-Pedregaltype theorem allows one to restrict the class of Young measures over which to minimize. This is explained in detail (for classical Young measures) in [36]. A similar application is possible for generalized Young measures.
The characterization of generalized BV-Young measures was first achieved in [28]. A different, "local" proof was given in [40], another improvement is in [25,Theorem 6.2]. All of these arguments crucially use Alberti's rank-one theorem [1] (see [31] for a short and elegant new proof) and thus, since this theorem is specific to BV, extensions to further BV-like spaces have been prohibited so far. The only partial result for a characterization beyond BV seems to be in [7], but that result is limited to first-order operators (which does not cover BD) and also additional technical conditions have to be assumed.
We now explain briefly the framework underlying this work and introduce some notation to state our main result; precise definitions are given in Section 2. Given an L 1 -bounded sequence of maps v j : → R N ( ⊂ R d ), the Fundamental Theorem of (generalized) Young measure theory states that there exists a subsequence of the v j 's (which we do not relabel) such that for all continuous f : × R N → R with the property that the recession function x ∈ , A ∈ R N exists, it holds that where (ν x ) x∈ , (ν ∞ x ) x∈ are parametrized families of probability measures on R N and ∂B N (the unit sphere in R N ), respectively, and λ ν is a positive, finite Borel measure on . Together, we call ν = (ν x , λ ν , ν ∞ x ) the (generalized) Young measure generated by the (subsequence of the) v j 's.
In plasticity theory [41,42,47], one often deals with sequences of uniformly L 1 -bounded symmetric gradients It is an important problem to characterize the (generalized) Young measures ν generated by such sequences (Eu j ). We call such ν BD-Young measures and write ν ∈ BDY( ), since all BD-functions [41,42,47] can be reached as weak* limits of sequences (u j ) as above. Recall that a function u ∈ L 1 ( ; R d ) lies in the space BD( ) of functions of bounded deformation if its distributional symmetrized derivative Eu is a bounded Radon measure on taking values in R d×d sym . Our main result is the following: Theorem 1.1. Let ν ∈ Y( ; R d×d sym ) be a (generalized) Young measure. Then, ν is a BD-Young measure, ν ∈ BDY( ), if and only if there exists u ∈ BD( ) with [ν] = Eu and for all symmetric-quasiconvex h ∈ C(R d×d sym ) with linear growth at infinity, the Jensen-type inequality holds at L d -almost every x ∈ .
Here, the generalized recession function h # : R N → R of a map h : R N → R with linear growth at infinity, i.e. |h(A)| ≤ C(1 + |A|) for some constant C > 0, is given as We remark that the use of the generalized recession function can in general not be avoided since not every quasiconvex function with linear growth at infinity has a (strong) recession function (and one needs to test with all those, but see [25,Theorem 6.2]). Further, a bounded Borel function f : For a suitable integrand f : × R d×d sym → R, the minimum principle can be seen as the extension-relaxation of the minimum principle (1. 2) The point is that (1.2) may not be solvable if f is not symmetric-quasiconvex, whereas (1.1) always has a solution. In this situation, our main Theorem 1.1 then gives (abstract) restrictions on the Young measures to be considered in (1.1). Another type of relaxation involving the symmetric-quasiconvex envelope of f is investigated in [5] within the framework of general linear PDE side-constraints.
Our proof of Theorem 1.1 roughly follows the "local" strategy developed in [40] for the characterization of BV-Young measures. The necessity part follows from a lower semicontinuity theorem, in this case the BD-lower semicontinuity result from [38], as usual. For the sufficiency part, we first characterize "special" Young measures that can be generated by sequences in BD, see Section 3. These "special" Young measures originate from a blow-up procedure and are called tangent Young measures. There are two types: regular and singular tangent Young measures, depending on whether regular (Lebesgue measure-like) effects or singular effects dominate around the blow-up point.
For the regular tangent Young measures the classical methods of Kinderlehrer & Pedregal [23,24] are applicable. In order to deal with singular tangent measures, we first need to strengthen the result on "good blow-ups" for Young measures with a BD-barycenter from [38], see Lemma 2.14, which is also interesting in its own right. We combine this lemma with the analogue of Alberti's rank-one theorem in BD from [16], which imposes strong constraints on the underlying BD-deformation (discussed in Remark 3.6). Glueing tangent BD-Young measures together, see Lemma 4.2, we then prove Theorem 1.1 in Section 4.
We stress that our argument crucially rests on the BD-analogue of Alberti's rank-one theorem recently proved by the authors in [16], see Theorem 2.12. The reason is that this result explains the local structure of singularities that can occur in BD-functions (more precisely, in the singular part of the symmetric derivative). A weaker version of this argument was already pivotal in the work [38]. However, to prove Theorem 1.1, the strong version of [16] is needed. Technically, in one of the proof steps to establish Theorem 1.1 we need to create "artificial concentrations" by compressing symmetric gradients in one direction. This is only possible if we know precisely what these singularities look like, see Lemma 3.5 and also Remark 3.6 for details. It is not clear to us if the use of Theorem 2.12 can be avoided to prove Theorem 1.1.
As another very useful technical tool, we utilize the BD-analogue of the surprising observation by Kirchheim and Kristensen [25] that the singular part of a BV-Young measure is unconstrained. Without this observation, a weaker characterization result could be established, where, however a second, singular Jensen-type inequality needs to be required. Indeed, it was shown in Theorem 4 of [38] that in the situation of our theorem automatically also the singular Jensen inequality holds. It is remarkable (and due to the observations in [25] alluded to above) that this is, however, not needed to prove the characterization result. The third central new ingredient in the proof is an argument yielding "very good" blow-ups at singular points, which are not only two-dimensional, but even one-dimensional (plus an affine part). This is achieved by iterating the blow-up construction (using the observation that "blow-ups of blow-ups are blow-ups"); see Lemma 2.14 for details.
As a (technical) application of Theorem 1.1, we show how our result can be used to split off an atomic part from a BD-Young measure in generating sequences, see Theorem 5.1.

Setup and Preliminary Results
In this section we recall all the notation and technical tools that will be employed in the subsequent sections. In particular, we collect many results from the framework of generalized Young measure, usually specialized to the BD-case.

Functions of Bounded Deformation
The space BD was introduced in [32,41,42,47] for applications in plasticity theory, much of the theory relevant to this work is developed in [3,6,13,26,46,47].
As a standing assumption throughout this whole work, let ⊂ R d be an open domain with Lipschitz boundary; in the following proofs we implicitly assume d ≥ 2, but the main results are (trivially) true also for d = 1 since then BV and BD agree and (symmetric-)quasiconvexity is just convexity. The space BD( ) of functions of bounded deformation is defined as the space of functions u ∈ L 1 ( ; R d ) such that the distributional symmetric derivative We split Eu according to the Lebesgue-Radon-Nikodým decomposition as where the approximate symmetrized gradient Eu is the Radon-Nikodým derivative of Eu with respect to Lebesgue measure and E s u is the singular part of Eu (with respect to L d ).
Since there is no Korn inequality in L 1 , see [12,25,35], it can be shown that BV( ; R d ) is a proper subspace of BD( ). See [13] for further results in this direction.
A rigid deformation is a skew-symmetric affine map r : We have the following Poincaré inequality: for each u ∈ BD( ) there exists a rigid deformation r such that where C = C( ) only depends on the domain . This is shown for example in [47] or see [46,Remark II.2.5].
Finally, we will also define the symmetric tensor product a b := (a ⊗ b + b ⊗ a)/2 = (ab T + ba T )/2 of two vectors a, b ∈ R d .

Symmetric-Quasiconvexity
The appropriate generalized convexity notion related to symmetrized gradients is the following: we call a bounded Borel function f : where D ⊂ R d is any bounded Lipschitz domain. Similar assertions to the ones for quasiconvex functions hold, cf. [18] and [10]. In particular, if f has linear growth at infinity, we may replace the space W 1,∞ 0 (D; R d ) in the above formula by W 1,1 0 (D; R d ). It can further be shown, see [20,Proposition 3.4], that any symmetric- The symmetric-quasiconvex envelope S Q f : This expression is either identically −∞ or finite and symmetric-quasiconvex. Analogously to the case for usual quasiconvexity (cf. [15,23]), for continuous f , the symmetric-quasiconvex envelope can be written as

Generalized Young Measures
The following theory is mostly from [2,28,38], where also proofs and examples can be found.
Let again ⊂ R d be a bounded Lipschitz domain. For f ∈ C( × R N ) and g ∈ C( × B N ), where B N denotes the open unit ball in R N , we let In particular, f ∈ E( ; R N ) has linear growth at infinity, i.e. there exists a constant M ≥ 0 (in fact, Furthermore, for all f ∈ E( ; R N ), the (strong) recession function f ∞ : × R N → R, defined as exists and takes finite values. Clearly, f ∞ is positively 1-homogeneous in A, that is For f ∈ C( × R N ) with linear growth at infinity, f ∞ may not exist, but we can always define the generalized recession function It is easy to see that f # is always positively 1-homogeneous and upper semicontinuous in its second argument. In many other works, f # is just called the "recession function", but here the distinction to our (strong) recession function f ∞ is important. for which we require that (iv) the map x → ν x is weakly* measurable with respect to L d , i.e. the function x is weakly* measurable with respect to λ ν ; and Equivalently to (i)-(vi), one may require The duality pairing between f ∈ E( ; R N ) and ν ∈ Y( ; R N ) is given as is a sequence of measures with sup j |γ j |( ) < ∞, then we say that the sequence (γ j ) generates a Young measure Here, γ s j is the singular part of γ j with respect to Lebesgue measure. Equivalently, Young measures" that are naturally associated with the γ j . Also, for ν ∈ Y( ; R N ) we define the barycenter as the measure The following is the central compactness result in Y( ; R N ): Then, (ν j ) is weakly* sequentially relatively compact in Y( ; R N ), i.e. there exists a subsequence (not relabeled) such that ν j * ν and ν ∈ Y( ; R N ).
is a sequence of measures with sup j |γ j |( ) < ∞ as above, then there exists a subsequence (not relabeled) and ν ∈ Y( ; R N ) By a standard density argument it suffices to check weak*-convergence of Young measures by testing with a countable set of integrands only, which is equivalent to the separability of the space E( ; R N ): Moreover, all h can be chosen Lipschitz continuous and each h has either compact support or is positively 1-homogeneous.

BD-Young Measures
is called a BD-Young measure, ν ∈ BDY( ), if it can be generated by a sequence of BD-symmetric derivatives. That is, for all ν ∈ BDY( ), there exists a (necessarily norm-bounded) sequence ; any such u is called an underlying deformation of ν.
The following results about BD-Young measures constitute a "calculus" for BD-Young measures, which will be used frequently in the sequel see [28,37,38] for proofs (the first reference treats BV-Young measures, but the proofs adapt lineby-line).

Lemma 2.3. (Good generating sequences)
Let ν ∈ BDY( ). Then: The proof of this result can be found in [28,Lemma 4]. (i) u agrees with an affine map on the boundary ∂ ; or Then, there exists a Young measureν ∈ BDY( ) acting on f ∈ E( ; R d×d sym ) as More precisely: with linear growth at infinity it holds that h,ν x = − h, ν y dy a.e.; (2) The concentration measure λν is a multiple of Lebesgue measure, λν = αL d , where α = λ ν ( )/| |;

Remark 2.5.
We remark that one may consider any averaged Young measure as in the preceding lemma to be defined on any bounded Lipschitz domain D ⊂ R d , so thatν ∈ BDY(D) and (2.4) is replaced by . This can be achieved by a covering argument analogous to the proof of Lemma 2.4 in [28, Proposition 7] (covering D with rescaled copies of ).
The proof of case (i) is contained in [28,Proposition 7], the proof of (ii) is similar, but requires an additional standard staircase (piecewise affine) construction to glue the rescaled versions of u together without incurring an additional jump part. Let and (u j ) be as in (ii). First, by Lemma 2.3 we may assume that there exists a For every j ∈ N let a jk ∈ R d be defined such that the similar rescaled sets jkl := a jkl + j −1 , k = 1, . . . , j, l = 1, . . . , j d−1 , form a cover of . We furthermore assume that the jkl are arranged in a regular grid with jkl (l = 1, . . . , j d−1 ) lying in the k'th slice in ξ -direction.
Furthermore, denote by γ ∈ R d the difference between the trace of u on the face of u in the positive ξ -direction and u in the negative ξ -direction; note that because of the assumption that u has the shape u(x) = ηh(x · ξ), γ is a constant vector. Then define For the weak derivative we get Note that the staircase term γ k/j is chosen precisely to annihilate the jumps over the slice boundaries in direction ξ ; over the other boundaries there are no jumps by the assumption on the shape of u. It can now be checked, following line-by-line the proof of [28,Proposition 7], that the w j generateν as required in the lemma. A special case is the following corollary:
We will also need the following approximation result, see [28,Proposition 8].
where ν C kl designates the averaged Young measure (as in Lemma 2.4) of the restriction ν C kl of ν to C kl .

Localization of Young Measures
The paper [38] proved two localization principles for BD-Young measures, leading to so-called "tangent Young measures"; here and in the following, for ease of notation, we leave out the dependence of the spaces on R d×d sym . We first define the following regular, or homogeneous, spaces of (tangent) Young measures for A 0 ∈ R d×d sym (Q being the standard unit cube).
The first localization principle then reads as follows: Proposition 2.8. (Localization at regular points) Let ν ∈ Y( ; R d×d sym ) be a Young measure. Then, for L d -almost every x 0 ∈ there exists a regular tangent Young measure which satisfies Here, is the push-forward of μ under T x 0 ,r n . A proof for the preceding proposition can be found in [38,Proposition 1]. We furthermore remark that σ in Proposition 2.8 is such that for all open sets U ⊂ Q with L d (∂U ) = 0, and all h ∈ C(R d×d ) such that the recession function h ∞ exists in the sense of (2.3), it holds that For the singular counterpart to Proposition 2.8, we first introduce the following spaces for any bounded Lipschitz domain D ⊂ R d (we again omit mention of R d×d sym for ease of notation): The duality pairing between E sing (D) and Y sing (D) is Furthermore, we say that the sequence of measures Additionally, if ν ∈ BDY( ), then σ ∈ BDY sing (D).
A proof of this fact can be found in [38, Proposition 2].

Good Singular Blow-Ups
In this section, as a preparation for the singular analogue of Proposition 3.1, we establish a result about good blow-ups for BD-Young measures in Lemma 2.14 below.
First, we recall from [38, Theorem 3] the following result. We here state it in a slightly different fashion, namely for Young measures with a BD-barycenter instead of BD-generated Young measures.

Remark 2.11.
In the preceding theorem, one sees easily that if ν ∈ BDY( ), then also σ ∈ BDY sing (D) (this is the original statement in [38,Theorem 3]). For us, however, this fact is not needed.
The proof is the same as in [38,Theorem 3], where the whole argument is only concerned with the barycenter and not the generating sequence. Moreover, we remark that [σ ](∂ D) can be achieved by a rescaling of the blow-up sequence r n ↓ 0 into αr n ↓ 0 for some α ∈ (0, 1) such that [σ ](∂(α D)) = 0 (assuming that 0 ∈ D).
The next ingredient we will need is the theorem on the singular structure of BD-functions, proved in [16]: This is the BD-analogue of the following celebrated Alberti rank-one theorem [1]: We will now state and prove a strengthened version of Lemma 2.10. For this, we first define suitable spaces.
In all of the following, let ξ ∈ S d−1 and denote by Q ξ the rotated unit cube (|Q ξ | = 1) with one face normal ξ . We first define one-directional versions of the spaces E sing , Y sing , BDY sing for A 0 ∈ R d×d sym \ {0}, ξ ∈ S d−1 ; as before, we henceforth leave out the dependence of the spaces on R d×d sym : Here, the ξ -directionality of λ σ means that for all Borel sets B ⊂ Q ξ it holds that Finally, the spaces Y = Eu for some u ∈ BD( ).
Remark 2.15. We note in passing that this improvement in fact allows one to slightly simplify the proof of the lower semicontinuity result in [38] as well.
and some a(x), b(x) ∈ R d . Indeed, denoting by λ * ν the singular part of λ ν with respect to |E s u|, the above holds at |E s u|-almost every x ∈ with a(x), b(x) = 0 by Theorem 2.12 and at λ * ν -almost every x ∈ with a(x) = b(x) = 0. From the previous Lemma 2.10 we get that there exists a singular tangent Young measure τ ∈ Y sing (D) to ν at λ s ν -almost every x 0 ∈ that satisfies (i), (ii) or (iii). We have that for any w ∈ BD(R d ) with Ew = [τ ], for some a, b ∈ R d . Indeed, by Proposition 2.9 we get τ ∞ y = ν ∞ x 0 for λ τ -almost every y ∈ D, which implies the first equality. Further, if x 0 is chosen such that (2.5) holds, we infer id, τ ∞ Step 1. If a, b are parallel, say a = b after rescaling, then the statement of the present lemma follows immediately with σ := τ , ξ := a/|a| = b/|b|, D = Q ξ (note that we can choose D in Lemma 2.10 as we like and we can decide beforehand whether a, b are parallel, see (2.5)). In this case, σ = τ ∈ Y sing (a a, a), as follows directly from Proposition 2.9 and Lemma 2.10 (iii).
Step 2. Only in the case id, ν ∞ x 0 = a b with a = b there is something left to prove. Without loss of generality we assume that a = e 1 and b = e 2 , which is possible after a change of variables. In this case, by Lemma 2.10 we have that for any w ∈ BD(Q) with Ew = [τ ] there exist functions g 1 , g 2 ∈ BV(R), w 0 ∈ R d , and a skew-symmetric matrix R ∈ R d×d skew such that w(y) = w 0 + g 1 (y 1 )e 2 + g 2 (y 2 )e 1 + Ry for a.e. y = (y 1 , . . . , y d ) ∈ R d .

Moreover,
Case (I). If either Dg 1 or Dg 2 are the zero measure, the conclusion of the theorem is trivially true with σ := τ , ξ := e 1 , η := e 2 , D the standard open unit cube. So, henceforth assume that both Dg 1 , Dg 2 are not the zero measure. In the following we denote by g 1 , g 2 the approximate derivatives of g 1 , g 2 , i.e. the densities of Dg 1 , Dg 2 with respect to Lebesgue measure.
Case (II). If D s g 1 , D s g 2 = 0, then we may use the regular localization principle, Proposition 2.8, to construct a non-zero regular tangent Young measure σ ∈ Y reg of τ at y 0 ∈ R d . We will argue below that in fact σ is a singular tangent Young measure to our original ν at x 0 and that [σ ] ∈ BD( ), see Step 3 of the proof.
Case (III). On the other hand, if D s g 1 = 0 (without loss of generality), then we claim we can find y 0 = (s 0 , t 0 , y 3 , . . . , y d ) ∈ R d with the property that there exists a non-zero singular tangent Young measure σ ∈ Y sing (Q) to τ at y 0 and that Indeed, notice that second and third conditions hold for L 1 -almost every t 0 because the set of Lebesgue points of g 2 has full Lebesgue measure in R and the Radon-Nikodým derivative of D s g 2 by L 1 is zero L 1 -almost everywhere. Hence, they hold for (|D s g 1 | ⊗ L d−1 )-almost every y 0 = (s 0 , t 0 , y 3 , . . . , y d ) ∈ R d by Fubini's theorem. By the singular localization principle for Young measures, Proposition 2.9, we know that the first property holds for almost every y ∈ R d with respect to λ s τ . By (2.6), and the fact that |D s g 1 | ⊗ L d−1 is singular with respect to Thus, |D s g 1 | ⊗ L d−1 = 0 is absolutely continuous with respect to |E s w|.
Furthermore, |E s w| is absolutely continuous with respect to λ s τ because Thus, the first condition also holds at (|D s g 1 | ⊗ L d−1 )-almost every y 0 and we find at least one y 0 = (s 0 , t 0 , y 3 , . . . , y d ) with the claimed properties.
Step 3. We observe that σ is still a singular tangent Young measure to ν at the original point x 0 if the latter was chosen suitably. Indeed, it can be easily checked that the property of being a singular tangent Young measure is preserved when passing to another "inner" (regular or singular) tangent Young measure; the only non-obvious fact here is that "tangent measures of tangent measures are tangent measures", but this is well-known and proved for example in Theorem 14.16 of [33].
The "inner" blow-up sequence of the BD-primitives of the barycenters has the form with r n ↓ 0 and constants c n = r −d n if we are in case (II) and if we are in case (III). Note that in both cases lim sup n→∞ r d n c n < ∞. To retain a BD-uniformly bounded sequence, it might also be necessary to add an (ndependent) rigid deformation to w (n) , but for ease of notation this is omitted above.
Then, w (n) * v in BD(Q) and v has the property that Ev = [σ ]. For Ew (n) we get Ew (n) = (e 1 e 2 ) r d n c n g 1 (s 0 + r n z 1 ) + g 2 (t 0 + r n z 2 ) L d (dz) Let ϕ ∈ C ∞ c (R d ) and choose R > 0 so large that supp ϕ ⊂⊂ Q(0, R) = (−R, R) d . Using the special properties of our choice of y 0 together with the fact that c n r −d n as n → ∞, we observe that and this goes to zero as n → ∞. Also, Hence, the e 2 -directional parts of Ew (n) in the limit converge to the fixed matrix β(e 1 e 2 ), where β = α lim n→∞ r d n c n (this limit exists after taking a subsequence). More precisely, in case (II), # r −d n and σ is a regular tangent Young measure to τ , thus potentially β = 0; otherwise, in case (III) it must hold that β = 0 (since c n ∼ λ s τ (Q(y 0 , r n )) −1 ). The e 1 -directional parts of Ew (n) clearly stay e 1 -directional under the operation of taking weak* limits. Thus, v is of the required form, with (ξ, η) = (e 1 , e 2 ).

Functional Analytic Properties of BDY reg , BDY sing
In this section we prove the following lemma about our tangent Young measure spaces: Lemma 2.16. The sets BDY reg and BDY sing (a b, ξ) are convex and weakly* closed (with respect to the topology induced as a subset of (E reg ) * and E sing (ξ ) * ) for all a, b ∈ R d \ {0}, ξ ∈ {a, b}.
Proof. We only prove the statements for BDY sing (a b, ξ) since they are more difficult (for regular BD-Young measures the argument is easier because our underlying deformation of the homogeneous Young measure is even affine). We furthermore assume without loss of generality that ξ = a.
Step 1: Weak*-closedness of BDY sing (a b, a). Let { f n } n∈N = {ϕ n ⊗h n } n∈N ⊂ E sing (a) be a countable set of integrands that determines Young measure convergence in Y sing (a b, a), this can be achieved by a reasoning analogous to Lemma 2.2 (only take ξ -directional ϕ n and positively 1-homogeneous h n ). Let σ be in the weak* closure of BDY sing (a b, a). Then, for every j ∈ N there exists σ j ∈ BDY sing (a b, a) with In particular, σ j * σ in E sing (a) * and also in Y(Q a ; R d×d sym ) since the sequence (σ j ) is compact in that space. Indeed, 1 ⊗ | q |, σ j is uniformly in j bounded, so we may use the compactness result from Lemma 2.1. It is not hard to check that the defining properties of Y sing (a b, a) (such as the a-directionality of λ σ , y → ν ∞ y ) are preserved under weak* limits, hence σ ∈ Y sing (a b, a).
We need to show also that σ is generated by a sequence of symmetric gradients, which follows by a diagonal argument: select for each j ∈ N a function for all n ≤ j. Then, adding a rigid deformation to the u j 's, Poincaré's inequality in BD, see (2.1), yields that there exists a (non-relabeled) subsequence such that Eu j Y → μ ∈ BDY(Q a ). Clearly, μ = σ by construction.
Step 2: Convexity of BDY sing (a b, a) assuming λ μ (∂ Q a ) = λ ν (∂ Q a ) = 0). Let μ, ν ∈ BDY sing (a b, a) be such that λ μ (∂ Q a ) = λ ν (∂ Q a ) = 0 and let θ ∈ (0, 1). By the approximation principle, Lemma 2.7, we have that both μ, ν are weak* limits of piecewise homogeneous and averaged Young measures. The partition with respect to which the approximations are piecewise constant can be chosen the same for both μ and ν (this can be seen from the proof of the averaging principle in [28,Section 5.3]). Thus, by the weak* closedness proved in the first step, it suffices to show the result for homogeneous, one-directional BD-Young measures.
Assume now that we have two homogeneous, one-directional BD-Young measuresμ,ν ∈ BDY sing (a b, a) with λ μ (∂ Q a ) = λ ν (∂ Q a ) = 0, which we assume to be defined on a cube with one face orthogonal to a. Indeed, by Remark 2.5 we can always assume that the averaged Young measuresμ,ν are defined on Q a (one can also argue by inspecting the proof of Lemma 2.7 to conclude that the subdivision may be chosen to consist of cubes only).
Step 3: Convexity of BDY sing (a b, a). To conclude the proof we show that the set of μ ∈ B DY sing (a b, a) such that λ μ (∂ Q a ) = 0 is weakly* dense in BDY sing (a b, a). Indeed, assume that a = e 1 , σ ∈ BDY sing (e 1 b, e 1 ) and that for (u j ) ⊂ BD(Q) ∩ C ∞ (Q; R d ) with u j | ∂ Q = bg(x 1 )| ∂ Q for some g ∈ BV(−1/2, 1/2), we have Eu j Y → σ . We consider g to be extended continuously to all of R. Then, define for α > 1, It is not hard to see that Eu α j Y → σ α for some σ α ∈ BDY sing (e 1 b, e 1 ) with λ σ (∂ Q a ) = 0 and σ α * σ as α ↓ 1. This and the previous step easily implies the convexity of BDY sing (a b, a).

Local Characterization
We first show the characterization result for tangent Young measures, i.e. those Young measures originating from Propositions 2.8 and 2.9 above.

Characterization for Regular Blow-Ups
With the definition of E reg , Y reg (A 0 ), BDY reg (A 0 ) from Section 2.5, we have the following result about the characterization at regular points: for all symmetric-quasiconvex h ∈ C(R d×d sym ) with linear growth at infinity. Then, σ ∈ BDY reg (A 0 ).
Our proof here is quite concise since it is very close to Kinderlehrer & Pedregal's original argument [23,24] and also essentially the same as the one given for [40, Proposition 3.2].

Proof.
Step 1. First, by Lemma 2.16, the set BDY reg (A 0 ) is weakly*-closed and convex (considered as a subset of (E reg ) * ).
We will show below that for every weakly*-closed affine half-space H in (E reg ) * with BDY reg (A 0 ) ⊂ H , we have σ ∈ H . Then the Hahn-Banach theorem will imply that σ ∈ BDY reg (A 0 ). Fix such a weakly* closed half-space H . By standard arguments from functional analysis, see for example [14,Theorem V.1.3], there exists f H ∈ E reg such that In particular, We will show f H , σ ≥ κ, whereby σ ∈ H .
Step 2. For the the symmetric-quasiconvex envelope S Q f H of f H it holds that Then, using the generalized Riemann-Lebesgue Lemma, Corollary 2.6, there exists which is a contradiction.

Characterization for Singular Blow-Ups
Here we will prove the singular analogue of Proposition 3.1.
The preceding proposition is surprising since it says that every singular Young measure in Y sing 0 (a b, ξ) is generated by a sequence of symmetric derivatives of BD-functions.
We first record the following lemma, which is a direct consequence of the main result in [25]: To prove this result we recall a version of the main result of [25]: sym → R be positively one-homogeneous and symmetric quasiconvex. Then, h ∞ is convex at every matrix a b for a, b ∈ R d , that is, there exists an affine function g : Proof of Lemma 3.3. By the preceding theorem, h is actually convex at matrices a b, that is, the Jensen inequality holds for measures with barycenter a b, such as our μ.
The following lemma on "artificial concentrations" will be crucial in the sequel: Lemma 3.5. Let ν ∈ BDY(S), where for some z 0 ∈ Q a , R > 0, a ∈ S d−1 , Assume further that there exists a sequence (v j ) ⊂ BD(S) with Ev j Y → ν and v j (x) = bg(x · a) on ∂ S for some b ∈ R d and g ∈ BV(R). Then, there existŝ ν ∈ BDY sing (a b, a) such that for all positively 1-homogeneous h ∈ C(R d×d sym ). The condition on the generating sequence is in particular satisfied if [ν] = Eu for some u ∈ BD(S) with the property that u(x) = bg(x · a) on ∂ S.

Proof. Note that by Lemma 2.3 if
Hence the second part of the statement follows from the first.
Up to a translation and a one-dimensional scaling, we can assume that x 0 = 0 and R = 1. Let and define w j ∈ B D(S) by where g(±1) is defined in the sense of trace. It can be seen that Ew j generates a Young measure μ ∈ BDY(Q a ) with μ x = δ 0 almost everywhere since Ew j → 0 in measure. Furthermore, for all positively 1-homogeneous h ∈ C(R d×d sym ), we get from the a-directionality of g that Now apply the averaging principle, Lemma 2.4, to μ to getν ∈ BDY sing (a b, a) with the property (3.2). Indeed, the maps y →ν ∞ y , x →ν ∞ y are constant (a.e.) and λν is a multiple of Lebesgue measure, so it only remains to check that [ν] = (a b)λν. To see the latter result, it suffices to observe using an integration by parts that (n being the outward unit normal) Since also λν is a multiple of Lebesgue measure, we conclude [ν] =(a b)λν.
Remark 3.6. The preceding result is in fact the reason why we need the singular structure theorem in BD, Theorem 2.12, as opposed to a mere rigidity result as in [38]. Indeed, the above Lemma will play a key role in the proof of Proposition 3.2 and in order to apply it one is forced to require in the definition of Y sing (a b, a) the a-directionality of λ σ . Lemma 2.14, which relies on Theorem 2.12, then implies that it is not restrictive to consider tangent Young measure lying in Y sing (a b, a), see Step 2 in the proof of Proposition 4.1.
(A) S Q f ε H (z k , q ) is finite, symmetric-quasiconvex, and positively 1-homogeneous; (B) There exists a "recovery sequence" (ψ The symmetric-quasiconvexity and the positive 1-homogeneity of S Q f ε H (z k , q ) are now easy to see by standard techniques, see for instance the appendix to [23], which concerns quasiconvexity, but the methods adapt.
To show (B), we first recall that from the fact that for separately convex functions an upper p-growth bound also implies a lower p-growth bound (with a different constant), where 1 ≤ p < ∞, see Lemma 2.5 in [27]. Thus, it follows that By definition, this sequence (ψ (k) j ) j satisfies (3.4). Further, we may estimate, using the symmetric-quasiconvexity and where we have also discarded some leading elements from the sequence (ψ (k) j ) j . Thus, by (3.6), which is (3.5).
Step 3. Now, with the z k 's chosen in each S k to satisfy (A), (B), we estimate as follows, using (3.3) and the Jensen-type inequality from Lemma 3.3, Here, for the last line we used f ε H ,μ (δ) ≥ f H ,μ (δ) ≥ κ, sinceμ (δ) ∈ H (recall that H was such that BDY sing (a b, a) ⊂ H ). Now, first let δ → 0, using that thê μ (δ) 's are uniformly in the Young measure-sense bounded (since the bound in (3.5) is independent of δ), and then let ε ↓ 0 to arrive at Hence, σ ∈ H and the Hahn-Banach argument described at the beginning of the proof yields the conclusion.

Characterization for Zero-Barycenter Singular Blow-Ups
Here, we use the following spaces (where Q = (−1/2, 1/2) d is the unit cube): Notice that for σ ∈ Y sing (0) we do not require one-directionality of y → σ ∞ y and λ σ . We also define the spaces Y  The proof of this fact proceeds essentially in the same way as the proof for Proposition 3.2 in the previous section with some straightforward modifications: (i) Wherever a direction a or ξ is needed, we use a, ξ = e 1 ; (ii) The proof of the analogue of Lemma 2.16 is exactly the same; (iii) In the proof of the Proposition 3.2, we can no longer assume that f H is onedirectional. Thus, we need to replace the slices S i partitioning Q a = Q with rescaled cubes Q i covering Q, for instance in a regular lattice; the same holds for the slices S(z 0 , r ). By averaging via Lemma 2.4, we may also conclude that we can get μ,μ (δ) ∈ BDY sing (0).

Proof of Theorem 1.1
First, we remark that the necessity part of our theorem is precisely the assertion of Theorem 4 of [38]. It remains to show the "sufficiency" part: holds for L d -almost every x ∈ , then ν ∈ BDY( ).
Proof. Note that we may additionally assume λ ν (∂ ) = 0 by embedding the problem into a larger domain and extending all involved maps by zero to this larger domain. This introduces an additional singular part, but this does not impinge the validity of (4.1) on the larger domain and nothing needs to be assumed on the singular part. We argue by considering regular and singular points separately.
Step 1. Let x 0 ∈ be a regular point, i.e. a point where the regular localization principle in Proposition 2.8 holds; this is the case for L d -almost every point of . From said result we get the existence of a regular tangent Young measure σ ∈ Y reg (P 0 ), where We claim that σ satisfies the Jensen-type inequality assumed in Proposition 3.1. Indeed, at L d -almost every y. Here we used (4.1) and the properties of regular blow-ups listed in Proposition 2.8. Thus, Proposition 3.1 yields that σ ∈ BDY reg (Q).
Step 2. By Lemma 2.14 at λ s ν -almost every x 0 ∈ , there exists a singular tangent Young measure σ ∈ Y sing (ξ η, ξ ) for some ξ ∈ S d−1 , η ∈ R d and with the properties listed in Proposition 2.9 and such that for some v ∈ BD(Q ξ ).
In particular, again by Lemma 2.14, there exists v ∈ BD(Q ξ ) with for some v 0 ∈ R d , β ∈ R, a function g ∈ BV(R), and a matrix R ∈ R d×d skew . Furthermore, we have that (by properties of blow-ups, see Theorem 2.44 in [4]) In particular, id, σ ∞ y = P 0 for λ σ -almost every y ∈ R d . Note that if P 0 = 0, then λ σ is one-directional since Ev = [Dg(x · ξ) + β](η ξ) is. Now, depending on whether P 0 = 0 or P 0 = a b (these are the only two possibilities by Lemma 2.14), our σ lies either in the space Y sing (P 0 , ξ) for ξ ∈ {a, b} or in the space Y sing (0). Also, we may assume that λ σ (∂ Q ξ ) = 0 by a simple rescaling argument (similar to the one described in Remark 2.11). Consequently, by either Proposition 3.2 or Proposition 3.7, we infer σ ∈ BDY sing (Q ξ ).
Our proposition, and thus Theorem 1.1, is now implied by the following lemma: there exists a (regular or singular) tangent Young measure σ x ∈ BDY(Q ξ(x) ) to ν at x for some ξ(x) ∈ S d−1 . Then, ν ∈ BDY( ).

Proof.
Step 1. We know that for every tangent Young measure σ , there exists a sequence of radii r n ↓ 0 and a sequence of constants c n > 0 such that σ (n) * σ for Here, Q(x 0 , r n ) = x 0 + r n Q and Q generically denotes the unit cube with one face normal to a or b if x 0 is a singular point with a b = 0 (see Lemma 2.14) or the standard unit cube if x 0 is a regular point or a singular point with a b = 0. We require also that λ ν (∂ Q(x 0 , r n )) = λ σ (∂ Q) = 0. We further define where T x 0 ,r n (x) := (x − x 0 )/r n and T x 0 ,r n # Eu := Eu • (T x 0 ,r n ) −1 = Eu(x 0 +r n q ) is the push-forward of Eu under T x 0 ,r n . Moreover, we can assume that by properties of blow-ups, see Lemma 3.1 of [39], there is v ∈ BD(Q) with [σ ] = Ev and such that u (n) → v strictly, where the error term E i can be estimated as x − x i r i , (4.11) and C and ω are as in (4.9). Here, we used that c −1 i = 1 Q i (x i ,r i ) ⊗ | q |, ν (cf. (2.7)) and (4.6).
Step 5.We will now show that ϕ (x)h (Ew k (x)) dx → ϕ ⊗ h , ν as k → ∞. (4.12) We start from x − x i r i dx =: I + I I (4.13) and we distinguish the cases where h has compact support or is positively 1homogeneous.
In the first case we use (4.8) and the error estimate (4.9) to get where E can be estimated by |E| ≤ e k | | + 1 ⊗ | q |, ν + Ew k L 1 and e k denotes a quantity that goes to 0 as k → ∞ and fixed. For the second term we have since the union of all Q i (x i , r i ) with x i ∈ S has asymptotically vanishing Lebesgue measure as k → ∞. Here, again,ê k denotes a quantity that goes to 0 as k → ∞ and fixed. Thus, combining (4.14) and (4.15) we have shown (4.12) for h compactly supported.
Let now h be positively 1-homogeneous. By using (4.8) and (4.9) the first term in (4.13) can be treated as in (4.14) to get where again e k → 0 as k → ∞ and fixed. For the second term we note that by (4.10) and (4.11) we have where as beforeê k → 0 as k → ∞ and fixed. Recalling that h = h ∞ by 1-homogeneity we deduce by (4.16) and (4.17) that (4.12) holds also in this case.

Atomic Parts of BD-Young Measures
As an application of the characterization theorem, we prove the following splitting result for generating sequences, a generalization of the result from Section 6 in [40] (the generalization can also be obtained for BV-Young measures): Theorem 5.1. Let ν ∈ BDY( ) with λ ν (∂ ) = 0 and v ∈ BD( ). Furthermore, assume that ν has E s v as an atomic part, that is