Existence of minimisers of variational problems posed in spaces of mixed smoothness
aa r X i v : . [ m a t h . A P ] F e b Existence of minimisers of variational problems posed inspaces of mixed smoothness
Adam Prosinski ∗ February 9, 2021
Abstract
The present work constitutes a first step towards establishing a systematic frame-work for treating variational problems that depend on a given input function througha mixture of its derivatives of different orders in different directions. For a fixed vector a := ( a , . . . , a N ) ∈ N N and u : R N ⊃ Ω → R n we denote by ∇ a u := ( ∂ α u ) h α, a − i =1 thematrix whose i -th row is composed of derivatives ∂ α u i of the i -th component of the map u , and where the multi-indices α satisfy h α, a − i = P Nj =1 α j a j = 1. We study functionalsof the form W a ,p (Ω; R n ) ∋ u ˆ Ω F ( ∇ a u ( x )) d x, where W a ,p (Ω; R n ) is an appropriate Sobolev space of mixed smoothness and F is theintegrand. We study existence of minimisers of such functionals under prescribed Dirich-let boundary conditions. We characterise coercivity, lower semicontiuity, and envelopesof relaxation of such functionals, in terms of an appropriate generalisation of Morrey’squasiconvexity. A classical problem in the calculus of variations is asserting existence of minimisers of inte-gral functionals of the form W ,p (Ω , R n ) ∋ u ´ Ω F ( ∇ u ) d x . Usually, the key difficultyis ensuring that the functional is sequentially lower semicontinuous in the appropriatetopology. Thanks to a long series of important contributions (see [1], [7], [27], [51], [57],[60] among others) we know that this depends on quasiconvexity of the integrand F . Inthe vector-valued case ( n >
1) quasiconvexity (introduced by Morrey in [61] and studiedin [2], [5], [8], [9], [24], [29], [31], [37], [48], [52], [62], [80]) is strictly weaker than ordinaryconvexity which is sufficient, but far from necessary, for lower semicontinuity when work-ing with gradients of Sobolev functions. This disparity is due to the special structure ofgradient vector fields, encompassed in the relation curl ∇ u = 0. This phenomenon is moregeneral than just gradients and, in the framework of Murat and Tartar’s compensatedcompactness ([63], [64] [81], [82], [83]) has led to another family of variational results (see,for instance, [3], [19], [39], [40], [74]) for functionals acting on vector fields v satisfying A v = 0 for a first-order constant-rank differential operator A .Without attempting to give a comprehensive account of the field (see instead thebooks [28], [38], [75]) let us underline that a common point of all these results, be it ∗ Carnegie Mellon University [email protected] radients, higher order gradients, or A -free fields, is that the partial differential operatorsinvolved are homogeneous of fixed order. This, however, need not always be the case inapplications. Pantographic sheets (see [32], [36], [84]) furnish an example of a recentlyintroduced metamaterial, characterised by energies of the form ´ | ∂ x u | + | ∂ yy u | d x , i.e.,with maximal derivatives of different orders in different directions. The aim of the presentpaper is to develop a systematic framework for studying such variational problems.We are interested in existence of minimisers of variational problems posed in Sobolevspaces of mixed smoothness. For a fixed vector a := ( a , . . . , a N ) with positive integercoordinates and a given function u : R N ⊃ Ω → R n we denote by ∇ a u := ( ∂ α u ) h α, a − i =1 the matrix whose i -th row is composed of derivatives ∂ α u i of the i -th component ofthe map u , and where the multi-indices α satisfy h α, a − i = P Nj =1 α j a j = 1. We studyfunctionals of the form W a ,p (Ω; R n ) ∋ u ˆ Ω F ( ∇ a u ( x )) d x, (1.1)where F is the integrand and W a ,p (Ω; R n ) is an appropriate Sobolev space of mixedsmoothness, the elements of which satisfy ∂ α u ∈ L p for all α with h α, a − i = 1.The central notion of this paper is that of a -quasiconvexity. We say that a function F : R n × m → [ −∞ , ∞ ) is a -quasiconvex if for every V ∈ R n × m one has F ( V ) inf u ∈ C ∞ c ( Q ; R n ) − ˆ Q F ( V + ∇ a u ( x )) d x. Our main results are on coercivity, lower semicontinuity, and relaxation of functionalsof the form (1.1). For continuous integrands F with | F ( V ) | C ( | V | p + 1) we prove inTheorem 4.3 that, having fixed a Dirichlet class W a ,pg (Ω), all minimising sequences of(1.1) are bounded if, and only if, there exists a constant c > V ∈ R n × m such that V F ( V ) − c | V | p is a -quasiconvex at V . Then, in Theorem 5.9, we showthat for continuous integrands F : R n × m → [0 , ∞ ) satisfying | F ( V ) | C ( | V | p + 1) thefunctional (1.1) is sequentially weakly lower semicontinuous on W a ,p (Ω) if, and only if, F is a -quasiconvex. Finally, in the last section, we study relaxations of (1.1) and in Theorem6.2 show that, if F is as before, then the sequentially weakly lower semicontinuous envelopeof (1.1) is again an integral functional andinf u j ⇀u (cid:26) lim inf j →∞ ˆ Ω F ( ∇ a u j ) d x (cid:27) = ˆ Ω Q F ( ∇ a u ( x )) d x, (1.2)where Q F is the a -quasiconvex envelope of F given by Q F ( V ) := inf ϕ ∈ C ∞ c (Ω) | Q | ˆ Q F ( V + ∇ a ϕ ( x )) d x. The infimum in (1.2) is taken over all sequences u j converging to u weakly in W a ,p (Ω),and Q F is the largest a -quasiconvex function that is no greater than F . This result isproven under the additional assumption that F is locally Lipschitz, or that it satisfies F ( V ) > c | V | p − C for some constants C, c > V ∈ R n × m . In the latter casewe are able, in Theorem 6.13, to remove the p -growth bound from above and obtain arelaxation formula also for integrands F that may take the value + ∞ .The starting point of this project is the theory of Sobolev spaces of mixed smoothness.The main question here is what other regularity and integrability properties follow when function u is assumed to be L p integrable, together with all its derivatives ∂ a i x i u forsome a = ( a , . . . , a N ). The theory of embeddings of spaces of mixed smoothness waslargely developed by Nikolskii, who started the study in [65]. Since then a number ofauthors have made important contributions to the theory of spaces of mixed smoothness,among which we recall [13], [14], [18], [21], [44], [49], [50], [69], [70], [77], [78], [79]. Letus note here that the list is far from being fully comprehensive. Instead we refer thereader to the two-volume book by Besov, Il’in, and Nikolskii (see [16] and [17]), whichwe will follow in most of the technical preliminaries. Let us note that, in their book,the authors call the spaces we work with anisotropic Sobolev spaces. For the purpose ofthe present article we have opted for the name Sobolev spaces of mixed smoothness toavoid confusion as to the nature of the anisotropy present in the problems we consider.Indeed, variational problems with different growth properties in different directions areoften called anisotropic in the existing literature.The arguments in the main part of the paper, i.e., the lower semicontinuity and re-laxation, are based on a Young measures approach. These measures were first introducedby Young in [85], [86], [87] and have since been used and studied by a number of au-thors, see [4], [6], [12], [51], [46], [47], and [58] among others. A classical reference thatcontains a much more complete bibliography is [67]. The prevalence of Young measuresin modern calculus of variations is due to the fact that they conveniently describe os-cillation effects (see [34] for a generalisation capable of describing concentration as well)that may occur in weakly, but not strongly, convergent sequences of vector fields. Thus,Young measures facilitate limit passages in nonlinear quantities, a crucial issue in thefield. Here, we are particularly interested in Young measures generated by sequences of a -gradients which, as shown in Theorems 3.16 and 3.17, can be characterised by dualitywith a -quasiconvex functions in the spirit of the Kinderlehrer-Pedregal ([46], [47]) resultfor classical gradients.Throughout the paper we make ample use of what has been done in the classicalgradient and A -free frameworks. Our main focus is on the new difficulties induced bythe mixed smoothness setting. To separate those from other technical issues, we workwith the model case of autonomous integrands F , that is ones that only depend on ∇ a u rather than lower order derivatives of u or on the spatial variable x . We are also veryliberal when it comes to assumptions on the domain Ω and we do not attempt to optimiseour results in that regard. We also limit ourselves to the reflexive regime W a ,p (Ω) with p ∈ (1 , ∞ ), although it would certainly be interesting to consider the case p = 1. Finally,whilst the present work establishes existence of minimisers, we do not say anything abouttheir regularity. This will be treated in an ongoing collaboration with Kristensen [55]. The present work is based on the author’s doctoral thesis [72] prepared under the super-vision of Prof. Jan Kristensen, whose support and guidance have been of immense help.The author would also like to thank Prof. Sir John Ball FRS, Prof. Gregory Seregin,Prof. Luc Nguyen, Prof. Gui-Qiang Chen, and Prof. Kewei Zhang, who have all acted asreferees for the thesis at various stages of its completion. The generous financial supportof Oxford EPSRC CDT in Partial Differential Equations, the Clarendon Fund, and StJohn’s College Oxford is gratefully acknowledged. Spaces of mixed smoothness
We begin by introducing the function spaces in which our variational problems are set. Wecollect basic facts about Sobolev spaces of mixed smoothness W a ,p which, while availablein the literature, might not be as classical and well-known as the corresponding theory ofthe usual Sobolev spaces W k,p . Let us note that, what we call Sobolev spaces of mixedsmoothness, is often referred to as ‘anisotropic Sobolev spaces’ in the literature that wecite. We have opted for the name ‘mixed smoothness’ as the term ‘anisotropic variationalproblems’ is already widespread and used to describe problems where the integrand andthe input functions exhibit different growth properties in derivatives in different directions.That is, ‘anisotropic variational problems’ usually refer to problems posed in the spaceW k, p with vector parameter p , rather than W a ,p with vector parameter a , which we areinterested in here. Nevertheless, our setting is certainly anisotropic and we shall use thisterm occasionally, particularly when talking about scaling. Any partial differentiation operator ∂ α := ∂ α ∂ α . . . ∂ α N N acting on functions mapping asubdomain of R N to R n may be identified with the multi-index α = ( α , . . . , α N ) ∈ Z N + ,where Z N + denotes the set of points in R N with non-negative integer coordinates.Specifying a set of derivatives A ⊂ Z N + and their desired integrability defines a Sobolev-like space — for example the classical W k,p Sobolev spaces correspond to A := { α ∈ Z N + : | α | k } with L p integrability on all the derivatives. In this work our principalassumption is that the functions in our spaces admit maximal pure derivatives in eachdirection, but the orders of these maximal derivatives remain arbitrary.Here and in all that follows, Ω is a bounded open Lipschitz subset of R N with | ∂ Ω | = 0,where | ∂ Ω | denotes the N -dimensional Lebesgue measure of the the boundary of Ω. Wedenote by C ∞ c (Ω , R n ) the space of smooth and compactly supported functions ϕ : Ω → R n .We often omit the target space when it is clear from the context and simply write C ∞ c (Ω).We fix a vector a = ( a , . . . , a N ) ∈ N N , where the respective entries a i denote thedesired maximal order of differentiability with respect to the x i coordinate. We fix anexponent p ∈ (1 , ∞ ), let a − := ( a − , . . . , a − N ), and introduce the following Definition 2.1.
For a bounded open set Ω ⊂ R N the Sobolev space W a ,p (Ω; R n ) is definedas the completion of C ∞ (Ω; R n ) ∩ { ϕ : P h α, a − i k ∂ α ϕ k L p (Ω; R n ) < ∞} with respect tothe norm k u k W a ,p := X h α, a − i k ∂ α u k L p (Ω; R n ) . We often omit the target space R n and write simply W a ,p (Ω) or even W a ,p if the domain Ω is clear from the context. We also denote by W a ,p (Ω; R n ) the completion of C ∞ c (Ω; R n ) in the same norm. The set { α ∈ Z N > : h α, a − i } has the important property that if α, β ∈ Z N > are twomulti-indices with β α (coordinate-wise) and α is in our set then so is β , which makesthe collection a smoothness in the language of Pe lczy´nski-Senator (see [69]). Moreover, allthe maximal elements of the smoothness lie on a common hyperplane { α : h α, a − i = 1 } ,thus allowing for convenient scaling, which we will discuss later. This hyperplane iscalled a pattern of homogeneity by Kazaniecki, Stolyarov, and Wojciechowski in [45]. Itis worth mentioning that their paper is the first one to introduce a simple version ofanisotropic quasiconvexity (to be discussed later) thus inspiring the present work. Let usobserve that, in our case, the hyperplane of homogeneity intersects all coordinate axes atinteger points (i.e. we have maximal pure derivatives in all directions), which is important or the structure of the relevant Sobolev spaces. Our exposition of the theory of thesespaces is based on the book [16] by Besov, Il’in, and Nikolskii. Finally, let us remarkthat the functions considered are, in general, vector-valued and we impose the samedifferentiability on all components of the functions considered. In general, it would beinteresting to allow for different smoothnesses in different components. This is, however,outside of the scope of the present paper. Proposition 2.2 (see [16]) . For p ∈ [1 , ∞ ) the space W a ,p (Ω; R n ) coincides with thespace of functions u ∈ L p (Ω; R n ) with distributional derivatives ∂ α u ∈ L p (Ω; R n ) for all h α, a − i . The spaces W a ,p (Ω) and W a ,p (Ω) are both separable Banach spaces. For p ∈ (1 , ∞ ) the two spaces are also reflexive. This is shown in [16], bar the reflexivity part, which may be immediately deducedby considering W a ,p (Ω; R n ) as a closed subspace of L h α, a − i L p ( R N ; R n ) through theembedding u ( ∂ α u ) h α, a − i . An important structural property of spaces of mixed smoothness is the existence of con-tinuous and compact embeddings, similar to the classical Sobolev embeddings. To beginwith, we note that, as in the classical case of, say W ,p (Ω), there are certain regularityassumptions that one must impose on the domain Ω. Definition 2.3.
Let b ∈ R N be a vector with non-zero coordinates. Fix h ∈ (0 , ∞ ) and ε ∈ (0 , ∞ ) . The set V ( b, h, ε ) := [
Let Ω ⊂ R N be open and let K ∈ N . Suppose that for k ∈ { , , . . . , K } there exist open sets Ω k and a -horns V k (with coefficients b k , h k , ε k depending on k ) suchthat Ω = K [ k =1 Ω k = K [ k =1 (Ω k + V k ) . Then we say that Ω satisfies the weak a -horn condition. Theorem 2.5 (see Theorem 9.5 in [16]) . Suppose that an open set Ω ⊂ R N satisfies theweak a -horn condition and let p ∈ (1 , ∞ ) . Then there exists a real number h ∈ (0 , ∞ ) depending on Ω and a constant C such that, for all h ∈ (0 , h ) and all u ∈ W a ,p (Ω) , onehas, for all multi-indices β ∈ Z N + with h β, a − i , that k ∂ β u k p C h −h β, a − i N X i =1 k ∂ a i i u k p + h −h β, a − i k u k p ! . Thus, on domains satisfying the weak a -horn condition (see [15] where the condi-tion was first studied) the intermediate derivatives are controlled by the maximal purederivatives and the function itself, so that the relevant Sobolev space is well behaved.In the isotropic case (i.e., a i = a j for all i, j ) the a -horn is in fact a cone andhorn conditions are equivalent to the, more familiar, cone conditions. In the genuinelyanisotropic scenario the a -horn condition is more surprising. For instance, (see [16]) thetwo-dimensional disc only satisfies the weak a -horn condition if a a a . Fortu-nately, there are no issues when working with rectangular domains, which we note in thefollowing: emma 2.6 (see [16]) . Any set of the form ( l , r ) × . . . × ( l N , r N ) ⊂ R N for some l i , r i ∈ R satisfies the weak a -horn condition. Before we proceed, let us note several important consequences of the embeddings:
Proposition 2.7.
Let Ω ⊂ R N be a bounded open set with a Lipschitz boundary. Thenthe embedding of W a ,p (Ω) into L p (Ω) is compact.Proof. For any a one has the inclusion W a ,p (Ω) ⊂ W ,p (Ω) into the standard Sobolevspace. Since the inclusion W ,p (Ω) ⊂ L p (Ω) is compact the proof is finished. Lemma 2.8.
Suppose that a bounded open set Ω ⊂ R N with a Lipschitz boundary satisfiesthe weak a -horn condition and let p ∈ (1 , ∞ ) . Then for any β with h β, a − i < themapping W a ,p (Ω) ֒ → L p (Ω) given by u ∂ β u is completely continuous, i.e., if u j ⇀ u in W a ,p (Ω) then ∂ β u j → ∂ β u in L p (Ω) .Proof. Considering u j − u instead of u j we may assume that u = 0. Fix a β with h β, a − i < h > h ∈ (0 , h ), one has k ∂ β u j k p C h −h β, a − i X h α, a − i =1 k ∂ α u k p + h −h β, a − i k u j k p . Proposition 2.7 implies that u j converges strongly to 0 in L p . Hence, there exists asequence h j ∈ (0 , h ) with h j → Ch −h β, a − i j k u j k p → j → ∞ . Finally,because u j is bounded in W a ,p , we know that (cid:16)P h α, a − i =1 k ∂ α u k p (cid:17) is bounded, andsince Ch −h β, a − i j → k ∂ β u j k p →
0, which ends the proof.Another consequence of Theorem 2.5 is
Corollary 2.9.
Let Ω ⊂ R N satisfy the weak a -horn condition. Then all of the following k u k := k u k p + N X i =1 k ∂ a i i u k p , k u k := k u k p + X h α, a − i =1 k ∂ α u k p , k u k := X h β, a − i k ∂ β u k p , (2.1) yield equivalent norms on W a ,p (Ω) . In closing this subsection let us note that the study of embeddings for spaces of mixedsmoothness has been started by Besov and Il’in in [15]. Together with Nikolskii, theseauthors have expanded, compiled, and clarified the theory in [16]. It is, however, not theonly relevant source. Similar questions have been studied, among others, by Boman (see[18]) who also treats the case p = ∞ , as well as Demidenko and Upsenskii (see [33]) whoworked on quasielliptic operators, a class important in regularity of solutions to mixedsmoothness variational problems, which we will return to in a forthcoming paper. Finally,in the particular case of rectangular sets (like in Lemma 2.6) a bounded extension maybe constructed using the Hestenes’ method (as observed by Burenkov and Fain in [22])which then, together with embeddings on the full space, yields existence of embeddingson rectangular domains as well. .3 Canonical Projection For u ∈ W a ,p we write ∇ a u for the a -gradient of u given by ∇ a u := ( ∂ α u ) h α, a − i =1 . Forfuture use let us denote the cardinality of the set { α ∈ Z + : h α, a − i = 1 } by m , so thatfor u ∈ W a ,p (Ω , R n ) the a -gradient is a map defined on Ω with values in R n × m . In whatfollows we will often need to carry out certain operations, for example truncations, on a -gradients of various functions. These are easy to do on mappings Ω → R n × m , but theyneed not preserve the a -gradient structure, and to remedy that we turn to CanonicalSobolev Projections following Pe lczy´nski’s work in [68]. The aim is to obtain an analogueof the Helmholtz decomposition for the mixed smoothness setting. We will then use it forregularizing generating sequences of certain Young measures, similarly to what has beendone by Fonseca and M¨uller in [40] in the case of A -free vector fields and A -quasiconvexity.There is a canonical embeddingW a ,p ( R N ; R n ) → M h α, a − i L p ( R N ; R n )given by u ( ∂ α u ) h α, a − i , but it is not surjective. With p = 2 one may define thecanonical projection of the target space onto the image of this embedding, i.e.,P a : M h α, a − i L ( R N ; R n ) → Im W a , ( R N ; R n ) → M h α, a − i L ( R N ; R n ) . (2.2)It has been shown in [68] (see Corollary 5.1 therein) that the projection from (2.2) is ofstrong type ( p, p ) for 1 < p < ∞ , thus one can extend it by continuity from M h α, a − i L ( R N ; R n ) → M h α, a − i L ( R N ; R n )to M h α, a − i L p ( R N ; R n ) → M h α, a − i L p ( R N ; R n ) . Lemma 2.10.
Fix any p ∈ (1 , ∞ ) and denote by P a the extension of the canonicalprojection discussed above. Then:i) the map P a is a bounded linear operator on L h α, a − i L p ( R N ; R n ) ;ii) for any V ∈ L h α, a − i L p we have P a (P a V ) = P a V ;iii) if the family { V j } ⊂ L h α, a − i L p is p -equiintegrable then so is { P a V j } .Proof. The first assertion is the content of Corollary 5.1 in [68]. Point ii) is, by definition,true in L h α, a − i L ( R N ; R n ). For a general exponent p let us fix V ∈ L h α, a − i L p ( R N ; R n )and a family V j ⊂ L h α, a − i L ( R N ; R n ) with k V − V j k p →
0. By continuity of P a (andthus of P a ◦ P a ) one hasP a V j → P a V and P a (P a V j ) → P a (P a V ) in L p ( R N ; R n ) , and since P a V j = P a (P a V j ) for each j the claim is proven.For the last part consider the standard truncation τ k given by τ k ( X ) := ( X if | X | k,k X | X | if | X | > k. (2.3)Then fix any sequence V j satisfying the assumptions of point iii). Since { τ k ( V j ) } isbounded in L ∞ and in L p we know, by continuity of P a as a map from L q to L q , that P a τ k ( V j ) } is bounded in any L q with p q < ∞ , so that this family is equiintegrable inL p . Then again, p -equiintegrability of { V j } itself yieldslim k →∞ sup j || V j − τ k ( V j ) || p = 0 , so again continuity of P a : L p → L p giveslim k →∞ sup j || P a ( V j − τ k ( V j )) || p = 0 , hence { P a ( V j ) } is p -equiintegrable as claimed. In what follows we will often need to use a specific anisotropic scaling. For a real number
R > v = ( v , v , . . . , v N ) ∈ R N we let R ⊙ v := ( R /a v , R /a v , . . . , R /a N v N ) . Here, and in all that follows, Q ⊂ R N will, unless otherwise specified, denote [ − , N .We let Q R ( x ) ⊂ R N be the open box centred at x ∈ R N and scaled according to therule just described, so that Q R ( x ) := { x ∈ R N : | x i − x i | a i < R for all 1 i N } . Equivalently, we could write Q R ( x ) = x + R ⊙ Q . We call R the anisotropic radius ofthe box Q R and x its centre. From now on we will always understand a ‘box’ to meana set of the form above, i.e., an anisotropically scaled and translated unit cube.Observe that, unless the scaling is in fact isotropic (that is, a i = a j for all i, j ), ourfamily of boxes is not of bounded eccentricity, i.e., there does not exist a constant c > Q R ( x ) in our collection is contained in some (Euclidean) ball B with | Q R ( x ) | > c | B | . Therefore, it is not obvious if standard results such as the Vitali coveringlemma, or the Lebesgue differentiation theorem, hold for balls replaced by anisotropicallyscaled boxes. Note that even sharper statements of the Vitali covering lemma, such asthe one in [76] by Saks (which only requires the eccentricity to be bounded along fixedsequences converging to a given point), or the one in [59] by Mejlbro and Topsøe (wherethe condition on the eccentricity is in integral form), are not directly applicable.Nevertheless, these results may be proven in a straightforward way — it suffices tofollow the proof of Vitali’s covering lemma given in [11], and the Lebesgue differentiationtheorem (which is what we are after) follows immediately. In fact, the result could alsobe deduced as a special case of the more general work by Calder´on and Torchinsky (see[25]), thus we omit the proof of the following: Theorem 2.11 (Anisotropic Lebesgue differentiation theorem) . Let f ∈ L loc ( R N ) . ForLebesgue almost every x ∈ R N one has lim sup R → | Q R ( x ) | ˆ Q R ( x ) | f ( x ) − f ( x ) | d x = 0 . .5 Polynomial approximation For a function f ∈ L ( Q R ( x )) we denote by ( f ) Q R ( x ) its average over Q R ( x ), i.e.,( f ) Q R ( x ) := − ˆ Q R ( x ) f ( x ) d x. We often write simply Q R if the center is not important or clear from the context. Lemma 2.12.
There exists a constant C such that, for any r > and any σ ∈ (0 , ) ,there exists a cut-off function η ∈ C ∞ c ( Q r ; [0 , which is identically equal to on Q (1 − σ ) r and satisfies k ∂ β η k L ∞ Cr −h β, a − i σ −| β | , for all multi-indices β with h β, a − i .Proof. It is enough to consider the case r = 1, as the general result will then follow byscaling. With r = 1 observe that the distance between the faces of Q and Q − σ along the x i axis is equal to 1 − (1 − σ ) /a i . For sufficiently small C the function Cσ + (1 − σ ) /a i is decreasing in σ , so there exists a constant C > σ ∈ (0 , ), we have1 − (1 − σ ) /a i > Cσ, for all i . Now it is enough to construct one dimensional cut-off functions η i ( x i ) thatrealise the desired cut-off along the particular axes and satisfy k ∂ k η i k L ∞ Cσ ) − k . Wethen conclude by setting η ( x ) := Q Ni =1 η i ( x i ).We recall the following version of the Poincar´e inequality in W a ,p proven by Dupontand Scott in [35], where, instead of requiring zero boundary values, we allow for correctionin terms of the kernel of the operator ∇ a . Proposition 2.13 (see Theorem 4.2 in [35]) . There exists a constant C such that forany function u ∈ W a ,p ( Q ) with p ∈ [1 , ∞ ) there exists a polynomial P u ∈ C ∞ ( Q ) with ∇ a P u ≡ satisfying k u − P u k W a ,p ( Q ) C k∇ a u k L p ( Q ) . Observe that, in the above, Q is fixed to be the unit cube, and we do not assertanything about approximations on other domains. However, we will only ever use thisresult on anisotropic boxes, and it is easy to see how to adjust the constant to scaling, asshown in the following: Corollary 2.14.
There exists a constant C such that for any function u ∈ W a ,p ( Q r ) with p ∈ [1 , ∞ ) there exists a polynomial P u ∈ C ∞ ( Q r ) with ∇ a P u ≡ ( ∇ a u ) Q r such thatfor any β with h β, a − i we have r − h β, a − i k ∂ β ( u − P u ) k L p ( Q r ) C k∇ a ( u − P u ) k L p ( Q r ) . The constant C does not depend on the function u nor the radius r .Proof. First of all, note that using our anisotropic rescaling we may reduce to the case r = 1. This also determines the scaling, i.e., the r − h β, a − i factor. Secondly, it is clearlyenough to prove the result for u ∈ C ∞ ( Q ), as the general case then follows from densityof smooth functions in W a ,p ( Q ). Observe that by considering e u ( x ) := u ( x ) − X h α, a − i =1 x α ( ∂ α u ) Q , e reduce our task to finding a polynomial e P e u with ∇ a e P e u ≡ k ∂ β ( e u − e P e u ) k L p ( Q ) C k∇ a e u k L p ( Q ) for all β with h β, a − i <
1, as the case h β, a − i = 1 is trivial. The existence of such a e P e u is the content of Proposition 2.13, which completes the proof. Proposition 2.15 (see Theorem 10.16 in [30]) . For any bounded open Lipschitz domain Ω , any u ∈ W a ,p (Ω) and any ε > there exist a function u ε ∈ W a ,pu (Ω) and a finite familyof disjoint boxes { Q ε,i } i such that ∇ a u ε is constant on each Q ε,i ⊂ Ω , (cid:12)(cid:12) Ω \ S i Q ε,i (cid:12)(cid:12) < ε ,and k u − u ε k W a ,p (Ω) < ε .Proof. Fix an arbitrary u ∈ W a ,p (Ω) and a parameter τ ∈ (0 ,
1) to be determined later.Decompose Ω, up to a set of measure zero, into a countable family of disjoint, open boxes { Q τ,i } of radii equal, or smaller than, τ . Select a finite subfamily of I boxes covering Ω upto a set of measure less than ε/ (cid:12)(cid:12)(cid:12) Ω \ S Ii =1 Q τ,i (cid:12)(cid:12)(cid:12) < ε/ Q τ,i with 1 i I . Let P τ,i denotethe polynomial approximating u on Q τ,i given by Corollary 2.14. Let σ ∈ (0 , /
2) be aparameter to be determined later. For every Q τ,i take a cut-off function η τ,i ∈ C ∞ c ( Q τ,i )identically equal to 1 on (1 − σ ) ⊙ Q τ,i , as in Lemma 2.12. Define v ( x ) := u ( x ) + I X i =1 η τ,i ( x ) ( P τ,i ( x ) − u ( x )) , so that ∇ a v is constant on each (1 − σ ) ⊙ Q τ,i and v ∈ W a ,pu (Ω). We may now calculate k u − v k p W a ,p (Ω) = X h β, a − i ˆ Ω | ∂ β ( u − v ) | p d x = X h β, a − i I X i =1 ˆ Q τ,i (cid:12)(cid:12)(cid:12) ∂ β ( η τ,i ( x ) ( u ( x ) − P τ,i ( x ))) (cid:12)(cid:12)(cid:12) p d x X h β, a − i I X i =1 ˆ Q τ,i X γ β | ∂ γ η τ,i ( x ) | p (cid:12)(cid:12)(cid:12) ∂ β − γ ( u ( x ) − P τ,i ( x )) (cid:12)(cid:12)(cid:12) p d x. Using the bounds on the derivatives of η we may write k u − v k p W a ,p (Ω) C X h β, a − i X γ β I X i =1 σ − p | β | τ − p h γ, a − i ˆ Q τ,i (cid:12)(cid:12)(cid:12) ∂ β − γ ( u ( x ) − P τ,i ( x )) (cid:12)(cid:12)(cid:12) p d x. Using the bound from Corollary 2.14 on each Q τ,i now yields k u − v k p W a ,p (Ω) C X h β, a − i X γ β I X i =1 σ − p | β | τ − p h γ, a − i τ p − p h β − γ, a − i k∇ a ( u − P τ,i ) k p L p ( Q τ,i ) . Thus, k u − v k p W a ,p (Ω) C X h β, a − i X γ β I X i =1 σ − p | β | τ p − p h β, a − i k∇ a ( u − P τ,i ) k p L p ( Q τ,i ) , nd finally, since h β, a − i σ < /
2, and τ <
1, we may write k u − v k p W a ,p (Ω) Cσ − p max j a j I X i =1 k∇ a ( u − P τ,i ) k p L p ( Q τ,i ) . Now it is time to choose the parameters σ and τ . Observe that | (1 − σ ) ⊙ Q τ,i || Q τ,i | = (1 − σ ) | a − | for any τ and any i . Thus, choosing σ small enough ensures that, with any τ , we willhave (cid:12)(cid:12) Ω \ (cid:0)S i (1 − σ ) ⊙ Q τ,i (cid:1)(cid:12)(cid:12) < ε . With σ fixed it is enough to choose τ small enough sothat I X i =1 k∇ a ( u − P τ,i ) k p L p ( Q τ,i ) C − σ p max j a j ε, which is possible as, thanks to Theorem 2.11, one can approximate ∇ a u in the L p normby its averages over a grid of boxes of sufficiently small radii. Using the corresponding v as u ε ends the proof. The main technical tool that we will use in studying lower semicontinuity of integral func-tionals defined on Sobolev spaces of mixed smoothness is the theory of Young measures(see [85], [86], [87], and [88]). These measures describe the behaviour of weakly converg-ing sequences more accurately than just their weak limits and facilitate limit passages innonlinear quantities.In this section we introduce, and study, the oscillation Young measures generatedby weakly convergent sequences of a -gradients. In doing so we follow the strategies ofKristensen in [54] (who studies classical gradients) and Fonseca and M¨uller in [40] (whowork with A -free vector fields). In what follows we focus on the adjustments required bythe mixed smoothness setting whilst skipping the technical parts that translate withoutmajor adaptations. We refer the reader to the author’s thesis [72], where a full step-by-step exposition may be found. We denote by M ( R n × m ) the set of all Radon measures on R n × m , and by P ( R n × m ) ⊂M ( R n × m ) the set of all probability measures on R n × m . We say that a function F : Ω × R n × m → ( −∞ , ∞ ] is a normal integrand if F is Borel measurable and, for every fixed x ∈ Ω, the function W F ( x, W ) is lower semicontinuous. Similarly, we say that a function F : Ω × R n × m → R is Carath´eodory if both F and − F are normal integrands. Finally,a map ν : Ω → M ( R n × m ) is said to be weak*-measurable if x
7→ h ν x , ϕ i is (Lebesgue)measurable for any continuous and compactly supported function ϕ : R n × m → R .We begin with the following version of the Fundamental Theorem of Young Measureswhich may be found, for example, in Pedregal’s book, (see [67], followed by a simple resulton translations. Theorem 3.1 (see [67]) . Let Ω ⊂ R N be a measurable set of finite measure and V j : Ω → R n × m be a bounded sequence of L p functions for some p ∈ [1 , ∞ ] . Then there exists asubsequence V j k and a weak ∗ -measurable map ν : Ω → M ( R n × m ) such that the followinghold:i) every ν x is a probability measure; i) if F : Ω × R n × m → R ∪ {∞} is a normal integrand bounded from below, then lim inf j →∞ ˆ Ω F ( x, V j k ( x )) d x > ˆ Ω F ( x ) d x, where F ( x ) := h ν x , F ( x, · ) i = ˆ R n × m F ( x, y ) d ν x ( y ); iii) if F : Ω × R n × m → R ∪ {∞} is Carath´eodory and bounded from below, then lim j →∞ ˆ Ω F ( x, V j k ( x )) d x = ˆ Ω F ( x ) d x < ∞ if and only if { F ( · , V j k ( · )) } is equiintegrable (in the usual L sense). In this case F ( · , V j k ( · )) ⇀ F in L (Ω) · The family { ν x } x ∈ Ω is called the (oscillation) Young measure generated by V j k . If thereexists some x ∈ Ω such that ν x = ν x for almost every x ∈ Ω then we say that ν is ahomogeneous Young measure and often identify the family { ν x } with the single measure ν x if there is no risk of confusion. Proposition 3.2 (see [67]) . If { V j } generates an oscillation Young measure ν and if W j → W in measure, then { V j + W j } generates the translated Young measure e ν x := δ W ( x ) ∗ ν x , where h δ U ∗ µ, ϕ i = h µ, ϕ ( · + U ) i for U ∈ R n × m and ϕ ∈ C ( R n × m ) . In particular, if W j → in measure, then { V j + W j } still generates ν . Similarly, if k V j − W j k p → for some p ∈ [1 , ∞ ] then both V j and W j generate the same Young measure. First, following the approach from Kristensen’s lecture notes (see [54]), we show how todecompose a given weakly convergent sequence of a -gradients into a p -equiintegrable os-cillation part supported away from the boundary, and a concentration part that convergesto 0 in measure.In this section we denote by f ∇ a u the full gradient of u given by f ∇ a u := ( ∂ α u ) h α, a − i and by c ∇ a u its lower gradient c ∇ a u := ( ∂ α u ) h α, a − i < , so that f ∇ a u = c ∇ a u ⊕ ∇ a u . Wealso set d := (cid:12)(cid:12)(cid:8) α : h α, a − i (cid:9)(cid:12)(cid:12) so that for u : Ω → R n we have f ∇ a u : Ω → R d × n and c ∇ a u : Ω → R ( d − m ) × n . Definition 3.3.
For p ∈ (1 , ∞ ) we say that a family of functions { V k } ⊂ L p (Ω , R n ) is p -equiintegrable if the family {| V k | p } ⊂ L (Ω; R ) is equiintegrable in the usual sense. Lemma 3.4.
Let Ω ⊂ R N be a bounded open Lipschitz set satisfying the weak a -horncondition. Then, for any sequence u j ⇀ in W a ,p (Ω) , there exists a sequence v j ∈ C ∞ c (Ω) such that the sequence ( f ∇ a u j − f ∇ a v j ) converges to in measure. In particular, if ∇ a u j (or,equivalently, f ∇ a u j ) generates some oscillation Young measure ν then ∇ a v j (respectively f ∇ a v j ) also generates ν . Furthermore, if { f ∇ a u j } is p -equiintegrable then so is { f ∇ a v j } . roof. Take an increasing family of smooth, open sub-domains Ω k ⋐ Ω k +1 ⋐ Ω (here ⋐ denotes compact inclusion) with S ∞ k =1 Ω k = Ω. Fix a family of cut-off functions ϕ k ∈ C ∞ c (Ω; [0 , ϕ k ≡ k and denote M k := || f ∇ a ϕ k || L ∞ (Ω) < ∞ .For any k, j we may write u j = ϕ k u j + (1 − ϕ k ) u j , and the goal is to show that thesecond term is small. We have ˆ Ω | f ∇ a ((1 − ϕ k ) u j ) | d x = ˆ Ω \ Ω k | f ∇ a ((1 − ϕ k ) u j ) | d x, because on Ω k the integrand is identically equal to 0. Differentiating the product wedistinguish between the case where all the derivatives fall on u j and the one where wealso differentiate (1 − ϕ k ), which yields ˆ Ω | f ∇ a ((1 − ϕ k ) u j ) | d x ˆ Ω \ Ω k (1 − ϕ k ) | f ∇ a u j | d x + ˆ Ω \ Ω k M k | c ∇ a u j | d x. Since the family {| f ∇ a u j |} is bounded in L p , it is uniformly integrable in L . Addingthe fact that | − ϕ k | | Ω \ Ω k | → k → ∞ , uniformly in j . On the other hand, Lemma 2.8 showsthat the lower gradients c ∇ a u j converge to 0 strongly in L p , thus in particular in L .Therefore, there exists a sequence k j → ∞ such that ´ Ω \ Ω kj M k j | c ∇ a u j | d x → j → ∞ . Combining the two we see that f ∇ a ((1 − ϕ k j ) u j → .Repeating the above reasoning with L norm replaced by L p we get ˆ Ω | f ∇ a ((1 − ϕ k ) u j ) | p d x C ˆ Ω \ Ω k (1 − ϕ k ) p | f ∇ a u j | p d x + C ˆ Ω \ Ω k M pk | c ∇ a u j | p d x, (3.1)with some absolute constant C . The first term is now bounded uniformly in k and j ,whereas for the second one we may select a sequence k ′ j such that it converges to 0.Hence, adjusting the first sequence k j (i.e., slowing it down if necessary) we obtain that(1 − ϕ k j ) u j is bounded in W a ,p .Combining the two we deduce that (1 − ϕ k j ) u j ⇀ a ,p , since the sequence isbounded and the only possible limit is 0, as the full gradient converges to 0 in L . Byconstruction, the sequence also converges to 0 in measure, thus setting v j := ϕ k j u j endsthe proof of the first part of the statement.For equiintegrability it is enough to notice that if { f ∇ a u j } is p -equiintegrable then thefirst term in (3.1) converges to 0 with k → ∞ , uniformly in j (as (1 − ϕ k ) is bounded inL ∞ and | Ω \ Ω k | → f ∇ a u j − f ∇ a v j converges to 0 strongly in L p ,which proves p -equiintegrability of f ∇ a v j .The following result is the key point of this section. It shows that (up to a subsequence)one may decompose a weakly convergent sequence into a p -equiintegrable part that carriesthe oscillation and a part converging to 0 in measure, which carries the concentration. Proposition 3.5.
Let Ω ⊂ R N be a bounded open Lipschitz domain satisfying the weak a -horn condition and let p ∈ (1 , ∞ ) . Suppose that u j ⇀ u in W a ,p (Ω; R n ) . Then, thereexists a subsequence u j k and sequences { g k } ⊂ C ∞ c (Ω; R n ) and { b k } ⊂ W a ,p (Ω; R n ) , bothweakly convergent to in W a ,p (Ω; R n ) , and such that the family f ∇ a g k is p -equiintegrable, f ∇ a b k → in measure, and u j k = u + g k + b k .Proof. By considering { u j − u } instead of { u j } we reduce to the case u ≡
0. Furthermore,taking a subsequence if necessary, we may assume that f ∇ a u j generates some oscillationYoung measure ν . Lemma 3.4 shows that we may also take u j = v j + b j with b j → n measure and v j ∈ W a ,p (Ω). Thus, in what follows we focus on decomposing v j ,remembering that f ∇ a v j generates the same Young measure as f ∇ a u j .For l ∈ N recall that the standard truncation τ l : R d × n → R d × n is given by τ l ( W ) := ( W if | W | l,l W | W | if | W | > l. Since the truncation is bounded and continuous we getlim l →∞ lim j →∞ ˆ Ω | τ l ( f ∇ a v j ) | p d x = lim j →∞ ˆ Ω ˆ R d × n | τ l ( W ) | p d ν x d x = ˆ Ω ˆ R d × n | · | p d ν x d x, where the first equality is due to Theorem 3.1 and the second one is an application of theMonotone Convergence Theorem. We infer that one can extract a sequence j l → ∞ suchthat lim l →∞ ˆ Ω | τ l ( f ∇ a v j l ) | p d x = ˆ Ω ˆ R d × n | · | p d ν x d x. Note that L p boundedness of f ∇ a v j l implies equiintegrability in L q for q ∈ [1 , p ), hence τ l ( f ∇ a v j l ) − f ∇ a v j l → q , and thus the two generate the same oscillationYoung measure ν . This, paired with the last equality and Theorem 3.1, implies that thefamily { τ l ( f ∇ a v j l } is p -equiintegrable.In this way we have constructed a sequence w l := τ l ( f ∇ a v j l ) that is p -equiintegrable andgenerates ν . The last thing we need to take care of is the fact that w l is not necessarilya full gradient of a W a ,p function. To remedy that, extend w l by 0 to the whole of R N and do the same with v j l , keeping the same notation for the extensions. Since wehad v j l ∈ W a ,p (Ω), its extension by 0 is in the space W a ,p ( R N ). Apply the canonicalprojection P a to w l to get the decomposition w l = f ∇ a g l + r l . We have k r l k L q = k w l − f ∇ a g l k L q k w l − f ∇ a v j l k L q + k f ∇ a g l − f ∇ a v j l k L q = k w l − f ∇ a v j l k L q + k P a ( w l − f ∇ a v j l ) k L q C k w l − f ∇ a v j l k L q → , where we have used P a ( f ∇ a v j l ) = f ∇ a v j l and the L q continuity of P a . In particular, thisgives r l → p -equiintegrability of { w l } yields the same for { f ∇ a g l } . Lastly, we restrict g l to Ω to get g l ∈ W a ,p (Ω) and apply thecut-off argument from Lemma 3.4 to end the proof.The following is a simple and useful corollary of the above. Corollary 3.6.
Let Ω be a bounded open Lipschitz domain satisfying the weak a -horncondition. Let ν be a W a ,p -gradient oscillation Young measure on Ω . Then there exists asequence u j ∈ W a ,p (Ω) generating ν and such that the family {∇ a u j } is p -equiintegrable.Furthermore, if the barycentre of ν is at all points of Ω , then the functions u j may bechosen in the space C ∞ c (Ω) . The final technical ingredient of this subsection is the following localisation result. Weomit the proof and instead refer the reader to [72] where a full argument may be found. Itis simply a matter of following Kristensen’s approach (see [54]) and replacing the technicaltools such as decomposition of generating sequences, polynomial approximations, andscalings, by their mixed smoothness counterparts stated and proven above. roposition 3.7. Let Ω ⊂ R N be an open and bounded domain. Fix < p < ∞ and let ν = ( ν x ) x ∈ Ω be a W a ,p -gradient Young measure on Ω . Then for L n -a.e. x ∈ Ω , ( ν x ) y ∈ Q is a homogeneous W a ,p -gradient Young measure on the unit cube Q . Its barycentre is ν x Q . This section aims to develop the mixed smoothness equivalent of quasiconvexity intro-duced first by Morrey in [61] (see also [28] for a broad list of references). For now, wepresent only a few background results that we will need to establish the dual characteri-sation of W a ,p -gradient Young measures. We will revisit a -quasiconvexity in more detailin the next section, when we study lower semicontinuity properties of functionals. Definition 3.8.
We say that a function g : R n × m → [ −∞ , ∞ ) is a -quasiconvex if forevery V ∈ R n × m one has g ( V ) inf u ∈ C ∞ c ( Q ; R n ) − ˆ Q g ( V + ∇ a u ( x )) d x. For functions that need not be a -quasiconvex we introduce the following: Definition 3.9.
For a measurable function g : R n × m → [ −∞ , ∞ ) we define the function Q g : R n × m → [ −∞ ; ∞ ) by Q g ( V ) := inf (cid:26) − ˆ Q g ( V + ∇ a u ( x )) d x : u ∈ C ∞ c ( Q ) (cid:27) . (3.2)The expression on the right-hand side of the above definition is often called Da-corogna’s formula in the standard first order gradient case (see [28]). In what follows wewill often refer to Q g as the a -quasiconvex envelope of g , and the next lemma justifies thisterminology, by asserting that Q g is the largest a -quasiconvex function that is smallerthan or equal to g . By definition Q g g , simply by testing the definition with u ≡ a -quasiconvex function that isno bigger than g must be no bigger than Q g . Thus it only remains to show that Q g is a -quasiconvex. This can be done following the approach of Fonseca and M¨uller from[40]. One only needs to use the anisotropic scaling we introduced earlier instead of theclassical isotropic one. Thus, for the sake of brevity, we simply state the result and referthe reader to [72] for the full argument. Lemma 3.10.
For a continuous function g : R n × m → R we have Q ( Q g ) = Q g, that is, Q g is indeed a -quasiconvex. W a ,p -gradient Young measures Our next aim is to obtain a dual characterisation of oscillation W a ,p -gradient Young mea-sures in terms of W a ,p -quasiconvex functions. The root of the principal results (Theorems3.16 and 3.17) goes back to the seminal work of Kinderlehrer and Pedregal (see [46] and[47]). Our approach, however, is based on [40] and the proofs we give are adaptations ofthe techniques presented therein — for the sake of brevity we focus on the main pointsthat require adaptation to our mixed smoothness setting, and refer the reader to [72] forthe full arguments. n the following we work with the space E p defined as E p := (cid:26) g ∈ C ( R n × m ) : lim | W |→∞ g ( W )1 + | W | p exists in R (cid:27) , and equipped with the norm k g k E p := sup W ∈ R n × m | g ( W ) | | W | p . It is not difficult to see that E p is a separable Banach space. Furthermore, the space ofprobability measures with finite p -th moment is a subset of E ∗ p through the pairing h ν, g i := ˆ R n × m g ( W ) d ν ( W ) , for ν ∈ { µ ∈ P ( R n × m ) : ´ R n × m | W | p d ν ( W ) < ∞} and g ∈ E p . In particular, the space ofhomogeneous W a ,p -gradient Young measures is a subset of E ∗ p . For future use, we notethe following technical result. Lemma 3.11.
Fix a bounded open Lipschitz set Ω ⊂ R N and a function ϕ ∈ W a ,p ( Q ) .Suppose that for every j ∈ N we have fixed a countable family { Q ji } i = { Q x ji ( r ji ) } i ofpairwise disjoint anisotropic boxes contained in Ω and with lim j →∞ sup i r ji = 0 . Assumefurthermore that lim j →∞ | Ω \ S i Q ji | = 0 . Define ϕ j := X i r ji ϕ (( r ji ) − ⊙ ( x − x ji )) . Then ϕ j ⇀ weakly in W a ,p (Ω) . The sequence ∇ a ϕ j is p -equiintegrable and generatesthe homogeneous Young measure ∇ a ϕ (cid:16) L N ¬ Q | Q | (cid:17) .Proof. First of all note that the function ϕ j is well-defined as an element of W a ,p (Ω). Tosee this observe that for every h β, a − i i we have k ∂ β [ r ji ϕ (( r ji ) − ⊙ x )] k L p ( Q ji ) = ( r ji ) −h β, a − i | Q ji || Q | k ∂ β ϕ k L p ( Q ) . This also shows that the sequence ϕ j converges strongly to 0 in L p (Ω) and that it isbounded in W a ,p (Ω), thus converges weakly to 0 in that space. The p -equiintegrabilityof {∇ a ϕ j } follows from the fact that for any M > j ˆ Ω |∇ a ϕ j | p |∇ a ϕ j | >M d x = | Ω || Q | ˆ Q |∇ a ϕ | p |∇ a ϕ | >M d x, which we get by a simple change of variables on each Q ji .To show that ∇ a ϕ j generates the desired Young measure let us fix arbitrary functions f ∈ C ∞ c (Ω) and g ∈ E p . Then ˆ Ω f ( x ) g ( ∇ a ϕ j ( x )) d x = X i ˆ Q ji f ( x ) g ( ∇ a ϕ j ( x )) d x. Since f is Lipschitz we may write (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ˆ Q ji ( f ( x ) − f ( x ji )) g ( ∇ a ϕ j ( x )) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ˆ Q ji g ( ∇ a ϕ j ( x )) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε j | Ω || Q | ˆ Q | g ( ∇ a ϕ ) | d x, here ε j = ε j ( f, sup i r ji ) → j → ∞ . Changing variables on each Q ji we may writethat X i ˆ Q ji f ( x ji ) g ( ∇ a ϕ j ( x )) d x = X i | Q ji || Q | f ( x ji ) ˆ Q g ( ∇ a ϕ ( y )) d y. Note that the term P i | Q ji | f ( x ji ) is a Riemann sum of f on the anisotropic boxes Q ji ,from which we deduce that it converges, as j → ∞ , to the integral ´ Ω f ( x ) d x . Thus,finally lim j →∞ ˆ Ω f ( x ) g ( ∇ a ϕ j ( x )) d x = (cid:18) ˆ Ω f ( x ) d x (cid:19) (cid:18) − ˆ Q g ( ∇ a ϕ ( y )) d y (cid:19) , and a standard density argument on f and g ends the proof.Since every bounded open set may be covered, up to a subset of measure 0, witharbitrarily small anisotropic boxes we get the following: Corollary 3.12.
For any ϕ ∈ W a ,p ( Q ) and any bounded open Lipschitz set Ω ∈ R N themeasure given by ∇ a ϕ (cid:16) L N ¬ Q | Q | (cid:17) is a homogeneous W a ,p -gradient Young measure on Ω . We denote the space of homogeneous W a ,p -gradient Young measures on Q withbarycentre 0 by H p ( Q ). Lemma 3.13.
The set H p is convex.Proof. Fix ν, µ ∈ H p and θ ∈ (0 , v j , u j ⊂ C ∞ c ( Q ) satisfy v j ⇀ u j ⇀ a ,p ( Q ) with ∇ a v j , ∇ a u j generating ν and µ respectively. Pick a sequence of smoothcut-off functions ϕ i ∈ C ∞ c ( Q ; [0 , ϕ i ր ( − , − θ ) × Q N − and such that |{ ϕ i = ( − , − θ ) × Q N − ) }| → i → ∞ . Here Q N − is the ( N −
1) dimensional cube.Define w ij := ϕ i v j + (1 − ϕ i ) u j . Then ∇ a w ij = ϕ i ∇ a v j + (1 − ϕ i ) ∇ a u j + R ( i, j ) , where R ( i, j ) is the remainder, consisting of the terms where we put some of the derivativeson ϕ i or (1 − ϕ i ). Due to Lemma 2.8 the lower order derivatives of v j and u j convergestronly to 0 in L p , thus k R ( i, j ) k L p = k∇ a w ij − ( ϕ i ∇ a v j + (1 − ϕ i ) ∇ a u j k L p C ( ϕ i )( k c ∇ a v j k L p + k c ∇ a u j k L p ) → j → ∞ for any fixed i (here C ( ϕ i ) is a finite constant that depends on the function ϕ i ). Therefore, we may select a subsequence j ( i ) → ∞ as i → ∞ such that k∇ a w ij ( i ) − ( ϕ i ∇ a v j ( i ) + (1 − ϕ i ) ∇ a u j ( i ) ) k L p → i → ∞ . Let w i := w ij ( i ) ∈ C ∞ c ( Q ). It is straightforward to see that ∇ a w i generates ( − , − θ ) × Q N − ν + ( − θ, × Q N − µ as its oscillation Young measure. This clearlyholds for ( − , − θ ) × Q N − ∇ a v j ( i ) + ( − θ, × Q N − ∇ a u j ( i ) and, by construction, thedifference of this sequence and ( ϕ i ∇ a v j ( i ) + (1 − ϕ i ) ∇ a u j ( i ) ) converges to 0 in measure,whilst the difference of ∇ a w i and the latter sequence converges to 0 in L p . Finally, w i iscompactly supported in Q , so we may extend it periodically to R N and define w ki ( x ) := R − k w i ( R k ⊙ x ) , where R k := k a · ... · a N . Then, by Lemma 3.11, for all ϕ ∈ C ∞ and ψ ∈ E p we havelim i →∞ lim k →∞ − ˆ Q ϕ ( x ) ψ ( ∇ a w ki ( x )) d x = lim j →∞ − ˆ Q ϕ ( x ) (cid:18) − ˆ Q ψ ( ∇ a w i ( y )) d y (cid:19) d x. inally, since we have already identified the measure generated by ∇ a w i , we may writelim i →∞ lim k →∞ − ˆ Q ϕ ( x ) ψ ( ∇ a w ki ( x )) d x = − ˆ Q ϕ ( x ) d x ( θ h ν, ψ i + (1 − θ ) h µ, ψ i ) . A standard density argument and a diagonal extraction in the separable spaces L ( Q )and C ( R n × m ) let us obtain a subsequence e w l ⊂ { w ki } ⊂ C ∞ with a -gradients generatingthe measure θν + (1 − θ ) µ , which ends the proof. Lemma 3.14.
The set H p is relatively closed in P ( R n × m ) ∩ { µ : ´ R N | W | p d µ < ∞} withrespect to the weak* topology on E ∗ p .Proof. Fix an arbitrary ν ∈ H p E ∗ ∩ P ( R n × m ) and let { f i } ⊂ C ∞ ( Q ), { g j } ⊂ C ∞ c ( R n × m )be countable dense subsets of L ( Q ) and C ( R n × m ) respectively. Take also f ( x ) ≡ g ( W ) := | W | p . By definition of the weak* topology, for any fixed g ∈ C ( R n × m ) thereexists a ν k ∈ H p with |h ν − ν k , g i| < k . Through a diagonal argument we may ensure that this is satisfied simultaneously for anyfinite set of g ’s, i.e., for any k ∈ N there exists a ν k ∈ H p such that |h ν − ν k , g j i| < k , for all j ∈ { , , . . . , k } . Since ν k ∈ H p we may find a sequence { w kj } ⊂ W a ,p ( Q ) with a -gradients generating ν k .Theorem 3.1 implies that, for any g ∈ C ∞ c ( R n × m ), we have g ( ∇ a w kj ) ⇀ h g, ν k i inL ( Q ). Another diagonal extraction and the triangle inequality let us establish existenceof a sequence { w k } ⊂ W a ,p ( Q ) such that (cid:12)(cid:12)(cid:12)(cid:12) h ν, g j i ˆ Q f i d x − ˆ Q f i g j ( ∇ a w k ) d x (cid:12)(cid:12)(cid:12)(cid:12) < k , for all 0 i, j k, (3.3)as all f i ’s are smooth and therefore bounded, so that they are admissible test functionsfor weak convergence in L .Setting i = j = 0 shows that {∇ a w k } is bounded in L p ( Q ), therefore we may find asubsequence generating some W a ,p -gradient Young measure µ . For notational simplicitywe assume that the entire sequence generates µ . From (3.3) and the fact that g j ( ∇ a w k ) ⇀ h µ, g j i in L as k → ∞ we infer that h ν, g j i ˆ Q f i dx = ˆ Q f i h µ, g j i dx, for all i, j. By density of f i in L p ( Q ) we deduce that h µ, g j i = h ν, g j i in (L p ) ∗ . In particular, theyare equal almost everywhere, so that for almost every x ∈ Q we have h µ x , g j i = h ν, g j i for all j . By density of { g j } in C ( R n × m ) we deduce that µ x = ν for almost all x , whichshows that µ is in fact homogeneous so ν = µ ∈ H p , which ends the proof.Because bounded continuous functions are a subset of E p we immediately get: Lemma 3.15.
If a sequence { ν j } ⊂ P ( R n × m ) ∩ E ∗ p converges to some ν ∈ P ( R n × m ) ∩ E ∗ p in the space E ∗ p then it also converges in the sense of weak convergence of probabilitymeasures. In particular, by the portmanteau theorem, we have lim j →∞ ˆ R n × m g d ν j = ˆ R n × m g d ν or all bounded and continuous functions g , and lim inf j →∞ ˆ R n × m g d ν j > ˆ R n × m g d ν for all lower semicontinuous functions g bounded from below. W a ,p -gradient Youngmeasures We are finally ready to show the two main results of this section:
Theorem 3.16.
A probability measure µ ∈ P ( R n × m ) is a homogeneous oscillation W a ,p -gradient Young measure with mean W if and only if µ satisfies ´ R n × m W d µ ( W ) = W , ´ R n × m | W | p d µ ( W ) < ∞ and ˆ R n × m g ( W ) d µ ( W ) > Q g ( W ) for all g ∈ E p . Theorem 3.17.
Fix a bounded open Lipschitz domain Ω satisfying the weak a -horncondition. Let { ν x } x ∈ Ω be a weak* measurable family of probability measures on R n × m .Then there exists a W a ,p (Ω) -bounded sequence { v n } ⊂ W a ,p (Ω) with {∇ a v n } generatingthe oscillation Young measure ν if and only if the following conditions hold:i) there exists v ∈ W a ,p (Ω) such that ∇ a v ( x ) = h ν x , Id i for a.e. x ∈ Ω; ii) ˆ Ω ˆ R n × m | W | p d ν x ( W ) d x < ∞ ; iii) for a.e. x ∈ Ω and all g ∈ E p we have h ν x , g i > Q g ( h ν x , Id i ) . The proof strategy is based on Fonseca and M¨uller’s approach from [40]. Havingestablished all the necessary technical ingredients above, there are no major difficulties inadapting their arguments. Below we present a proof for the case of homogeneous measuresand, for the sake of brevity, refer the reader to [72] for the full argument in the generalcase.
Proof. (of Theorem 3.16) First observe that it is enough to consider the case W = 0, aswe may always take a translation. For the proof of this case we argue by contradiction.Suppose that ν ∈ P ( R n × m ) satisfies ´ R n × m W d ν = 0 , ´ R n × m | W | p d ν ( W ) < ∞ , ´ R n × m g ( W ) d ν ( W ) > Q g (0) for all g ∈ E p , but ν H p . The p -th moment assumption on ν implies that ν ∈ E ∗ p . By Lemmas 3.13and 3.14 and the Hahn-Banach separation theorem, there exist g ∈ E p and α ∈ R suchthat Q g (0) h ν, g i < α h µ, g i for all µ ∈ H p . or any ϕ ∈ C ∞ c ( Q ) the measure ( ∇ a ϕ ) L N ¬ Q | Q | is an element of H p (see Corollary 3.12)so that − ˆ Q g ( ∇ a ϕ ( x )) d x > α. Taking infimum over all such ϕ ’s yields Q g (0) = inf ϕ ∈ C ∞ c ( Q ) − ˆ Q g ( ∇ a ϕ ( x )) d x > α, which contradicts Q g (0) h ν, g i < α and shows that ν is indeed in the set H p .For the reverse implication let ϕ k ∈ C ∞ c be such that {∇ a ϕ k } is a p -equiintegrablesequence generating ν (which is possible thanks to Corollary 3.6). Then for all k we have ˆ R n × m g ( ∇ a ϕ k ( x )) d x > Q g (0)by definition of Q g . On the other hand, p -equiintegrability of {∇ a ϕ k } and the growthbound on g imply that we may use point iii) of Theorem 3.1 to deduce thatlim k →∞ ˆ R n × m g ( ∇ a ϕ k ( x )) d x = ˆ R n × m g d ν, as for a continuous function the boundedness from below assumption is irrelevant, sincewe may simply consider the positive and negative parts of g separately. This and theprevious estimate finish the proof. We are now ready to move on to the core part of the paper, i.e., the study of existence ofsolutions to variational problems posed in Sobolev spaces of mixed smoothness. To usethe Direct Method we need two ingredients: compactness of sequences of minimisers andsequential lower semicontinuity of the given functional. In this section we focus on thefirst part and study coercivity of the relevant functionals. Our first goal is Theorem 4.3,which is a generalisation to the mixed smoothness framework of a recent result due toChen and Kristensen from [26], that was further improved by Gmeineder and Kristensenin [43].Here F : R n × m → R is a continuous integrand satisfying the growth condition | F ( X ) | C ( | X | p + 1) (4.1)for some C ∈ (0 , ∞ ) and all X ∈ R n × m . For a non-empty bounded open set Ω ⊂ R N wedenote by I( · , Ω) : W a ,p (Ω) → R the mapping defined by I( u, Ω) := ´ Ω F ( ∇ a u ) d x . To beable to properly introduce a Dirichlet problem for this functional we define, for a fixed g ∈ W a ,p ( R N ), the Dirichlet classW a ,pg (Ω) := { g + ϕ : ϕ ∈ W a ,p (Ω) } , Observe that we require the boundary datum g to be defined on the whole of R N , ratherthan just Ω ⊂ R N , thus bypassing possible trace issues. For q ∈ [1 , p ] we introduce, as in[26], the following notions: Definition 4.1.
We say that I( · , Ω) is L q coercive on W a ,pg (Ω) if for any sequence u j ∈ W a ,pg (Ω) with k∇ a u j k L q → ∞ one has I( u j , Ω) → ∞ . efinition 4.2. We say that I( · , Ω) is L q mean coercive on W a ,pg (Ω) if for any u ∈ W a ,pg (Ω) we have I( u, Ω) > C k∇ a u k q L q − C for some strictly positive constants C , C independent of u , but not necessarily of g . As in [26] (see also Proposition 3.1 in [43]), our main goal here is the following:
Theorem 4.3.
Let F : R n × m → R be a continuous integrand satisfying (4.1) . Then, forany q ∈ [1 , p ] the following are equivalent:a) For any bounded open Lipschitz set Ω ⊂ R N and any boundary datum g ∈ W a ,p ( R N ) the functional I( · , Ω) is L q coercive on W a ,pg (Ω) .b) There exist a non-empty bounded open Lipschitz set Ω ⊂ R N and a boundary datum g ∈ W a ,p ( R N ) such that the functional I( · , Ω) is L q coercive on W a ,pg (Ω) .c) For any bounded open Lipschitz set Ω ⊂ R N and any boundary datum g ∈ W a ,p ( R N ) the functional I( · , Ω) is L q mean coercive on W a ,pg (Ω) .d) There exist a non-empty bounded open Lipschitz set Ω ⊂ R N and a boundary datum g ∈ W a ,p ( R N ) such that the functional I( · , Ω) is L q mean coercive on W a ,pg (Ω) .e) For any bounded open Lipschitz set Ω ⊂ R N and any boundary datum g ∈ W a ,p ( R N ) all W a ,pg (Ω) minimising sequences for the functional I( · , Ω) are bounded in W a ,qg (Ω) .f ) There exist a non-empty bounded open Lipschitz set Ω ⊂ R N and a boundary datum g ∈ W a ,p ( R N ) such that all W a ,pg (Ω) minimising sequences for the functional I( · , Ω) are bounded in W a ,qg (Ω) .g) There exist a constant c > and a point X ∈ R n × m such that the integrand X F ( X ) − c | X | q is a -quasiconvex at X . Simply put, the above asserts that L q coercivity, L q mean coercivity, and boundednessof all minimising sequences are mutually equivalent, and it is enough to check either ofthese on a particular choice of domain and boundary datum. Furthermore, coercivitymay be characterized in terms of a -quasiconvexity of F ( · ) − c | · | q at a single point. Lemma 4.4 (see Proposition 3.1 in [26]) . Under the hypotheses of Theorem 4.3 its pointa) is equivalent to b), and point c) is equivalent to d).Proof.
The fact that a) is equivalent to b) in the first order gradient case is the contentof Proposition 3.1 in [26]. The proof of c) being equivalent to d) is similar — we will dothis here omitting the first part, as all that needs to be changed in the argument of [26]is the scaling factor.It is obvious that c) implies d), thus we only need to prove the other implication.Let Ω and g be the domain and the boundary datum for which I( · , Ω ) is L q meancoercive on W a ,pg (Ω ). Fix an arbitrary non-empty bounded open Lipschitz set Ω ⊂ R N and a boundary datum g ∈ W a ,p ( R N ). Take boxes Q r ( x ) ⋐ Ω (for some x ∈ Ω ) and Q R (0) ⋑ Ω, and fix cut-off functions ϕ ∈ C ∞ c ( Q r ( x )) and ρ ∈ C ∞ c ( Q R (0)) satisfying Q r ( x ) ϕ Q r ( x ) and Ω ρ Q R (0) . Any u ∈ W a ,pg (Ω) may be seen as a function in W a ,p ( R N ) if we extend it by u = g outsideΩ, simply from our definition of W a ,pg (Ω). We may then cut-off outside Q R (0) by setting e w := ρu ∈ W a ,p ( Q R (0)). Define w ( x ) := R − e w ( R ⊙ x ) ∈ W a ,p ( Q (0)) and set v ( x ) := (1 − ϕ ( x )) g ( x ) + rw ( r − ⊙ ( x − x )) for x ∈ Ω . learly v ∈ W a ,pg (Ω ) and we may calculate, using a simple change of variables, that k∇ a v k q L q (Ω ) = ˆ Ω \ Q r ( x ) |∇ a (1 − ϕ ( x )) g | q d x + ( rR ) | a − | ˆ Q R (0) \ Ω |∇ a ( ρg ) | q d x + ( rR ) | a − | k∇ a u k q L q (Ω) , so that k∇ a v k q L q (Ω ) = D + D k∇ a u k q L q (Ω) , where D i do not depend on u and D > ˆ Ω F ( ∇ a v ) d x = ˆ Ω \ Q r ( x ) F ( ∇ a (1 − ϕ ( x )) g ) d x + ( rR ) | a − | ˆ Q R (0) \ Ω F ( ∇ a ( ρg )) d x + ˆ Ω F ( ∇ a u ) d x ! . Thus, I( u, Ω) differs from I( v, Ω ) by a constant and a scaling. Since the same is truefor the L q norms of u and v we easily conclude that I( u, Ω) > − c + c k u k q L q (Ω) for some c i >
0, which ends the proof.
Definition 4.5.
Let F : R n × m → R be a continuous integrand satisfying the growthcondition (4.1) . For q ∈ [1 , p ] and a non-empty bounded and open Lipschitz subset Ω ⊂ R N we define, for t > , θ ( t ) = θ Ω q ( t ) := inf (cid:26) − ˆ Ω F ( ∇ a ϕ ) d x : ϕ ∈ W a ,p (Ω , R n ) , − ˆ Ω |∇ a ϕ | q d x > t (cid:27) . Lemma 4.6 (see Lemma 2.1 in [26]) . For any open, bounded, and non-empty domains ω, Ω ⊂ R N and any function ϕ ∈ C ∞ c (Ω) there exists a function ψ ∈ C ∞ c ( ω ) such that thepushforward measures [ ∇ a ϕ ] (cid:16) L N ↾ Ω L N (Ω) (cid:17) and [ ∇ a ψ ] (cid:16) L N ↾ ω L N ( ω ) (cid:17) are equal.Proof. This is proven using a standard exhaustion argument, as in Lemma 2.1 in [26].We omit the proof here, as the only difference with the aforementioned paper is that oneneeds to change the scaling to our anisotropic variant.
Corollary 4.7.
The unit cube Q in the definition of the a -quasiconvex envelope may bereplaced by any other domain without changing the resulting envelope. That is, for anyopen bounded Lipschitz domain Ω ⊂ R N we have Q F ( · ) = inf ϕ ∈ C ∞ c (Ω) − ˆ Ω F ( · + ∇ a ϕ ( x )) d x. The next result asserts that the quasiconvex envelope of an integrand of p -growthis either degenerate and equal to −∞ everywhere, or it inherits the original integrand’sgrowth bounds. Let us remark that in the context of classical quasiconvexity this istypically shown using rank-one convexity, which is lacking in our case. Therefore, theusual proofs do not generalise to the mixed smoothness setting, hence we present a newargument that we believe is, in a sense, more natural and straightforward. Lemma 4.8.
Suppose that F is a continuous integrand satisfying (4.1) . Then Q F iseither identically equal to −∞ or it is real-valued everywhere and satisfies the growthcondition (4.1) , albeit possibly with a larger constant. roof. Assume for contradiction that Q F is finite at some point X ∈ R n × m , but it doesnot satisfy the p growth condition. Let D be the constant with which F satisfies the p growth assumption (4.1). Since Q F F , it follows that the p growth bound must fail forthe negative part of Q F . Thus, there exists a sequence of points X j ∈ R n × m such that Q F ( X j ) < − j ( | X j | p + 1). From this and Corollary 4.7 we infer existence of a sequenceof functions ϕ j ∈ C ∞ c ( Q / ) such that − ˆ Q / F ( X j + ∇ a ϕ j ) d x < − j ( | X j | p + 1) . Fix a cut-off function ρ ∈ C ∞ c ( Q ) with ρ ≡ Q / . For each j let P j ( x ) := P h α, a − i =1 ( X j − X ) x α and set ψ j := ρP j + ϕ j ∈ C ∞ c ( Q ). Then Q F ( X ) lim inf j →∞ | Q | − ˆ Q F ( X + ∇ a ψ j ) d x = lim inf j →∞ | Q | − ˆ Q / F ( X j + ∇ a ϕ j ) d x + | Q | − ˆ Q \ Q / F ( X + ∇ a ( ρP j )) d x lim inf j →∞ (cid:18) − (cid:19) j ( | X j | p + 1) + − ˆ Q D ( | X + ∇ a ( ρP j ) | p + 1) d x, where we have used the definition of ψ j , the definition of X j , and the p -growth boundon F respectively. We note that | X + ∇ a ( ρP J ) | p p − | X | p + 2 p − |∇ a ( ρP j ) | p , and byconstruction | ∂ α P j ( x ) | C | X j − X | with a uniform constant C for all x ∈ Q and all α with h α, a − i
1, thus |∇ a ( ρP j ) | p C | X j − X | p , since ρ is just a fixed C ∞ c function.Plugging this into our inequality we get Q F ( X ) lim inf j →∞ (cid:18) − (cid:19) j ( | X j | p + 1) + C ( | X | p + | X j − X | p + 1) = −∞ , which contradicts the assumption that Q F ( X ) is finite. Lemma 4.9. If F is a continuous integrand satisfying the p -growth condition (4.1) andsuch that its a -quasiconvex envelope is not identically equal to −∞ , then the functional u
7→ − ˆ Ω Q F ( ∇ a u ) d x is sequentially upper semicontinuous along sequences u j converging to a given u in the W a ,p norm and with ∇ a u j → ∇ a u almost everywhere.Proof. By Lemma 4.8, Q F satisfies the p -growth condition (4.1) as well (perhaps with adifferent constant C ). Furthermore, since X − ´ Ω F ( X + ∇ a ϕ ( x )) d x is continuous forany ϕ ∈ W a ,p we see that Q F is a pointwise infimum of a family of continuous functions,thus Q F is upper semicontinuous. Since the functions C (1 + |∇ a u j | p ) − Q F ( ∇ a u j ) areall non-negative, we get, by Fatou’s lemma, thatlim inf j →∞ − ˆ Ω C (1 + |∇ a u j | p ) − Q F ( ∇ a u j ) d x > − ˆ Ω lim inf j →∞ C (1 + |∇ a u j | p ) − Q F ( ∇ a u j ) d x. Rearranging and using strong L p convergence of ∇ a u j yieldslim sup j →∞ − ˆ Ω Q F ( ∇ a u j ) d x − ˆ Ω lim sup j →∞ Q F ( ∇ a u j ) d x. Finally, ∇ a u j → ∇ a u almost everywhere by assumption, and Q F is upper semicon-tinuous, thus lim sup j →∞ Q F ( ∇ a u j ) Q F ( ∇ a u ) almost everywhere, which ends theproof. he following result justifies the omission of the underlying set Ω in our notation θ ( t )by showing that the auxiliary function θ does not depend on the choice of Ω. Moreover, itshows that θ only depends on Q F rather than F itself, thus showing that F is L q (mean)coercive if and only if Q F is. Lemma 4.10 (see Lemma 3.1 in [26]) . Let ω, Ω ⊂ R n × m be non-empty bounded openLipschitz subsets of R N , and define θ Ω q ( t ) as above. Define also θ qc ( t ) := inf (cid:26) − ˆ ω Q F ( ∇ a ϕ ) d x : ϕ ∈ W a ,p ( ω, R n ) , − ˆ ω |∇ a ϕ | q d x > t (cid:27) . Then θ Ω q ( t ) = θ qc ( t ) for all t > .Proof. The argument follows that of [26] with the key differences being that, due to lackof rank-one convexity, we cannot assert continuity of Q F (see however Lemma 6.1), andthat our polynomial approximation requires a countable partition.The case t = 0 is the content of Corollary 4.7. Another easy case is when Q F isidentically equal to −∞ . Then one can, for example, choose two disjoint open subsetsΩ , Ω ⋐ Ω and construct functions ϕ i ∈ C ∞ c (Ω i ) and use ϕ to satisfy the restriction − ´ Ω i |∇ a ϕ i | q d x > t , whilst using ϕ to make the integral ´ Ω F ( ∇ a ϕ ) d x as small as onewishes, with the last point being possible thanks to the fact that Q F (0) = −∞ . Thus, inthe following we assume that t > Q F > −∞ .Let us fix t >
0. For any ε > ϕ ∈ C ∞ c (Ω) satisfying − ˆ Ω |∇ a ϕ | q d x > t and θ qc ( t ) + ε > − ˆ Ω Q F ( ∇ a ϕ ) d x. To see this, it is enough to observe that the existence of e ϕ ∈ W a ,p (Ω) satisfying these twoinequalities, the second with a smaller ε ′ , is guaranteed by the definition of θ qc . To get thesame with ϕ ∈ C ∞ c (Ω), it is enough to observe that e ϕ may be approximated in W a ,p (Ω)by a sequence ϕ j ∈ C ∞ c (Ω) with k∇ a ϕ j k L q > k∇ a e ϕ k L q for all j and ∇ a ϕ j → ∇ a e ϕ almosteverywhere. Then we simply use Lemma 4.9 and conclude.From here it is easy to pass to a function ψ ∈ W a ,p (Ω) satisfying the above with2 ε instead of ε in the second inequality and for which ∇ a ψ is piecewise constant ona large part of Ω. We simply apply Proposition 2.15 to ϕ ∈ C ∞ c (Ω) (preserving theboundary values) and obtain a sequence ϕ j approximating ϕ in W a ,p and such thatthe measure of the complement of the set of boxes T , denoted τ j , on which ∇ a ϕ j isconstant goes to 0 as j goes to infinity. We may then rescale the sequence to satisfythe condition − ´ Ω |∇ a ϕ j | q d x > t , pass to a subsequence for which the gradients convergealmost everywhere, and finally use Lemma 4.9 again to deduce that elements ϕ j for largeenough j ’s satisfy the desired inequalities, still with an ε . Without loss of generalityassume that this holds for all j . Since ∇ a ϕ j is strongly L p convergent it is also p -equiintegrable, thus (as Q F satisfies the p -growth bound) Q F ( ∇ a ϕ j ) and F ( ∇ a ϕ j ) areequiintegrable as well. Therefore, we may find a j such that the measure of Ω \ S τ j T issmall enough so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω \ S τj T Q F ( ∇ a ϕ j ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε | Ω | and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ Ω \ S τj T F ( ∇ a ϕ j ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε | Ω | . We set ψ := ϕ j and τ = τ j . By Corollary 4.7 we may, for each T ∈ τ , find a function ϕ T ∈ C ∞ c ( T ) satisfying − ˆ T F ( ∇ a ψ + ∇ a ϕ T ) d x < Q F ( ∇ a ψ ) + ε. (4.2) e set ϕ := P T ∈ τ ϕ T . Since the sum is finite this function is well defined and belongsto the class C ∞ c (Ω). We then have θ qc ( t ) + 2 ε > − ˆ Ω Q F ( ∇ a ψ ) d x = | Ω | − ˆ S τ Q F ( ∇ a ψ ) d x + ˆ Ω \ S τ Q F ( ∇ a ψ ) d x ! > | Ω | − ˆ S τ F ( ∇ a ψ + ∇ a ϕ ) d x − ε > − ˆ Ω F ( ∇ a ψ + ∇ a ϕ ) d x − ε. Here the first inequality is from the construction of ψ , the second one results from esti-mating the remainder integral (cid:12)(cid:12)(cid:12) ´ Ω \ S τ Q F ( ∇ a ψ ) d x (cid:12)(cid:12)(cid:12) and using (4.2) on T ∈ τ , and thelast one is just an estimate on (cid:12)(cid:12)(cid:12) ´ Ω \ S τ T F ( ∇ a ψ ) d x (cid:12)(cid:12)(cid:12) paired with the fact that ϕ ≡ \ S τ T . Finally, since ϕ ∈ C ∞ c (Ω), Jensen’s inequality gives − ˆ Ω |∇ a ψ + ∇ a ϕ | q d x > − ˆ Ω |∇ a ψ | q d x > t. Since ε > ω = Ω then θ qc = θ Ω q . To pass to anarbitrary ω we need to realise the distribution of ∇ a ( ψ + ϕ ) on ω , which follows easily fromLemma 4.6 — observe that this also shows that we preserve the moment restrictions. Lemma 4.11 (see Proposition 3.2 in [26]) . Let F : R n × m → R be a continuous in-tegrand satisfying the growth condition (4.1) . Then its associated auxiliary function θ := θ Ω q : [0 , ∞ ) → R ∪ {−∞} is convex.Proof. We omit the proof as our case only differs from that of [26] by a scaling exponent.We are now ready to finish the proof of the main result of this section.
Proof of Theorem 4.3, part 2.
We have already seen that a) and b) are equivalent, andso are c) and d). That c) implies a) is obvious. We will now show that a) implies d). Ifa) holds, i.e., if F is L q coercive, then its auxiliary function θ satisfies lim t →∞ ϕ ( t ) = ∞ .This, paired with the fact that θ is convex (by Lemma 4.11), implies that there exist c > c ∈ R such that θ ( t ) > c t + c for all t >
0. Thus, taking Ω to be anyadmissible domain and imposing zero boundary conditions, we may take t = k∇ a ϕ k q L q (Ω) for any ϕ ∈ W a ,p (Ω) and conclude that F is L q mean coercive. Thus, we conclude thatthe conditions a) – d) are all equivalent.The fact that c) implies e) and f) is immediate. For the other direction let us startwith a simple case of Ω = Q , and boundary datum g such that ∇ a g is some constant X ∈ R n × m . Let us assume that in this case all minimising sequences are bounded,we now wish to prove that F is L q mean coercive at X on Q . Consider the auxiliaryfunction e θ of the translated integrand e F ( · ) := F ( X + · ), which clearly still satisfies theL p growth bound. The assumption that all minimising sequences are bounded easilyimplies that the infimum of our variational problem, equal to e θ (0), cannot be −∞ . Ifit was, one could take an arbitrary (bounded) minimising sequence ϕ j , rescale it to besupported on a smaller cube, and then use the remaining room to make the W a ,q norm ofthe resulting sequence arbitrarily large without spoiling the minimising property, thanksto the p growth assumption on F .Clearly e θ is non-decreasing, and convex as shown in Lemma 4.11. If e θ was boundedfrom above we would deduce existence of minimising sequences of arbitrarily large W a ,q norms, which would contradict our assumption. Thus, e θ is real-valued, unbounded, and onvex, and so we deduce, as previously, that F is L q mean coercive at z on Q , whichimplies point d) of our theorem, and thus all the points a) - d).Now let us take an arbitrary Ω and g and assume, for contradiction, that F is notmean coercive, but all minimising sequences are bounded. As previously, the infimumof the variational problem has to be a real number. Take any minimising sequence u j and note that, due to Proposition 2.15, we may approximate any u j with a sequence u εj ∈ W a ,pu j = W a ,pu , elements of which are piecewise polynomial on a finite family ofdisjoint open boxes, covering Ω up to a set of measure ε , and converge to u j in theW a ,p norm. Since F is a continuous integrand of p growth, the map u ´ Ω F ( ∇ a u ) iscontinuous in W a ,p , thus extracting a diagonal subsequence we may assume that our (notrenamed) minimising sequence u j is such that for each j there exists a finite family { Q ji } i of pairwise disjoint boxes with (cid:12)(cid:12) Ω \ S i Q ji (cid:12)(cid:12) < /j and with ∇ a u j constant on each Q ji .By the previous step, since we assume that F is not L q mean coercive we may, for anyopen box and any a -polynomial boundary datum, construct an unbounded minimisingsequence. Doing so on each Q ji with boundary datum u j (which is indeed an a -polynomialon Q ji ) we may construct functions ϕ ji ∈ W a ,p ( Q ji ) such that j (cid:12)(cid:12)(cid:12) Q ji (cid:12)(cid:12)(cid:12) < ˆ Q ji (cid:12)(cid:12)(cid:12) ∇ a ( u j + ϕ ji ) (cid:12)(cid:12)(cid:12) q d x, and ˆ Q ji F ( ∇ a ( u j + ϕ ji )) d x ˆ Q ji F ( ∇ a u j ) + 1 /j d x. Extending each ϕ ji by zero outside Q ji and letting v j := u j + P i ϕ ji yields a sequence v j ∈ W a ,pu (Ω) that clearly still minimizes the functional, but is unbounded in W a ,q , andthus we have a contradiction, which proves that f) implies c), thus the points a) throughf) are equivalent.That g) implies d) is easy. Assuming that F − δ |·| q is quasiconvex at some X ∈ R n × m we may write F ( X ) − δ | X | q − ˆ Ω F ( X + ∇ a ϕ ( x )) − δ | X + ∇ a ϕ ( x ) | d x for all ϕ ∈ C ∞ c (Ω). This is exactly L q mean coercivity on W a ,pg (Ω) with g a polynomialsuch that ∇ a g = X , c = δ > c = F ( X ) − δ | X | q .As the last step we prove that c) implies g). Assume that F is L q mean coercive,take Ω to be any admissible domain, and let the boundary condition be g ≡
0. Ifthe corresponding coercivity constants are c and c then take any δ ∈ (0 , c ) and put G ( X ) := F ( X ) − δ | X | q for X ∈ R n × m . Clearly, the auxiliary function of G is boundedfrom below by ( c − δ ) t + c and, as always, G > Q G . From Lemma 4.10 we know thatthe auxiliary functions of G and Q G are the same, thus we may write − ˆ Ω G ( ∇ a ϕ ) d x > − ˆ Ω Q G ( ∇ a ϕ ) d x > ( c − δ ) − ˆ Ω |∇ a ϕ | q d x + c for all ϕ ∈ W a ,p (Ω). Thanks to Corollary 4.7 we may find a sequence ϕ j ⊂ C ∞ c (Ω) forwhich Q G (0) − ˆ Ω Q G ( ∇ a ϕ j ) d x − ˆ Ω G ( ∇ a ϕ j ) d x ց Q G (0) . Since we have picked δ < c it is easy to conclude that G is still L q mean coercive. Since − ´ Ω G ( ∇ a ϕ j ) d x is bounded we infer that the sequence ∇ a ϕ j is bounded in L q . As in [26]we consider the following probability measures on R n × m ν j := ( ∇ a ϕ j ) (cid:18) L N ↾ Ω | Ω | (cid:19) nd observe that they have uniformly bounded q -th moments. Thus, passing to a sub-sequence if necessary, we may assume that ν j ∗ ⇀ ν in C ( R n × m ) ∗ , where ν is someprobability measure on R n × m with finite q th moment. Setting H ( X ) := G ( X ) − Q G ( X ),which is a non-negative and lower semicontinuous function, we may use the portmanteautheorem to write 0 ˆ R n × m H d ν lim inf j →∞ ˆ R n × m H d ν j = 0 , thus H = 0 on the support of ν . However, then G = Q G on the support of ν , i.e., F − δ |·| q is a -quasiconvex on the support of ν , and since ν is a probability measure its support isnon-empty, which ends the proof. The central part of our existence theory is sequential weak lower semicontinuity of inte-gral functionals acting on Sobolev spaces of mixed smoothness. We will show that thisproperty is equivalent to a -quasiconvexity of the integrand, with the precise variant of a -quasiconvexity depending on the regularity of the integrand and on the space on whichthe functional is defined. We begin with the definitions of (closed) W a ,p -quasiconvexityand a short discussion of the relationship between the various notions.Similarly to Ball and Murat in [9] we define the following: Definition 5.1.
We say that a function F : R n × m → R is W a ,p -quasiconvex if for every X ∈ R n × m one has F ( X ) inf u ∈ W a ,p ( Q ; R n ) − ˆ Q F ( X + ∇ a u ( x )) d x. Observe that this definition differs from that of a -quasiconvexity only by replacingthe test space C ∞ c by W a ,p . In fact, we have the following: Lemma 5.2 (see [9]) . Suppose that F : R n × m → R is continuous and satisfies | F ( X ) | C (1 + | X | p ) for every X ∈ R n × m . Then F is W a ,p -quasiconvex if and only if it is a -quasiconvex. In the classical case of first order gradients this has been shown by Ball and Muratin [9] through a simple application of Fatou’s lemma. The same proof works in our case,thus we skip it and note that the above and Lemma 3.10 immediately imply the following.
Corollary 5.3.
Suppose that F : R n × m → R is continuous and satisfies | F ( X ) | C (1 + | X | p ) for every X ∈ R n × m . Then Q F is W a ,p -quasiconvex. Let us note that the growth assumption on F cannot, in general, be removed fromthe statement of Lemma 5.2. In other words, W a ,p and W a ,q -quasiconvexity are differentnotions in general. The previously mentioned paper [9] contains an example of a func-tion that is W ,p quasiconvex only for sufficiently large exponents p , thus showing theimportance of the relevant test space.Finally, for future use, we introduce a third notion (analogous to the one introducedby Pedregal in [66] and later studied by Kristensen in [53]) that will be particularly usefulfor dealing with extended-real valued integrands. Definition 5.4.
We say that a function F : R n × m → ( −∞ , ∞ ] is closed W a ,p -quasiconvexif F is lower semicontinuous and Jensen’s inequality holds for F and every homogeneousoscillation W a ,p -gradient Young measure, i.e., F ( X ) inf ν ∈ H pX ˆ R n × m F ( W ) d ν ( W ) , here H pX is the set of all homogenous W a ,p -gradient Young measures with mean X . As in Theorem 3.16, we note that replacing ν ∈ H pX with δ X ∗ µ for µ ∈ H p allows usto rewrite the inequality in the definition of closed W a ,p -quasiconvexity as F ( X ) inf µ ∈ H p ˆ R n × m F ( X + W ) d µ ( W ) , for all X ∈ R n × m .Before we proceed, we note the following lemma which, in the case of classical gradi-ents, was first proven by Ball and Zhang in [10]: Lemma 5.5.
Suppose that F : R n × m → R is a continuous integrand satisfying | F ( X ) | C ( | X | p + 1) for some constant C and for all X ∈ R n × m . Then F is a -quasiconvex if, andonly if, it is W a ,p -quasiconvex and if, and only if, it is closed W a ,p -quasiconvex.Proof. Under these assumptions the equivalence between a -quasiconvexity and W a ,p -quasiconvexity is the content of Lemma 5.2. To prove the equivalence of closed W a ,p -quasiconvexity with these notions let us first observe that clearly closed W a ,p -quasiconvexityimplies a -quasiconvexity, as for every ϕ ∈ C ∞ c ( Q ) the measure given by ( ∇ a ϕ ) L N ¬ Q | Q | isa homogeneous W a ,p -gradient Young measure (see Corollary 3.12). Note that this partdoes not use any assumptions on F — indeed, closed W a ,p -quasiconvexity is the strongestof the three notions.For the other direction fix a point X ∈ R n × m and an arbitrary µ ∈ H p and let ϕ j ∈ W a ,p ( Q ) be such that the sequence {∇ a ϕ j } is p -equintegrable and generates µ ,which is possible thanks to Corollary 3.6. Due to the growth conditions on F we knowthat the family { F ( X + ∇ a ϕ j ) } is equiintegrable as well, so that Theorem 3.1 shows that ˆ R n × m F ( X + W ) d µ ( W ) = lim j →∞ − ˆ Q F ( X + ∇ a ϕ j ( x )) d x > F ( X ) , where the last inequality is due to W a ,p -quasiconvexity of F . Since X and ν were arbitrary,this ends the proof. Observe that we do not need to assume that F is bounded frombelow to apply Theorem 3.1, as we may simply consider its positive and negative partsseparately.In the following we establish an initial result that identifies closed W a ,p -quasiconvexityas a sufficient condition for lower semicontinuity of integral functionals in the mixedsmoothness setting. Lemma 5.6.
Let Ω be a bounded open Lipschitz domain. Suppose that F : R n × m → [0 , ∞ ] is a closed W a ,p -quasiconvex integrand. Then the functional I( u ) := ˆ Ω F ( ∇ a u ) d x is sequentially weakly lower semicontinuous on W a ,p (Ω) .Proof. Fix an arbitrary u ∈ W a ,p (Ω) and a sequence u j ⇀ u in W a ,p (Ω). We need toshow that lim inf I( u j ) > I( u ). Since u j is weakly convergent in W a ,p (Ω) it is also boundedin that space. Passing to a subsequence if necessary, we may assume that the lim inf isa true limit and, passing to a further subsequence, that ∇ a u j generates some W a ,p -gradient Young measure ν = { ν x } x ∈ Ω . Observe that, since p ∈ (1 , ∞ ), the barycentre of ν x is ∇ a u ( x ) for almost every x in Ω. By Theorem 3.1 we know thatlim j →∞ I( u j ) = lim j →∞ ˆ Ω F ( ∇ a u j ) d x > ˆ Ω ˆ R n × m F d ν x d x. roposition 3.7 tells us that, for Lebesgue almost every x , the measure ν x is a homoge-neous W a ,p -gradient Young measure. Hence F , as a closed W a ,p -quasiconvex function,satisfies Jensen’s inequality when tested with ν x for almost every x , which ends the proof,as we now havelim j →∞ I( u j ) > ˆ Ω ˆ R n × m F d ν x d x > ˆ Ω F ( ν x ) d x = ˆ Ω F ( ∇ a u ( x )) d x = I( u ) . Corollary 5.7.
Since the proof above is based on localisation, we immediately get that,under the assumptions of the above lemma, the functional W a ,p (Ω) ∋ u ˆ A F ( ∇ a u ) d x is sequentially weakly lower semicontinuous on W a ,p (Ω) for any measurable subset A ⊂ Ω . Lemma 5.8.
Let Ω be a bounded open Lipschitz domain satisfying the weak a -horncondition. Suppose that F : R n × m → ( −∞ , ∞ ] is a measurable integrand and that itsassociated functional I( u ) := ´ Ω F ( ∇ a u ) d x is sequentially weakly lower semicontinuouson W a ,p (Ω) . Then F is W a ,p -quasiconvex.Proof. Fix an arbitrary ϕ ∈ W a ,p ( Q ) and a X ∈ R n × m . Let u be an a -polynomial with ∇ a u ≡ X . For an arbitrary k ∈ N we may cover Ω, up to a set of measure zero, with acountable family of anisotropic boxes of the form { x ji + r ji ⊙ Q } i , with r ji < /j for all j, i . Let ϕ ji ( x ) := ( r ji ) ϕ (( r ji ) − ⊙ ( x − x j )) and set ϕ j := P ∞ i =1 ϕ ji . Then ϕ j convergesweakly to 0 in W a ,p (Ω) (see Lemma 3.11), thus u + ϕ j ⇀ u in W a ,p (Ω), so thatlim inf j →∞ ˆ Ω F ( ∇ a ( u + ϕ j ) d x = lim inf j →∞ ˆ Ω F ( X + ∇ a ϕ j ) d x > ˆ Ω F ( ∇ a u ) d x = | Ω | F ( X ) . However, for every j we have, by a change variables, ˆ Ω F ( X + ∇ a ϕ j ) d x = | Ω |− ˆ Q F ( X + ∇ a ϕ ) d x. which ends the proof.Putting together the content of Lemmas 5.5, 5.6, and 5.8 we obtain the following: Theorem 5.9.
Let Ω be a bounded open Lipschitz domain satisfying the weak a -horncondition. Suppose that F : R n × m → [0 , ∞ ) is a continuous integrand satisfying the p -growth condition | F ( X ) | C ( | X | p + 1) for some constant C and all X ∈ R n × m . Thenthe functional I( u ) := ˆ Ω F ( ∇ a u ) d x is sequentially weakly lower semicontinuous on W a ,p if and only if F is W a ,p -quasiconvex. We have already identified W a ,p -quasiconvexity as an equivalent condition for lower semi-continuity of integral functionals in the mixed smoothness framework. When a functionallacks lower semicontinuity one typically studies its relaxation, defined as the sequentiallyweakly lower semicontinuous envelope of the original functional, i.e., throughI( u ) := inf u j ⇀u (cid:26) lim inf j →∞ I( u j ) (cid:27) , here the infimum is taken over all sequences u j converging to u in the appropriate sense.In our case this means u j ⇀ u in W a ,p (Ω). With this definition the relaxed functional Iis sequentially lower semicontinuous with respect to weak convergence in W a ,p (Ω). Theaim of this section is to show that, under appropriate assumptions, the relaxation isagain an integral functional with integrand given by a -quasiconvexification of the originalone. In the classical setting of first order gradients and under ( p, p ) growth conditionson the integrand, this was first done by Dacorogna in [27]. The main results of thissection are contained in Theorems 6.2 and 6.13, which deal with the p -growth case andthe extended-real valued case, respectively. p -growth case Lemma 6.1.
Suppose that F : R n × m → [0 , ∞ ) is a continuous and L p coercive integrandwith F ( X ) C ( | X | p + 1) . Assume further that F is locally Lipschitz with | F ( X ) − F ( W ) | D (1 + | X | p − + | W | p − ) | X − W | , (6.1) for all X, W ∈ R n × m and some D ∈ R . Then Q F is continuous.Proof. For a continuous integrand of p -growth the mapping X − ´ Q F ( X + ∇ a ϕ ) d x is continuous for any fixed ϕ ∈ C ∞ c ( Q ). Thus, by the Dacorogna formula (3.2), weimmediately see that the quasiconvex envelope is upper semicontinuous, as it is a pointwiseinfimum of a family of continuous functions.By Theorem 4.3 we know that the functional induced by F is L p mean coercive withany boundary datum. However, the constants describing the coercivity may depend onthe datum — if this was not the case then F would have to satisfy pointwise coercivitybounds of the form F ( X ) > C | X | p − C − , which we do not assume. Nevertheless, underthe assumption of local Lipschitz continuity of F , it may be shown that the constantsmay be chosen uniformly on compact sets. To prove that, let us take X to be a point atwhich F − c | · | p , with some positive constant c , is a -quasiconvex — such a point existsby Theorem 4.3. For an arbitrary X ∈ R n × m and any ϕ ∈ C ∞ c ( Q ) we may, using (6.1),write − ˆ Q | F ( X + ∇ a ϕ ) − F ( X + ∇ a ϕ ) | d x D − ˆ Q (cid:0) | X + ∇ a ϕ | p − + | X + ∇ a ϕ | p − (cid:1) | X − X | d x. Rearranging we get − ˆ Q | F ( X + ∇ a ϕ ) − F ( X + ∇ a ϕ ) | d x c + c − ˆ Q |∇ a ϕ | p − d x, where the constants c i depend on X but may be chosen uniformly on compact sets. Theseconstants also depend on D and X , but, since the integrand is fixed, this dependenceis not important here. Rearranging and using the strict a -quasiconvexity at X given byTheorem 4.3 we may now write − ˆ Q F ( X + ∇ a ϕ ) d x > c − ˆ Q |∇ a ϕ | p d x − c − c − ˆ Q |∇ a ϕ | p − d x, with a different constant c . Using weighted Young’s inequality we get − ˆ Q F ( X + ∇ a ϕ ) d x > c − ˆ Q |∇ a ϕ | p d x − c , here, again, the constant c has changed, but it is still independent of ϕ and may bechosen locally uniformly in X . Thus, we have shown that for any compact set K thereexists a constant c > − ˆ Q F ( X + ∇ a ϕ ) d x > c − ˆ Q |∇ a ϕ | p d x − c − , (6.2)for all X ∈ K and all ϕ ∈ C ∞ c ( Q ).To show that Q F is lower semicontinuous take an arbitrary point X ∈ R n × m anda sequence X j → X . Assume, without loss of generality, that lim inf j →∞ Q F ( X j ) =lim j →∞ Q F ( X j ) < ∞ . For every j we may, by the Dacorogna formula, find a function ϕ j ∈ C ∞ c ( Q ) such that Q F ( X j ) > − ˆ Q F ( X j + ∇ a ϕ j ) d x. Since X j is a convergent sequence, it is contained in a compact set. Similarly, the sequence Q F ( X j ) is bounded, thus, based on what we have just proven in (6.2), we deduce thatthe family {∇ a ϕ j } is bounded in L p . Using the Lipschitz bound (6.1), we may write − ˆ Q F ( X + ∇ a ϕ j ) d x − ˆ Q F ( X j + ∇ a ϕ j ) d x + D − ˆ Q (cid:0) | X | p − + | X j | p − + |∇ a ϕ j | p − (cid:1) | X − X j | d x. (6.3)Now, | X − X j | converges to 0 as j → ∞ , and the integral − ˆ Q (cid:0) | X | p − + | X j | p − + |∇ a ϕ j | p − (cid:1) d x is bounded, hence the last term in (6.3) goes to 0 with j . Thus Q F ( X ) lim inf j →∞ − ˆ Q F ( X + ∇ a ϕ j ) d x lim inf j →∞ − ˆ Q F ( X j + ∇ a ϕ j ) d x = lim inf j →∞ Q F ( X j ) , which ends the proof.Let us remark that the above could be easily generalised to the A -free setting ofcompensated compactness due to Murat and Tartar (see [63], [64] by Murat and [81],[82], [83] by Tartar). It is known that in general, when the characteristic cone of A does not span the entire space, A -quasiconvex envelopes of smooth functions need notbe continuous (see, for example, Remark 3.5 in [40]). Our proof shows that this is not aproblem if one restricts to coercive integrands satisfying the bound (6.1). That being said,in the context of A -quasiconvexity, this may also be resolved in a different way (see theauthor’s recent collaboration with Rait¸˘a [73])), however that approach cannot be appliedin the mixed smoothness case, and thus we do not elaborate on it further.Our main relaxation result is the following: Theorem 6.2.
Let Ω be a bounded open Lipschitz domain satisfying the weak a -horncondition. Supppose that F : R n × m → [0 , ∞ ) is a continuous and L p coercive integrandwith F ( X ) C ( | X | p + 1) . Assume furthermore that F satisfies F ( X ) > D | X | p − D − (6.4) r | F ( X ) − F ( W ) | D (1 + | X | p − + | W | p − ) | X − W | , (6.5) for all X, W ∈ R n × m . Then the sequentially weakly lower semicontinuous envelope of thefunctional I F is given by I F ( u ) := inf u j ⇀u (cid:26) lim inf j →∞ I F ( u j ) (cid:27) = ˆ Ω Q F ( ∇ a u ( x )) d x = I Q F ( u ) , where the infimum is taken over all sequences u j converging to u weakly in W a ,p (Ω) .Proof. When F satisfies the first of the two alternative conditions we have provided, i.e.,when F is of p -growth from below as well as from above, this result is a corollary of a moregeneral relaxation result (for extended real-valued integrands) that we will demonstratenext, thus we postpone this part of the proof.Under the assumption that F is locally Lipschitz, we have shown in Lemma 6.1,that Q F is continuous. We also know, from Lemma 4.8, that Q F satisfies the same p -growth assumption as F . Lemma 3.10 tells us that Q F is a -quasiconvex, which in viewof the continuity and growth bounds implies, by Lemma 5.5, that it is W a ,p -quasiconvex.Finally, Lemma 5.6 shows that the functional induced by Q F is indeed sequentially weaklylower semicontinuous on W a ,p (Ω). This translates toI F ( u ) > I Q F ( u ) , and so it remains to prove the reverse inequality.Proceeding similarly to a proof in Dacorogna’s book [28] (see Theorem 9.1 therein)let us start with the simple case of Ω = Q and u with ∇ a u = X for some constant X ∈ R n × m . By the Dacorogna formula (3.2) there exists a sequence ϕ j ∈ C ∞ c ( Q ) with ˆ Q F ( ∇ a u + ∇ a ϕ j ) d x → ˆ Q Q F ( ∇ a u ) d x. It only remains to show that the sequence ϕ j may be chosen in such a way as to satisfy ϕ j ⇀ a ,p ( Q ). The argument here is essentially a simpler version of the one in theproof of Lemma 3.11. For a fixed j extend ϕ j periodically and consider the sequence ϕ kj ( x ) := r − k ϕ j ( r k ⊙ x ) , with r k := k a · ... · a N . Then ∇ a ϕ kj preserves the integral, i.e., ˆ Q F ( ∇ a u + ∇ a ϕ kj ) d x = ˆ Q F ( ∇ a u + ∇ a ϕ j ) d x, for every k , and ϕ kj ⇀ a ,p ( Q ) with k → ∞ . Thus, a diagonal extraction argumentends the proof in this basic case.For the general case let us fix an arbitrary function u ∈ W a ,p (Ω). Using Proposition2.15 we may find a sequence v j ∈ W a ,pu (Ω) with v j → u in W a ,p (Ω) and such that foreach j there exists a finite family of anisotropic boxes { Q ji } i such that ∇ a v j is constanton each Q ji and (cid:12)(cid:12) Ω \ S i Q ji (cid:12)(cid:12) → j → ∞ . We know, from Lemma 6.1, that Q F iscontinuous, and from Lemma 4.8 that it satisfies the p -growth bound. Thus ˆ Ω Q F ( ∇ a v j ) d x → ˆ Ω Q F ( ∇ a u ) d x. ince v j converges to u strongly in W a ,p , the sequence ∇ a v j is p -equiintegrable. Byassumption and Lemma 4.8 we know that both F and Q F satisfy the p -growth boundfrom above, so that ˆ Ω \ S i Q ji F ( ∇ a v j ) d x → ˆ Ω \ S i Q ji Q F ( ∇ a v j ) d x → . In particular, ˆ S i Q ji Q F ( ∇ a v j ) d x + ˆ Ω \ S i Q ji F ( ∇ a v j ) d x → ˆ Ω Q F ( ∇ a u ) d x. Using the previous step we may, for any fixed Q ji , find a sequence ϕ ji,k ∈ C ∞ c ( Q ji ) with ϕ ji,k ⇀ a ,p (Ω) as k → ∞ and such that ˆ Q ji F ( ∇ a v j + ∇ a ϕ ji,k ) d x → ˆ Q ji Q F ( ∇ a v j ) d x as k → ∞ . Letting ϕ jk := P i ϕ ji,k ∈ C ∞ c (Ω), we have ϕ jk ⇀ a ,p (Ω) as k → ∞ and ˆ S i Q ji F ( ∇ a v j + ∇ a ϕ jk ) d x → ˆ S i Q ji Q F ( ∇ a v j ) d x. A standard diagonal extraction argument allows us to construct a, non-relabelled, diag-onal sequence ϕ j ∈ C ∞ c (Ω) with ϕ j ⇀ a ,p (Ω) and such that ˆ Ω F ( ∇ a v j + ∇ a ϕ j ) d x → ˆ Ω Q F ( ∇ a u ) d x. Therefore, setting u j := v j + ϕ j , shows thatlim inf j →∞ I F ( u j ) I Q F ( u ) , and thus ends the proof.Observe that we only use the locally Lipschitz assumption on F in order to deducecontinuity of Q F from Lemma 6.1. If continuity can be ensured in a different way thenwe can dispense with the assumption (6.5). In the classical setting of first order gradientsit is known that quasiconvex functions are convex along rank-one directions, and thesespan the entire space, so that one may deduce continuity from directional convexity. Wehave already mentioned that, in general, there is no good analogue of rank-one convexityin the mixed smoothness setting. However, if a is such that all multi-indices α with h α, a − i = 1 are of the same parity, this has been resolved by Kazaniecki, Stolyarov,and Wojciechowski in [45]. Under this assumption they have shown that a -quasiconvexfunctions are convex along directions of the form X h α, a − i =1 i | α | + | α | x α b i e α,i . (6.6)Here x ∈ R N and b ∈ R n are arbitrary vectors, i = − e α,i is the canonical basis of R n × m , and α is an arbitrary multi-index on the hyperplane of homogeneity, i.e., with h α , a − i = 1. When all the multi-indices have the same parity, the coefficients in (6.6)are real and vectors of this form span R n × m , thus continuity follows from directionalconvexity. Thus, we obtain the following: orollary 6.3. If a − is such that all the multi-indices α with h α, a − i = 1 have or-ders of the same parity then the conclusion of Theorem 6.13 holds without the additionalassumptions (6.4) or (6.5) . If, on the other hand, the parities of | α | do not match, then the coefficients in (6.6)complexify, and we do not get any directional convexity. It is still possible to use the cal-culation leading to determining the form of the vectors in (6.6) to show that a -quasiconvexfunctions are (pluri)subharmonic in a certain sense (see the discussion in [45]), but wehave not yet been able to use that to strengthen our relaxation results. We believe,however, that this should be studied further and intend to do so in future work. W a ,p -quasiconvex envelope The p -growth assumption on the integrand was crucial in the results of the previous sub-sections. Indeed, when F is not of p -growth the notions of W a ,p -quasiconvexity and closedW a ,p -quasiconvexity need not coincide, and this is precisely the reason for introducing thisstricter notion of closed W a ,p -quasiconvexity. An explicit example, in the isotropic settingof first order gradients, of an integrand that is quasiconvex but not closed quasiconvexmay be found in Example 1.3 in [53].The first issue we run into is that, for an extended real-valued integrand, the formula(3.2) need not yield a closed W a ,p -quasiconvex function. The purpose of this subsectionis to establish a formula that does. We start with the natural definition of the closedW a ,p -quasiconvex envelope. Definition 6.4.
For a measurable function F : R n × m → ( −∞ , ∞ ] we define its closed W a ,p -quasiconvex envelope by F ( X ) := sup { G ( X ) : G F, G is closed a - p quasiconvex } . Our goal is the following:
Proposition 6.5.
For any p ∈ (1 , ∞ ) , the closed W a ,p -quasiconvex envelope of a lowersemicontinuous function F : R n × m → [0 , ∞ ] satisfying the growth condition F ( X ) > c | X | p for some constant c > is given by F ( X ) = inf ν ∈ H p h F ( · + X ) , ν i = inf ν ∈ H pX h F, ν i . Moreover, the function F is indeed closed W a ,p -quasiconvex. The reason we put the lower growth assumption in this result is that in a number ofplaces in the proof we will use an argument that selects, at each X ∈ R n × m a measure ν X ∈ H pX which (nearly) achieves h F, ν X i = F ( X ). The main idea of the proof is thatthen, with any fixed ν ∈ H p the measure µ defined by d µ := d ν X d ν is again a homo-geneous W a ,p -gradient Young measure. To prove that we need to ensure that it has afinite p -th moment, which is where coercivity comes into the picture. For now we donot know whether it is possible to relax this assumption, nevertheless it is in line withthe relaxation result we prove next. As in the case of integrands of p -growth, to prove arelaxation formula we need some sort of a coercivity assumption on the integrand. If wewere to relax the pointwise coercivity to simply L p or L p -mean coercivity we would needa Lipschitz assumption on our integrand of the form appearing in Theorem 6.13, whichwe cannot have if we wish to allow F to take the value + ∞ . Thus, for now at least,we content ourselves with including the lower-growth bound also in our formula for theclosed quasiconvex envelope. We note that the lower growth assumption is also presentin the relaxation result in Kristensen’s paper [53], on which we base our relaxation proof. efore we proceed to the proof let us recall the following classical result due to Kura-towski and Ryll-Nardzewski: Theorem 6.6 (see [56]) . Let X be a metric space and Y be a separable and completemetric space. Fix a multi-valued function G : X → Y . If for any closed set K ⊂ Y theset { x ∈ X : G ( x ) ∩ K = ∅} is Borel measurable then G admits a measurable selector,i.e., there exists a Borel measurable function g : X → Y such that for all x ∈ X we have g ( x ) ∈ G ( x ) .Proof of Proposition 6.5. The argument that we present closely follows the one in a recentpaper by the author (see [71]), where it was employed in the context of A -quasiconvexity.Denote R ( X ) := inf ν ∈ H p h F ( · + X ) , ν i . Clearly for any ν ∈ H p and X ∈ R n × m we have F ( X ) h F ( · + X ) , ν i , therefore takingthe infimum over ν ∈ H p yields F ( X ) R ( X ) , hence showing that R is closed W a ,p -quasiconvex will give the reverse inequality and endthe proof, as one immediately gets R F by testing with ν := δ ∈ H p .To show that R is lower semicontinuous fix X ∈ R n × m , a sequence X j → X , and an ε >
0. We will show that ε + lim inf j →∞ R ( X j ) > R ( X ) . Without loss of generality assume that lim j →∞ R ( X j ) = lim inf j →∞ R ( X j ) < ∞ , and let M be such that R ( X j ) + ε M for all j . By definition of R , for each X j there exists ν j ∈ H p with M > R ( X j ) + ε > h F ( · + X j ) , ν j i . Our growth assumption on F and boundedness of | X j | (as a convergent sequence) give M > ˆ R n × m c | X + X j | p d ν j > C (cid:18) ˆ R n × m | X | p d ν j − (cid:19) , which yields sup j ´ R n × m | X | p d ν j < ∞ . We see that the family { ν j } is bounded in E ∗ p ,therefore we may extract a weakly*-convergent subsequence from it — without loss ofgenerality assume that the whole sequence converges, i.e., ν j ∗ ⇀ ν in E ∗ p . By Lemma 3.14we have ν ∈ H p . Moreover δ X j ∗ ν j ∗ ⇀ δ X ∗ ν . Since F is lower semicontinuous andbounded from below we have ε + lim inf j →∞ R ( X j ) > lim inf j →∞ h F, δ X j ∗ ν j i > h F, δ X ∗ ν i = ˆ R n × m F ( · + X ) d ν > R ( X ) , where the last inequality comes from the definition of R and the fact that ν ∈ H p . Since ε > R is in fact lower semicontinuous.It now remains to show that R satisfies Jensen’s inequality with respect to homo-geneous oscillation W a ,p -gradient Young measures. To that end fix X ∈ R n × m and ν ∈ H pX . We wish to show that R ( X ) ´ R n × m R d ν. Without loss of generality we mayassume that ´ R n × m R d ν < ∞ . Fix an ε > R , for all X ∈ R n × m there exists ν X ∈ H p satisfying h F ( · + X ) , ν X i ε + R ( X ) , o that, for now only formally, ˆ R n × m (cid:18) ˆ R n × m F ( · + X ) d ν X (cid:19) d ν ( X ) ε + ˆ R n × m R d ν. Now — if we manage to show that ν X may be chosen in such a way that X ν X isweak* measurable and that the measure µ defined by duality as h g, µ i := ˆ R n × m (cid:18) ˆ R n × m g ( · + X ) d ν X (cid:19) d ν ( X ) (6.7)is a homogeneous W a ,p -gradient Young measure with mean X then the claim will follow,as by definition h F, µ i > R ( X ).Note that weak* measurability of X ν X only means Lebesgue measurability of X ´ R n × m g ( · + X ) d ν X , which need not be enough to integrate this function withrespect to ν . However, if we manage to get Borel measurability of the function in questionthen the construction is justified, as ν is a Radon (hence Borel) measure — we will callsuch a map Borel weak* measurable. It is clear that if one makes sense of the integrationon the right-hand side of (6.7) then it defines a linear functional on C ( R n × m ). Itsboundedness follows from the fact that all ν X , ν are probability measures, thus showingthat the functional is given by some finite Radon measure µ .For the measurable selection part we define a multifunction G given by G ( X ) := (cid:26) µ ∈ H p : ˆ R n × m F ( · + X ) d µ ε + R ( X ) (cid:27) . For the measurable selection result we intend to use (see Theorem 6.6) we need G to takevalues in 2 Y for some complete metric space Y . For that we define, for a given M > M := { X ∈ R n × m : | X | < M, R ( X ) M } . Observe that since we assumed R to be integrable with respect to ν , we have that X ∈ S ∞ M =1 Ω M for ν -a.e. X ∈ R n × m . Let us fix M ∈ N . Then, for any X ∈ Ω M and any µ ∈ G ( X ), we have ˆ R n × m F ( W + X ) d µ ( W ) ε + R ( X ) ε + R ( X ) M + 2 ε. The factor 2 in front of ε is not important here, we only put it there to allow for someroom in later parts of the argument. Due to the growth assumption on F we have ˆ R n × m F ( W + X ) d µ ( W ) > C ˆ R n × m | W + X | p d µ ( W ) > C ˆ R n × m | W | p d µ ( W ) − C − | X | p . Finally ´ R n × m | W | p d µ ( W ) C M , holds for all µ ∈ G ( X ), with the constant C M depend-ing only on M (and ε ). Therefore, we may consider our operator G as a map Ω M → Y M ,where Y M := (cid:26) µ ∈ H p : ˆ R n × m | W | p d µ C M (cid:27) . The set Y M may be equipped with the weak* topology inherited from E ∗ p . Since we puta uniform bound on the p -th moments (so also on the norm in E ∗ p ), this topology ismetrisable in a complete and separable manner when restricted to Y M . To prove that,first recall that due to Lemma 3.14 H p is weak* closed in E ∗ p . Since |·| p ∈ E p we know that he map µ ´ R n × m | W | p d µ is weak* continuous, thus Y M is weak* closed and bounded.The Banach-Alaoglu Theorem (see for example Theorem 3.16 in [20]) then implies that Y M is weak* compact. Since E p is clearly separable we deduce that the weak* topology on Y M is metrisable (see Theorem 3.28 in [20]). Finally, compact metric spaces are completeand separable, thus proving our claim. Lemma 6.7.
For any X ∈ Ω M the set G ( X ) is non-empty and closed.Proof. The fact that G ( X ) = ∅ comes straight from the definition of R . To show that itis closed it is enough to show that it is sequentially closed. Let us then fix a sequence { µ j } ⊂ G ( X ) and assume that it converges weak* in E ∗ p to some µ ∈ Y M . Since thefunction F is lower semicontinuous and bounded from below we get by Lemma 3.15 that R ( X ) + ε > lim inf j →∞ ˆ R n × m F ( · + X ) d µ j > ˆ R n × m F ( · + X ) d µ, so µ ∈ G ( X ), which ends the proof. Lemma 6.8.
For any non-empty closed set O ⊂ Y M the set { X ∈ Ω M : G ( X ) ∩ O = ∅} is Borel measurable.Proof. First note that we may rewrite the set in question as ∞ \ k =1 (cid:26) X ∈ Ω M : inf µ ∈ O ˆ R n × m F ( · + X ) d µ R ( X ) + ε (1 + 2 − k ) (cid:27) . Hence, it is enough to show that the sets (cid:26) X ∈ Ω M : inf µ ∈ O ˆ R n × m F ( · + X ) d µ R ( X ) + ε (1 + 2 − k ) (cid:27) are all Borel measurable. Define U ( X ) := inf µ ∈ O ˆ R n × m F ( · + X ) d µ. We claim that U is lower semicontinuous. Let X j → X . We need to show thatlim inf j →∞ U ( X j ) > U ( X ) . Without loss of generality assume that the lim inf is a truelimit and that it is finite, i.e.,lim j →∞ U ( X j ) = lim inf j →∞ U ( X j ) < ∞ . By definition of U , for each k there exists a measure µ j ∈ O with ˆ R n × m F ( · + X j ) d µ j U ( X j ) + 1 /k. Therefore lim j →∞ ˆ R n × m F ( · + X j ) d µ j = lim j →∞ U ( X j ) . Since the set O is a closed subset of a compact space Y M we may extract an E ∗ p weak*convergent subsequence from µ j . Without loss of generality assume that the entire se-quence µ j converges weak* to some µ ∈ O . This, combined with X j → X , implies that e have δ X j ∗ µ j ∗ ⇀ δ X ∗ µ in the sense of probability measures. Therefore, since F islower semicontinuous, the portmanteau theorem yieldslim inf j →∞ ˆ R n × m F ( · + X j ) d µ j = lim inf j →∞ ˆ R n × m F d (cid:0) δ X j ∗ µ j (cid:1) > ˆ R n × m F d ( δ X ∗ µ ) = ˆ R n × m F ( · + X ) d µ > U ( X ) , which shows that U is indeed lower semicontinuous. Since the set (cid:26) X ∈ Ω M : inf µ ∈ O ˆ R n × m F ( · + X ) d µ R ( X ) + ε (1 + 2 − k ) (cid:27) is the same as { X ∈ Ω M : U ( X ) R ( X ) + ε (1 + 2 − k ) } , and both U and R are lower semicontinuous (hence Borel measurable) the set in questionis Borel measurable as well, which ends the proof.Now, thanks to Lemmas 6.7 and 6.8 we may use Theorem 6.6 to deduce the existenceof a weak* measurable map ν M : Ω M → H p such that for any X ∈ Ω M the measure ν MX satisfies ˆ R n × m F ( · + X ) dν MX ε + R ( X ) . Finally let us define the map e ν : R n × m → H p by e ν X := ( ν MX for X ∈ Ω M \ Ω M − e µ for X S ∞ M =1 Ω M , where e µ is some arbitrary element of the (non-empty) set H p . Observe that the choiceof e µ does not matter, as we have already observed that the set R n × m \ S ∞ M =1 Ω M is of ν measure 0. This set is also Borel since we already know that R is Borel measurable, henceeach Ω M is Borel. Clearly the map e ν is Borel weak* measurable, i.e., it is a measurablemap from R n × m equipped with the Borel σ -algebra into H p equipped with the weak*topology inherited from E ∗ p . Therefore, we may define µ ∈ ( C ( R n × m )) ∗ as in (6.7). Itonly remains to show that µ ∈ H p .Positivity of µ results immediately from positivity of all ν X and ν . In the same waywe show that µ is a probability measure, as h , µ i = ˆ R n × m (cid:18) ˆ R n × m ν X (cid:19) d ν ( X ) = ˆ R n × m ν ( X ) = 1 , since all measures considered are probability measures. To prove that µ has a finite p -thmoment we write h|·| p , µ i = ˆ R n × m (cid:18) ˆ R n × m |· + X | p d ν X (cid:19) d ν ( X ) . Using the growth assumption on F we get ˆ R n × m |· + X | p d ν X C ˆ R n × m F ( · + X ) d ν X C ( R ( X ) + ε ) , where the last inequality is satisfied for ν -a.e. X . Integrating with respect to ν gives h|·| p , µ i C (cid:18) ε + ˆ R n × m R ( X ) d ν ( X ) (cid:19) < ∞ , ince, by assumption, R is integrable with respect to ν . Lastly, it remains to show that µ satisfies the inequality in Theorem 3.16. Fix any continuous functions g : R n × m → R with | g ( v ) | C (1 + | v | p ) for some constant C . We have h µ, g i = ˆ R n × m (cid:18) ˆ R n × m g ( · + X ) d ν X (cid:19) d ν ( X ) > ˆ R n × m Q g ( X ) d ν ( X ) > Q ( Q g )( X ) = Q g ( X ) , where the first inequality comes from the fact that all ν X ’s are Young measures with mean0, the second one from the respective property of ν , and the last equality from Lemma3.10. This shows that we indeed have µ ∈ H pX and ends the proof, as discussed earlier(see eq. (6.7)). We begin by defining a relaxed notion of convergence for vector fields that are nearly (upto an L p -small error) a -gradients of functions in W a ,p . The notion is reminiscent of theone often used in the A -free setting, where instead of working with sequences that satisfythe constraint exactly, i.e., with A V j = 0, one only requires A V j → − ,p ,see for example [40]. In the case of standard first order gradients this corresponds to thecondition curl V j → − ,p (Ω) investigated in [53]. Definition 6.9.
We say that a sequence of vector fields V j ∈ L p (Ω; R n × m ) is a se-quence of approximate W a ,p gradients if there exist sequences u j ∈ W a ,p (Ω; R n ) and v j ∈ L p (Ω; R n × m ) such that V j = ∇ a u j + v j and v j → strongly in L p . Following [54] we introduce the following notion of convergence:
Definition 6.10.
We say that a sequence of vector fields V j ∈ L p (Ω; R n × m ) convergesto V in the sense of approximate W a ,p gradients if V j converges to V weakly in L p and ( V j − V ) is a sequence of approximate W a ,p gradients. In such a case we write V j → a - p V . Proposition 3.2 immediately implies the following:
Lemma 6.11.
Assume that Ω satisfies the weak a -horn condition. Suppose that a se-quence V j = ∇ a u j + v j ∈ L p (Ω; R n × m ) of approximate W a ,p gradients converges weakly to in L p and generates an oscillation Young measure ν . Then {∇ a u j } generates the sameYoung measure ν . In particular, any oscillation Young measure generated by a sequenceof approximate W a ,p gradients is an oscillation W a ,p -gradient Young measure. Corollary 6.12.
Let Ω be a bounded open Lipschitz domain satisfying the weak a -horncondition. Suppose that F : R n × m → ( −∞ , ∞ ] is bounded from below and closed W a ,p -quasiconvex. Then the functional I( V ) := ˆ Ω F ( V ) d x is sequentially lower semicontinuous with respect to approximate W a ,p gradient conver-gence.Proof. Since, by Lemma 6.11 the Young measures generated by sequences of approximateW a ,p gradients are W a ,p -gradient Young measures, the argument of Lemma 5.6 carriesthrough unchanged. he main result here is the following: Theorem 6.13. If F : R n × m → ( −∞ , ∞ ] is a continuous integrand satisfying F ( X ) > C | X | p − C − for some C > then the sequentially (with respect to approximate W a ,p gradient convergence) weakly lower semicontinuous envelope of the functional I F is givenby I F [ V ] := inf V j → a - p V (cid:26) lim inf j →∞ I F [ V j ] (cid:27) = ˆ Ω F ( V ( x )) d x, where the infimum is taken over all sequences V j converging to V in the sense of ap-proximate W a ,p gradient convergence. As before, F denotes the closed W a ,p -quasiconvexenvelope of F .Proof. Corollary 6.12 guarantees that I F [ V ] > ´ Ω F ( V ( x )) d x , thus we only need to provethe opposite inequality. If F is identically equal + ∞ then there is nothing to show, so wemay restrict to proper integrands. Using a translation we may, without loss of generality,assume F ( X ) > C | X | p . In any case, the fact that F is bounded from below immediatelyimplies the same for F . Fix any V ∈ L p . Without loss of generality we may assume ´ Ω F ( V ( x )) d x < ∞ , as otherwise there is nothing to prove. Fix an ε > F ( V ( x )) < ∞ a.e. in Ω . Therefore, using Proposition 6.5, we may finda family of homogeneous oscillation W a ,p -gradient Young measures { ν x } x ∈ Ω with mean0 and such that, for almost every x ∈ Ω, we have F ( V ( x )) + ε > ˆ R n × m F ( · + V ( x )) d ν x . (6.8)Using exactly the same argument as in the proof of Proposition 6.5 we may ensure weak*measurability of x → ν x . We intend to show that ν is a suitable Young measure usingTheorem 3.17. Recall that we need to prove the following:i) there exists v ∈ W a ,p (Ω) such that ∇ a v ( x ) = h ν x , Id i for a.e. x ∈ Ω;ii) ˆ Ω ˆ R n × m | W | p d ν x ( W ) d x < ∞ ;iii) for a.e. x ∈ Ω and all continuous functions g : R n × m → R satisfying | g ( W ) | C (1 + | W | p ) for some positive constant C one has h ν x , g i > Q g ( h ν x , Id i ) . The first point is clearly satisfied, as all our measures are of mean 0. The second onemay be checked in the same way as in the already mentioned proof of Proposition 6.5,using the growth assumption on F . Finally, the third point results immediately from thefact that all ν x ’s are, by definition, elements of H p , so we may use Theorem 3.16.This shows that ν is indeed generated by some p -equiintegrable family {∇ a w j } with w j ∈ W a ,p (Ω) and w j ⇀ a ,p . For a given M ∈ N consider F M ( z ) := min( F ( z ) , M ( | z | p + 1)) . Clearly, for each M , the function F M is continuous and the family { F M ( V + ∇ a w j ) } j is p -equiintegrable, due to the same property of { V + ∇ a w j } . Theorem 3.1 then yields ˆ Ω F M ( V + ∇ a w j ) d x → ˆ Ω (cid:18) ˆ R n × m F M ( V ( x ) + · ) d ν x (cid:19) d x. n the other hand, since F M F and ν x are non-negative and satisfy (6.8), we have ˆ Ω (cid:18) ˆ R n × m F M ( V ( x ) + · ) d ν x (cid:19) d x ˆ Ω (cid:18) ˆ R n × m F ( V ( x ) + · ) d ν x (cid:19) d x ˆ Ω F ( V ( x )) d x + ε. From this we deduce, through a diagonal extraction, that there exists a sequence j ( M ) ∈ N with lim M →∞ j ( M ) = ∞ such that for all M one has ˆ Ω F M ( V + ∇ a w j ( M ) ) d x ˆ Ω F ( V ( x )) d x + 2 ε. (6.9)Define the set G M := (cid:8) x ∈ Ω : F ( V ( x ) + ∇ a w j ( M ) ( x )) M ( | V ( x ) + ∇ a w j ( M ) ( x ) | p + 1) (cid:9) , and fix some X ∈ R n × m for which F ( X ) < ∞ — such a point exists, as F is proper.Next define a vector field W M in such a way that V ( x ) + W M ( x ) = ( V ( x ) + ∇ a w j ( M ) ( x )) G M + X G cM . (6.10)We claim that { V + W M } M is an admissible vector field in the I F [ V ] problem. For thatit is enough to show that k V + W M − ( V + ∇ a w j ( M ) ) k L p (Ω) →
0. By definition we have k V + W M − ( V + ∇ a w j ( M ) ) k L p (Ω) = k V + W M − ( V + ∇ a w j ( M ) ) k L p ( G cM ) k X k L p ( G cM ) + M − (cid:18) ˆ Ω F M ( V + ∇ a w j ( M ) ) d x (cid:19) /p , where the last inequality comes from the definition of the set G cM and extending theintegral to all of Ω. Now, (6.9) yields M − (cid:18) ˆ Ω F M ( V + ∇ a w j ( M ) ) d x (cid:19) /p M − (cid:18) ˆ Ω F ( V ( x )) d x + 2 ε (cid:19) /p , thus showing the desired convergence to 0 in L p , as k X k L p ( G cM ) → G cM tends to 0, because F ( V ( x ) + ∇ a w j ( M ) ( x )) > M on G cM , and we have a uniform (with respect to M ) bound on the integral of the function in ques-tion. This implies in particular that V + W M converges to V in the sense of approximateW a ,p gradients convergence. Therefore, we haveI F [ V ] lim inf M →∞ ˆ Ω F ( V + W M ) d x = lim inf M →∞ ˆ G M F M ( V + w j ( M ) ) d x + ˆ G cM F ( X ) d x lim inf M →∞ ˆ Ω F ( V ( x )) d x + 2 ε = ˆ Ω F ( V ( x )) d x + 2 ε, where the last inequality results from (6.9) and the measure of G cM tending to 0. Since ε > eferences [1] E. Acerbi, N. Fusco , Semicontinuity problems in the calculus of variations,
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