Relaxation of nonlinear elastic energies involving deformed configuration and applications to nematic elastomers
aa r X i v : . [ m a t h . A P ] J un Relaxation of nonlinear elastic energies involving deformedconfiguration and applications to nematic elastomers
Carlos Mora-Corral and Marcos Oliva
Department of Mathematics, Faculty of Sciences, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain
July 25, 2018
Abstract
We start from a variational model for nematic elastomers that involves two energies:mechanical and nematic. The first one consists of a nonlinear elastic energy which isinfluenced by the orientation of the molecules of the nematic elastomer. The nematicenergy is an Oseen–Frank energy in the deformed configuration. The constraint of thepositivity of the determinant of the deformation gradient is imposed. The functionals arenot assumed to have the usual polyconvexity or quasiconvexity assumptions to be lowersemicontinuous. We instead compute its relaxation, that is, the lower semicontinuousenvelope, which turns out to be the quasiconvexification of the mechanical term plus thetangential quasiconvexification of the nematic term. The main assumptions are that thequasiconvexification of the mechanical term is polyconvex and that the deformation is inthe Sobolev space W ,p (with p > n − n the dimension of the space) and does notpresent cavitation. Keywords: nonlinear elasticity; nematic elastomers; relaxation; deformed configuration.
Liquid crystal elastomers are hybrid materials that combine the orientational order of liquidcrystals with the elastic properties of rubber-like solids. They are constituted by a networkof long, crosslinked polymer chains. It is this cross-linking what differentiates a liquid crystalelastomer from an ordinary liquid crystal polymer. In the inner structure of these elastomers,some elongated rigid monomer units (called mesogens ) are incorporated to the polymer chain.As any liquid crystal, it can have several phases, according to its internal ordering; they areusually classified in nematic, smectic and cholesteric. In the nematic phase, which is thestudy of this work, the molecules self-align to have a long-range directional order. In fact,most nematic liquid crystals are uniaxial: they have one axis that is longer and preferred.When we assume that the degree of order is fixed (along space and time), the order can bedescribed by a unit vector field ~n , indicating the preferred axis: this leads to the Oseen–Franktheory. In fact, if their degree of order is not fixed, then the more elaborated Landau–de1ennes’ Q -tensor theory is used instead. Classic references for liquid crystals are [22, 49], andone specifically for liquid crystal elastomers is [53].In the small deformation regime, the director field ~n can be defined in the referenceconfiguration, but when large deformations are present, it has to be evaluated at points in thedeformed configuration (see [25, 11]). Thus, while in hyperelasticity [5] one usually assumesthat the mechanical energy of a deformation u : Ω → R n is of the form(1.1) ˆ Ω W ( Du ( x )) d x, (where Ω ⊂ R n represents the body in its reference configuration), the coupling of rubber elas-ticity with the orientational order of the molecules produces a strong anisotropic behaviour,and the energy is given by(1.2) I mec ( u, ~n ) := ˆ Ω W ( Du ( x ) , ~n ( u ( x ))) d x. In this way, the energy density not only depends on the deformation gradient Du but it is alsoinfluenced by the director field ~n evaluated in the deformed configuration. Normally [25, 11],given an elastic-energy function W and a fixed degree of amplitude α >
0, one takes(1.3) W ( F, ~n ) = W (cid:0)(cid:0) α − ~n ⊗ ~n + √ α ( I − ~n ⊗ ~n ) (cid:1) F (cid:1) . The material parameter α describes the amount of local distortion, and the tensor α − ~n ⊗ ~n + √ α ( I − ~n ⊗ ~n ) represents a volume-preserving uniaxial stretch of amplitude α − alongthe direction ~n ; here I denotes the identity matrix. In this work, however, we allow for ageneral dependence of ~n , so that W is not necessarily of the form (1.3). In fact, for the sakeof generality, in this article we work in dimension n , despite the physically relevant case is, ofcourse, n = 3.The vector field ~n takes values in the unit sphere S n − , although, because of the head-to-tail symmetry of the nematics (i.e., the fact that ~n is indistinguishable from − ~n ; see, e.g., [22]),it should take values in the real projective space of dimension n −
1; see [10] for a comparisonbetween the two models. Still, we adopt the more usual approach of S n − and, in order totake into account the head-to-tail symmetry, the energy density W : R n × n × S n − → [0 , ∞ ] of(1.2) has to satisfy W ( F, ~n ) = W ( F, − ~n ) for all F ∈ R n × n and ~n ∈ S n − . It must also meetthe principle of objectivity, but in this work we will not use that assumption.The model that we adopt for the nematic elastomers is, with some small generalizations,that of Barchiesi and DeSimone [11] (see also [25, 2] for earlier studies and [12] for a laterslight generalization, which in fact is the starting point of this work). Accordingly, the energy I associated to the deformation u and the director ~n is the sum of two contributions: I = I mec + I nem , where I mec is as in (1.2), and(1.4) I nem ( u, ~n ) := ˆ u (Ω) V ( ~n ( y ) , D~n ( y )) d y. The term I mec is, as explained above, the mechanical energy of the deformation, where theeffect of the orientation of the molecules is taken into account. The term I nem , the nematic2nergy, is an Oseen–Frank energy in the deformed configuration; we have denoted by D~n the gradient of ~n . It is important that the function V that appears in (1.4) has an explicitdependence on ~n , since the most typical Oseen–Frank energy is (in dimension 3) of the form(1.5) K (div ~n ) + K ( ~n · curl ~n ) + K | ~n × curl ~n | + ( K + K ) (cid:0) tr( D~n ) − (div ~n ) (cid:1) , for some constants K , . . . , K , although sometimes the easier particular case(1.6) K | D~n | is used, which is the so-called one-constant approximation and corresponds to the choice K = K = K , K = 0. In general, the role of the energy density V is to penalize variationsof the nematic director, and, more precisely, the main types of distortion in a nematic: splay,twist and bend. We recall that, although formula (1.5) is usually applied when ~n is definedin the reference configuration Ω, it is also a valid model when ~n is defined in the deformedconfiguration u (Ω). In this case, the head-to-tail symmetry requests V ( ~n, G ) = V ( − ~n, − G )for all arguments ( ~n, G ) where V is defined.Existence of minimizers for the functional I was proved first in [11] and then generalizedin [12], for W of the form (1.3) and V being (1.6). In any case, it was clear from the proofthat the key hypotheses were the polyconvexity of W and the quasiconvexity of V . Theseassumptions imply the lower semicontinuity of both functionals I mec and I nem , and, togetherwith suitable coercivity assumptions, the direct method of the calculus of variations guaranteesthe existence of minimizers. The main difficulty in that analysis were the composition ~n ◦ u in the term I mec (since composition is not continuous in general with respect to the weaktopology) and the fact that the domain of integration in I nem depends on u . Those obstacleswere overcome by the use of a local invertibility property for the class of deformations u inthe admissible set.In this work we remove the conditions leading to the lower semicontinuity: the function W is not polyconvex (not even quasiconvex) and V is not quasiconvex (in fact, not tangentiallyquasiconvex , which is the natural convexity assumption in this context; see below). Then,minimizers may not exist, and the usual approach is the computation of a relaxed (or effective )energy. Relaxation typically indicates the formation of microstructure; see, e.g., [8, 43, 17] inthe context of elasticity, and [24, 51, 15] for nematic elastomers.Since the result of Dacorogna [19], we know that under p -growth conditions (where p is theexponent of the Sobolev space W ,p where the problem is set), the relaxation of a functionalof the form (1.1) is(1.7) ˆ Ω W qc ( Du ( x )) d x, where W qc is the quasiconvexification of W . However, a p -growth condition is incompatiblewith the standard assumption in nonlinear elasticity in which it is required that W is infinityin matrices F with det F W ( F ) → ∞ as det F → . W satisfying (1.8). The conclusion is that (1.7) is indeed the relaxation of (1.1), whereas themain assumptions are that W qc is polyconvex, and that the exponent p of the Sobolev spacewhere the problem is set satisfies p > n .When Ω ′ ⊂ R n is a fixed domain, the relaxation of an energy of the form ˆ Ω ′ V ( D~n ( y )) d y when ~n takes values in the unit sphere (or, in general, in a manifold) was proved in Dacorogna et al. [21] to be ˆ Ω ′ V tqc ( D~n ( y )) d y, where V tqc is the tangential quasiconvexification of V (see Section 3 for the definition). In ourcase, however, the domain of integration u (Ω) in I nem varies along the minimizing sequenceor the test functions, so the result of [21] is not directly applicable. Our function V also hasan extra dependence on ~n , but this is not a problem because it is a lower-order perturbation(see [3]).Finally, it is immediate to see from the definition that the relaxation of a sum is less thanor equal to the sum of the relaxations, so knowing the relaxation of each term I mec and I nem is insufficient to compute the relaxation of I , unless we have an extra condition implying thatthe two processes of relaxation do not interfere.In this paper we prove that the relaxation of I is I ∗ := I ∗ mec + I ∗ nem , with I ∗ mec ( u, ~n ) := ˆ Ω W qc ( Du ( x ) , ~n ( u ( x ))) d x, I ∗ nem ( u, ~n ) := ˆ u (Ω) V tqc ( ~n ( y ) , D~n ( y )) d y, where W qc is the quasiconvexification of W in the first variable and, as in [18], W qc is assumedto be polyconvex. The exponent p of the Sobolev space where u lies satisfies p > n −
1, whichconstitutes an improvement of the result of [18]. In the next paragraphs we comment on themain ideas of the proof.A relaxation result is usually proved in two steps: a lower bound and an upper bound. Thelower bound inequality consists in proving that the functional I ∗ is lower semicontinuous, andthe proof of this fact is a slight generalization of that of [12]. Hence, the bulk of the proof ofthe relaxation result relies, as in [18], in the upper bound, which amounts to the constructionof a recovery sequence : for each ( u, ~n ) we must find a sequence { ( u j , ~n j ) } j ∈ N such that u j → u in L (Ω , R n ), ~n j → ~n in L (in a precise sense, since the domain of definition of each ~n j varies)and I ( u j , ~n j ) → I ∗ ( u, ~n ) as j → ∞ .We start with the term I mec . We recall from [18] that the reason to choose u to be in theSobolev space W ,p with p > n is because this space makes the determinant of the gradientweakly continuous in L , i.e., if u j ⇀ u as j → ∞ in W ,p with det Du j > j ∈ N then det Du j ⇀ det Du in L . Functions in W ,p with p > n also enjoy nice properties suchas the continuity (this is Morrey’s [40] embedding theorem for p > n and was proved in [50]4or p = n under the assumption det Du > W ,p with p > n −
1, as well asextra regularity properties of such functions, provided some additional conditions hold; see[5, 52, 44, 45, 33, 35]. In fact, the possibility of lowering the exponent from p > n to p > n − etal. [12], where it was defined a class A p of functions u ∈ W ,p ( p > n −
1) with det
Du > cavitation occurs (cavitation is the formation of voids in thematerial, see [45]). This class contains the familiar classes A p,q , studied in [5, 52, 44], formedby the Sobolev maps u in W ,p such that cof Du ∈ L q and det Du > p > n − q > nn − . It was proved in [12] that many properties that W ,n enjoy also hold in A p . Themost important ones for this work are the weak continuity of the determinant and the localinvertibility, which states that for a.e. x ∈ Ω there is r > u is invertible in B ( x, r ).This local invertibility property is the key to analyzing functionals like I that involve bothreference and deformed configurations. Thus, the recovery sequence { u j } j ∈ N for u and, hence,the treatment of the term I mec is an adaptation of the construction of [18] but using sometools of [12]. As a direct corollary of our study we obtain that the relaxation result of [18]can be extended to the functions in the class A p (choosing W not depending on ~n and V = 0,even though V = 0 does not satisfy our assumptions). They key idea is to modify the valueof a given u in balls, so that in those balls u is replaced by a certain composition u ◦ v in sucha way that the orientation-preserving condition remains and that the modified function stillbelongs to A p . Moreover, the image of u coincides with the image of the modified function. Inthis way, we construct a sequence { u j } j ∈ N in A p such that u j (Ω) = u (Ω) for all j ∈ N , u j → u in L (Ω , R n ) and I mec ( u j , ~n ) → I ∗ mec ( u, ~n ) as j → ∞ . At this point, we ought to mention thatthe image u (Ω) requires a precise definition, since u is, in principle defined a.e., and u (Ω)must be open so that ~n is in the Sobolev space W ,s ( u (Ω) , S n − ). These technicalities weresolved in [12].The term I nem is tackled as in [21] with the use of the tangential convexification. Inprinciple, the only obstruction to apply their result directly is that the domain u (Ω) mayvary along the recovery sequence, but, as explained in the previous paragraph, the recoverysequence { u j } j ∈ N constructed for u satisfies that u j (Ω) = u (Ω). This equality is also thereason why the two processes of relaxation do not interfere and we have that the relaxationof I is the sum of the relaxations, i.e., I ∗ = I ∗ mec + I ∗ nem .Although the motivation of this work is the model for nematic elastomers explained above,the techniques presented here should be useful for other models involving reference and de-formed configurations, like those in magnetoelasticity (see [46, 38, 12]) or the Landau–deGennes model for liquid crystal elastomers (see [14, 12]). In this respect, this work seems tobe the first study where the relaxation in the deformed configuration has been performed.The article is structured as follows. Section 2 establishes the definitions and notationsused throughout the paper. Section 3 reviews the concepts of polyconvexity, quasiconvexityand tangential quasiconvexity. In Section 4 we define the class A p and recall some resultsfrom [12] that will be used in the paper. We also show some new results in the class A p in order to prove that the recovery sequence to be constructed in Section 7 is indeed in A p .5ection 5 proves the lower bound inequality, as well as the existence of minimizers for I ∗ . InSection 6 we recall three auxiliary results from [18] about the product of L functions andthe chain rule for Sobolev functions. Section 7 is the core of the paper: we prove the upperbound inequality by the construction of a recovery sequence. The paper finishes with Section8, where the relaxation result is established as a consequence of the results of Sections 5 and7. In this section we establish the general notation and definitions used in the paper. Wepostpone the definitions regarding the class A p to Section 4.We will work in dimension n >
2. In all the paper, Ω is a non-empty bounded open set of R n , which represents the body in its reference configuration.The closure of a set A is denoted by ¯ A and its boundary by ∂A . Given two sets U, V of R n , we will write U ⊂⊂ V if U is bounded and ¯ U ⊂ V . The open ball of radius r > x ∈ R n is denoted by B ( x, r ).Given a square matrix A ∈ R n × n , its determinant is denoted by det A . The adjugatematrix adj A ∈ R n × n satisfies (det A ) I = A adj A , where I denotes the identity matrix. Thetranspose of adj A is the cofactor cof A . If A is invertible, its inverse is denoted by A − .The inner (dot) product of vectors and of matrices will be denoted by · and their associatednorms are denoted by |·| . Given a, b ∈ R n , the tensor product a ⊗ b is the n × n matrix whosecomponent ( i, j ) is a i b j . The set R n × n + denotes the subset of matrices in R n × n with positivedeterminant, while SL ( n ) ⊂ R n × n is the set of matrices with determinant one. The set S n − denotes the subset of unit vectors in R n .The symbol . is used to indicate that the quantity of the left-hand side is less than orequal to a positive constant (whose precise value is not important) times the right-hand side.This constant is, of course, independent of the main quantity to estimate, which should beclear from the context.The Lebesgue measure in R n is denoted by |·| , and the ( n − H n − . For 1 p ∞ , the Lebesgue L p and Sobolev W ,p spaces are defined inthe usual way. So are the functions of class C k , for k a positive integer of infinity, and theirversions C kc of compact support. The derivative of a Sobolev or C k function u is written Du .The conjugate exponent of p is p ′ . We will indicate the domain and target space, as in, forexample, L p (Ω , R n ), except if the target space is R , in which case we will simply write L p (Ω);the corresponding norm is written k·k L p (Ω , R n ) . Given S ⊂ R n , the space L p (Ω , S ) denotes theset of u ∈ L p (Ω , R n ) such that u ( x ) ∈ S for a.e. x ∈ Ω, and analogously for other functionspaces. Weak convergence in L p or W ,p is indicated by ⇀ , while ∗ ⇀ is the symbol for weak ∗ convergence in L ∞ . Strong or a.e. convergence is denoted by → . Given a measurable set A the symbol ffl A denotes the integral in A divided by the measure of A . The identity functionin R n is denoted by id. 6 Polyconvexity, quasiconvexity and tangential quasiconvexity
Quasiconvexity is a central concept in the calculus of variations, since, under suitable growthassumptions, it is necessary and sufficient for the lower semicontinuity of functionals of theform (1.1) in the weak topology of W ,p (see the pioneering results of [39, 1] or the monograph[20]). However, no lower semicontinuity results have been proved so far for quasiconvexintegrands W satisfying (1.8). Here is where the concept of polyconvexity comes into play (see,e.g., [5, 7, 20]). Let τ be the number of minors of an n × n matrix; we call R τ + := R τ − × (0 , ∞ )and denote by M ( F ) ∈ R τ the collection of all the minors of an F ∈ R n × n in a given ordersuch that its last component is det F ; we denote by M ( F ) ∈ R τ − the collection of all theminors of an F ∈ R n × n except the determinant, in a given order. For the sake of clarity, inthe following definition of polyconvexity, we single out three cases, according to whether thedomain of definition is the set of all matrices or only those with positive determinant or onlythose with determinant one. Definition 3.1. a) A Borel function W : SL ( n ) → R ∪ {∞} is polyconvex if there exists aconvex function Φ : R τ − → R ∪ {∞} such that W ( F ) = Φ( M ( F )) for all F ∈ SL ( n ) .b) A Borel function W : R n × n + → R ∪ {∞} is polyconvex if there exists a convex function Φ : R τ + → R ∪ {∞} such that W ( F ) = Φ( M ( F )) for all F ∈ R n × n + .c) A Borel function W : R n × n → R ∪ {∞} is polyconvex if there exists a convex function Φ : R τ → R ∪ {∞} such that W ( F ) = Φ( M ( F )) for all F ∈ R n × n . We remark that if a W : SL ( n ) → R ∪ {∞} or W : R n × n + → R ∪ {∞} is polyconvex, thenits extension by infinity to R n × n is also polyconvex.In our study, we will deal with functions W with values in R ∪{∞} defined in SL ( n ) × S n − , R n × n + × S n − or R n × n × S n − . We will say that they are polyconvex in the first variable (or,in short, polyconvex) if W ( · , ~n ) is polyconvex for all ~n ∈ S n − .We now recall the classical concept of quasiconvexity. Its definition is done so that thefunction can take infinite values (see, e.g., [9]). Definition 3.2.
A Borel function W : R n × n → R ∪ {∞} is quasiconvex if for all F ∈ R n × n and all ϕ ∈ W , ∞ ( B (0 , , R n ) with ϕ ( x ) = F x on ∂B (0 , , we have W ( F ) B (0 , W ( Dϕ ) d x. The equality ϕ ( x ) = F x on ∂B (0 ,
1) is understood in the sense of traces. A Borel function W : SL ( n ) → R ∪ {∞} or W : R n × n + → R ∪ {∞} is quasiconvex if its extension by infinityis quasiconvex.When W takes always finite values, there are some possible equivalent definitions of its quasiconvexification (see, e.g., [20]), but when W is infinity in some parts of its domain,the definitions are no longer equivalent. We adopt that of [18], which is the natural onecorresponding to Definition 3.2 and reads as follows.7 efinition 3.3. The quasiconvexification W qc : R n × n → R ∪ {∞} of a Borel function W : R n × n → R ∪ {∞} is defined as W qc ( F ) := inf ( B (0 , W ( Dϕ ) d x : ϕ ∈ W , ∞ ( B (0 , , R n ) , ϕ ( x ) = F x on ∂B (0 , ) . For functions W : R n × n × S n − → R ∪ {∞} , its quasiconvexification W qc refers to thefirst variable. It is well known that a finite-valued quasiconvex function is rank-one convex;in particular, it is continuous. When the function takes infinite values, this fact was provedin [26]. For functions W : R n × n × S n − → R ∪ {∞} , the corresponding continuity result is asfollows. Proposition 3.4.
Assume that W : R n × n + × S n − → [0 , ∞ ) is continuous and there exists an h : [0 , → [0 , ∞ ) with lim t → h ( t ) = 0 such that for all F ∈ R n × n + and ~n, ~m ∈ S n − , (3.1) | W ( F, ~n ) − W ( F, ~m ) | h ( | ~n − ~m | ) W ( F, ~n ) . Extend W by infinity outside R n × n + × S n − . Then W qc | R n × n + × S n − is continuous.Proof. First we prove that for each G ∈ R n × n + there exists M G > ~n, ~ℓ ∈ S n − ,(3.2) (cid:12)(cid:12)(cid:12) W qc ( G, ~n ) − W qc ( G, ~ℓ ) (cid:12)(cid:12)(cid:12) M G h ( | ~n − ~ℓ | ) . Indeed, fix ε > ~m ∈ S n − let ψ ~m ∈ W , ∞ ( B (0 , , R n ) be such that ψ ( x ) = Gx on ∂B (0 ,
1) and B (0 , W ( Dψ ~m , ~m ) d x W qc ( G, ~m ) + ε. Define M G = sup ~m ∈ S n − W ( G, ~m ), which satisfies M G < ∞ thanks to the continuity of W .Moreover, for each ~m ∈ S n − , W qc ( G, ~m ) W ( G, ~m ) M G , so sup ~m ∈ S n − B (0 , W ( Dψ ~m , ~m ) d x M G + ε. Now, for all ~n, ~ℓ ∈ S n − , W qc ( G, ~n ) − W qc ( G, ~ℓ ) B (0 , h W ( Dψ ~ℓ , ~n ) − W ( Dψ ~ℓ , ~ℓ ) i d x + ε h ( | ~n − ~ℓ | ) B (0 , W ( Dψ ~ℓ , ~ℓ ) d x + ε h ( | ~n − ~ℓ | ) ( M G + ε ) + ε. As this is true for all ε > W qc ( G, ~n ) − W qc ( G, ~ℓ ) M G h ( | ~n − ~ℓ | ) , F ∈ R n × n + and ~ℓ ∈ S n − and fix ε >
0. By [26, Th. 2.4 and Prop. 2.3], W qc ( · , ~ℓ )is continuous. Therefore, there exists δ > G ∈ R n × n + satisfies | G − F | δ then (cid:12)(cid:12)(cid:12) W qc ( G, ~ℓ ) − W qc ( F, ~ℓ ) (cid:12)(cid:12)(cid:12) ε, so for all ~n ∈ S n − we have, using (3.2) and the triangle inequality, (cid:12)(cid:12)(cid:12) W qc ( G, ~n ) − W qc ( F, ~ℓ ) (cid:12)(cid:12)(cid:12) M G h ( | ~n − ~ℓ | ) + ε M F,δ h ( | ~n − ~ℓ | ) + ε, where M F,δ := sup (cid:8) M G : G ∈ R n × n + , | G − F | δ (cid:9) , which is finite because of the continuityof W . This concludes the proof.The proof under incompressibility is analogous and will be omitted. Its statement is asfollows. Proposition 3.5.
Assume that W : SL ( n ) × S n − → [0 , ∞ ) is continuous and there existsan h : [0 , → [0 , ∞ ) with lim t → h ( t ) = 0 such that for all F ∈ SL ( n ) and ~n, ~m ∈ S n − ,inequality (3.1) holds. Extend W by infinity outside SL ( n ) × S n − . Then W qc | SL ( n ) × S n − iscontinuous. We now explain the concept of tangential quasiconvexity and tangential quasiconvexifi-cation . For this, we fix a C manifold M embedded in R n (although we will always take M = S n − ); all concepts of tangential are referred to the manifold M . For each z ∈ M wedenote the tangent space of M at z by T z M . Given a Sobolev function ~n defined in an openset U ⊂ R n such that ~n ( y ) ∈ M for a.e. y ∈ U , we have that D~n ( y ) ∈ ( T ~n ( y ) M ) n for a.e. y ∈ U . Therefore, the function V of (1.4) need only be defined in T n M := { ( z, ζ ) : z ∈ M , ζ ∈ ( T z M ) n } . Thus, we consider a Borel function V : T n M → [0 , ∞ ). The following definition is due toDacorogna et al. [21] when V does not depend on the first variable. The natural definitionfor a V defined in the whole T n M is straightforward (see [3]). Definition 3.6.
Let V : T n M → [0 , ∞ ) be a Borel function.a) V is tangentially quasiconvex if for all ( z, ζ ) ∈ T n M and all ϕ ∈ W , ∞ ( B (0 , , T z M ) with ϕ ( y ) = ζy on ∂B (0 , we have V ( z, ζ ) B (0 , V ( z, Dϕ ( y )) d y. b) The tangential quasiconvexification V tqc : T n M → [0 , ∞ ) of V is V tqc ( z, ζ ):= inf ( B (0 , V ( z, Dϕ ( y )) d y : ϕ ∈ W , ∞ ( B (0 , , T z M ) , ϕ ( y ) = ζy on ∂B (0 , ) . ϕ ( y ) = ζy on ∂B (0 ,
1) is understood in the sense of traces and we areregarding ζ as an n × n matrix. Note that the fact ϕ ∈ W , ∞ ( B (0 , , T z M ) implies Dϕ ( y ) ∈ ( T z M ) n for a.e. y ∈ B (0 , B (0 ,
1) as domain of integration is irrelevant.From the definitions, it is immediate to check that V tqc is tangentially quasiconvex andthat V is tangentially quasiconvex if and only if V = V tqc .The next proposition and theorem summarize the main results of [21]; again, the formu-lation is adapted to cover a dependence of V on the first variable as well (see [3]). Proposition 3.7. a) For each z ∈ M , let P z ∈ R n × n be the matrix corresponding to theorthogonal projection from R n onto T z M . Define ¯ V : M × R n × n → [0 , ∞ ) as ¯ V ( z, ζ ) := V ( z, P z ζ ) and let ¯ V qc be the quasiconvexification of ¯ V with respect to the second variable. Then V tqc = ¯ V qc | T n M .b) Let M = S n − . Define ¯ V : S n − × R n × n → [0 , ∞ ) as ¯ V ( z, ζ ) := V ( z, ( I − z ⊗ z ) ζ ) and let ¯ V qc be the quasiconvexification of ¯ V with respect to the second variable. Then V tqc = ¯ V qc | T n S n − . Theorem 3.8.
Let Ω ′ ⊂ R n be open and bounded. Let s > . Let V : T n M → [0 , ∞ ) becontinuous and satisfy V ( z, ζ ) C (1 + | ζ | s ) , ( z, ζ ) ∈ T n M for some C > . Let ~n ∈ W ,s (Ω ′ , M ) . The following hold:a) If V is tangentially quasiconvex then, for any sequence { ~n j } j ∈ N ⊂ W ,s (Ω ′ , M ) convergingweakly to ~n in W ,s (Ω ′ , M ) , we have ˆ Ω ′ V ( ~n ( y ) , D~n ( y )) d y lim inf j →∞ ˆ Ω ′ V ( ~n j ( y ) , D~n j ( y )) d y. b) inf (cid:26) lim inf j →∞ ˆ Ω V ( ~n j ( y ) , D~n j ( y )) d y : ~n j ⇀ ~n in W ,s (Ω ′ , M ) (cid:27) = ˆ Ω ′ V tqc ( ~n ( y ) , D~n ( y )) d y. As commented in [41], using Proposition 3.7, we find that V is tangentially quasiconvexif and only if it is the restriction of a quasiconvex function (in the second variable) ¯ V : M × R n × n → [0 , ∞ ). Since finite-valued quasiconvex functions are continuous (because theyare rank-one convex), we infer that any tangentially quasiconvex V : T n M → [0 , ∞ ) iscontinuous in the second variable. 10 Class A p In this section we define the class A p of functions that will be the object of this work. Its mainaim is to present the results showing that, similarly to what occurs in Sobolev spaces, undersome additional conditions the cut-and-paste of functions in the class A p is still in the class A p (Lemma 4.8) and the composition of an orientation-preserving Lipschitz function with afunction of class A p is still in A p (Lemma 4.10). The reader not interested in the technicalitiesof the class A p may omit this section and admit Lemmas 4.8 and 4.10.The class A p consists, roughly, in the set of u ∈ W ,p (Ω , R n ) such that det Du > cavitation occurs. Cavitation is the formation of voids in some materials in extension (see[29] for the physical process and [45, 47, 16, 33, 34, 35] for some mathematical developments).The class A p was originally defined in M¨uller [42], then used by Giaquinta et al. [30], and inBarchiesi et al. [12] it was proved the local invertibility and extra regularity properties.This section consists of two subsections. In Subsection 4.1 we define the class A p , togetherwith many associated concepts, and state the known results that will be useful in Subsection4.2, where we prove the new results needed for the construction of the recovery sequence inSection 7. This subsection presents the definition of A p and its related concepts. It also states the resultsthat are useful in Subsection 4.2 in order to prove Lemmas 4.8 and 4.10. Definition 4.1.
A function u : Ω → R n is said to be injective a.e. in a subset A of Ω if thereexists a set N ⊂ A such that | N | = 0 and u | A \ N is injective. We will use the following result.
Proposition 4.2.
Given u ∈ W , (Ω , R n ) with det Du > a.e., there exists a measurable set Ω ⊂ Ω with | Ω \ Ω | = 0 such that:a) u | Ω satisfies the change of variables formula.b) If for some A ⊂ Ω the restriction u | A is injective a.e., then u | A ∩ Ω is injective. Part a) is due to [31] (see also [45, Prop. 2.6]). Part b) is due to [34, Lemma 3]. The setΩ is not uniquely defined; it can be given a precise definition (see, [45, 16, 34]) but this isnot important in the sequel: given a u we just fix any such Ω .For any measurable set A of Ω, we define the geometric image of A under u as u ( A ∩ Ω ),and we denote it by im G ( u, A ).We will use the topological degree for continuous functions (see, e.g., [23, 27]): if U ⊂ R n is a bounded open set, u : ¯ U → R n is continuous and y ∈ R n \ u ( ∂U ), we denote by deg( u, U, y )the degree of u in U at y . If u : ∂U → R n is continuous, its degree deg( u, U, · ) is defined asthe degree of any continuous extension ¯ u : ¯ U → R n , which exists thanks to Tietze’s theoremand does not depend on the extension due to the homotopy-invariance of the degree (see, e.g.,[23, Th. 3.1.(d6)], [27, Th. 2.4]). If u ∈ W ,p ( ∂U, R n ) with p > n −
1, by Morrey’s embedding,11 has a continuous representative. We define the degree of u in U , written deg( u, U, · ), as thedegree of its continuous representative.Now, if u ∈ W ,p (Ω , R n ) we define u ∗ as its precise representative (see, e.g., [54]): u ∗ ( x ) := lim r → B ( x,r ) u ( z ) d z, if that limit exists, and u ∗ is undefined elsewhere. It is well known that the above limit existsexcept on a set of p -capacity zero.Next, we define the topological image (introduced by ˇSver´ak [52]; see also [45]). Definition 4.3.
Let u ∈ W ,p (Ω , R n ) with p > n − .a) Given an open U ⊂⊂ Ω such that u ∗ ∈ W ,p ( ∂U, R n ) , we define im T ( u, U ) , the topologicalimage of U under u , as the set of y ∈ R n \ u ( ∂U ) such that deg( u ∗ , U, y ) = 0 .b) We define im T ( u, Ω) , the topological image of Ω under u , as the union of im T ( u, U ) when U runs over all open U ⊂⊂ Ω such that u ∗ ∈ W ,p ( ∂U, R n ) . Thanks to the continuity of the topological degree for continuous functions we have thatim T ( u, U ) is an open set, and so is im T ( u, Ω), as a union of open sets. Moreover,im T ( u, Ω) = [ i ∈ N im T ( u, U i )for every family { U i } i ∈ N such that Ω = S i ∈ N U i , U i ⊂⊂ Ω and u ∗ ∈ W ,p ( ∂U i , R n ). Definition 4.4.
Let u ∈ W , (Ω , R n ) and q > . Suppose that det Du ∈ L (Ω) and cof Du ∈ L q (Ω , R n × n ) . For φ ∈ W ,q ′ (Ω) ∩ L ∞ (Ω) and g ∈ C c ( R n , R n ) define E Ω ( u, φ, g ) := ˆ Ω [cof Du ( x ) · ( g ( u ( x )) ⊗ Dφ ( x )) + det Du ( x ) φ ( x ) div g ( u ( x ))] d x. Now we present the class of functions with which we will work in the rest of the chapter.
Definition 4.5.
For each p > n − and q > , we define A p,q (Ω) as the set of u ∈ W ,p (Ω , R n ) , such that det Du ∈ L (Ω) , cof Du ∈ L q (Ω , R n × n ) , det Du > a.e. and (4.1) E Ω ( u, φ, g ) = 0 , for all φ ∈ C c (Ω) and g ∈ C c ( R n , R n ) . We define A p (Ω) = A p, (Ω) . We denote by A p (Ω) the set of functions u ∈ A p (Ω) that satisfy det Du = 1 a.e. If the domain Ω is clear from the context, we will sometimes abbreviate the notation to A p,q , A p and A p .Observe that u ∈ W ,p implies cof Du ∈ L pn − , so A p (Ω) = A p,t (Ω) for t ∈ [1 , pn − ].Moreover, thanks to the result of [44] we have that if u ∈ W ,p satisfies cof Du ∈ L q anddet Du > p > n − q > nn − then u ∈ A p,q .The following local invertibility result is a particular case of [12, Cor. 4.7].12 roposition 4.6. Let u ∈ A p (Ω) . Then, for a.e. x ∈ Ω there exists r > such that u isinjective a.e. in B ( x, r ) . If u is injective a.e. in some U ⊂⊂ Ω then, by Proposition 4.2, u is injective in U ∩ Ω .Therefore u : U ∩ Ω → im G ( u, U ) is a bijection. If, in addition, u ∗ ∈ W ,p ( ∂U, R n ) then,thanks to [12, Th. 4.1], | im T ( u, U ) \ im G ( u, U ) | = | im G ( u, U ) \ im T ( u, U ) | = 0and, hence, the next definition of local inverse of a function in the class A p makes sense. Definition 4.7.
Let u ∈ A p (Ω) and U ⊂⊂ Ω be such that u is injective a.e. in U and u ∗ ∈ W ,p ( ∂U, R n ) . The inverse ( u | U ) − : im T ( u, U ) → R n is defined a.e. as ( u | U ) − ( y ) = x ,for each y ∈ im G ( u, U ) , and where x ∈ U ∩ Ω satisfies u ( x ) = y . By [12, Prop. 5.3] we have( u | U ) − ∈ W , (im T ( u, U ) , R n ) and D ( u | U ) − = (cid:0) Du ◦ ( u | U ) − (cid:1) − a.e. In this subsection we provide some auxiliary results for functions in A p . To be precise, forthe recovery sequence of Section 7 a typical construction is to cut and paste functions in A p ,as well as to compose a Lipschitz function with one in A p . The main aim of this subsectionis to show that, under suitable assumptions, these two operations make a new function stillin A p .The following lemma shows that when we paste two functions in the class A p that coincidein a neighborhood of a sphere, the resulting function is also in A p . Note that it is not sufficientthat the two functions coincide on the sphere, because a cavity may appear at a point of thesphere, and, hence, the resulting function will not be in A p (this phenomenom is known as cavitation at the boundary ; see [45, 47, 48, 32, 37]). Lemma 4.8.
Let p > n − and q > . Let B, B ′ be open sets such that B ′ ⊂⊂ B ⊂⊂ Ω .Assume u ∈ A p,q (Ω) , v ∈ A p,q ( B ) and u = v a.e. in B \ B ′ . Then the function w := ( v in B ′ ,u in Ω \ B ′ is in A p,q (Ω) . If, in addition, u ∈ A p (Ω) and v ∈ A p ( B ) , then w ∈ A p (Ω) .Proof. All the conditions in the definition of A p,q are immediate to check except (4.1), so let φ ∈ C c (Ω) and g ∈ C c ( R n , R n ). Fix an η ∈ C c (Ω) with support in B such that η = 1 in B ′ .Then, as η φ ∈ C c ( B ), E Ω ( w, φ, g ) = E Ω ( w, η φ, g ) + E Ω ( w, (1 − η ) φ, g ) = E B ( w, η φ, g ) + E Ω \ ¯ B ′ ( w, (1 − η ) φ, g )= E B ( v, η φ, g ) + E Ω \ ¯ B ′ ( u, (1 − η ) φ, g ) = 0 + E Ω ( u, (1 − η ) φ, g ) = 0 + 0 = 0 . This concludes the proof also in the case u ∈ A p (Ω) and v ∈ A p ( B ).13n the next lemma we see that E Ω ( u, φ, g ) is also zero for u ∈ A p,q and φ in the correctSobolev space. Lemma 4.9.
Let p > n − and q > . Let u ∈ A p,q (Ω) , g ∈ C c ( R n , R n ) and φ ∈ W ,q ′ (Ω) ∩ L ∞ (Ω) . Then E Ω ( u, φ, g ) = 0 .Proof. Let { φ j } j ∈ N be a sequence in C c (Ω) such that φ j → φ in W ,q ′ (Ω) and φ j ∗ ⇀ φ in L ∞ (Ω) as j → ∞ . This sequence can be constructed as follows: first one takes a sequence { ˜ φ j } j ∈ N in C c (Ω) such that ˜ φ j → φ in W ,q ′ (Ω) and a.e., and D ˜ φ j → Dφ a.e. Then, onedefines ¯ φ j = max { ˜ φ j , k φ k L ∞ + 1 } . It is easy to check that ¯ φ j → φ in W ,q ′ (Ω) and ¯ φ j ∗ ⇀ φ in L ∞ (Ω). Then, one takes φ j as a suitable mollification of ¯ φ j . When such φ j have beenconstructed, we have E Ω ( u, φ j , g ) = 0 for all j ∈ N , andlim j →∞ ˆ Ω cof Du ( x ) · ( g ( u ( x )) ⊗ Dφ j ( x )) + det Du ( x ) φ j ( x ) div g ( u ( x )) d x = ˆ Ω cof Du ( x ) · ( g ( u ( x )) ⊗ Dφ ( x )) + det Du ( x ) φ ( x ) div g ( u ( x )) d x, so E Ω ( u, φ, g ) = 0.We prove that the composition of a function in the class A p,q with a Lipschitz functionsatisfying some conditions is still in the class A p,q . The assumptions may look artificial, butwe will see in Section 7 that they will all be satisfied. Lemma 4.10.
Let p > n − and q > . Let u ∈ A p,q (Ω) , B ⊂⊂ Ω a ball, ρ : B → ¯ B Lipschitzsuch that ρ | ∂B = id | ∂B , det Dρ > a.e. and ´ Ω (det Dρ ) − q ′ d x < ∞ . Define z := ( u ◦ ρ in B,u in Ω \ B. Assume that z ∈ W ,p (Ω , R n ) , Dz = ( Du ◦ ρ ) Dρ in B , det Dz ∈ L (Ω) and cof Dz ∈ L q (Ω , R n × n ) . Then z ∈ A p,q (Ω) . If, in addition, u ∈ A p (Ω) and det Dρ = 1 a.e., then z ∈ A p (Ω) .Proof. By definition of A p,q , to prove z ∈ A p,q (Ω) we only have to show that E Ω ( z, φ, g ) = 0for all φ ∈ C c (Ω) and g ∈ C c ( R n , R n ). We have E Ω ( z, φ, g ) = E Ω \ ¯ B ( u, φ, g ) + E B ( u ◦ ρ, φ, g ) . By [52, Th. 8] or [36, Th. 3.3], we have ρ − ∈ W , ( B, R n ) and Dρ − ( y ) = Dρ ( ρ − ( y )) − , det Dρ ( ρ − ( y )) = 1det Dρ − ( y ) , cof Dρ ( ρ − ( y )) = Dρ − ( y ) T det Dρ − ( y )for a.e. y ∈ B , so, by a change of variables, E B ( u ◦ ρ, φ, g ) = ˆ B (cid:2) cof( Du ( y )) Dρ − ( y ) T · ( g ( u ( y )) ⊗ Dφ ( ρ − ( y )))+ det( Du ( y )) φ ( ρ − ( y )) div g ( u ( y )) (cid:3) d y. (4.2) 14y the chain rule (see, e.g., [54, Th. 2.1.11]) we get that φ ◦ ρ − ∈ W , ( B ) and D ( φ ◦ ρ − ) =( Dφ ◦ ρ − ) Dρ − in B . In fact, φ ◦ ρ − ∈ W ,q ′ ( B ) since, changing variables and using thefact that ρ and φ are Lipschitz, we get (cid:13)(cid:13) D ( φ ◦ ρ − ) (cid:13)(cid:13) q ′ L q ′ ( B ) . (cid:13)(cid:13) Dρ − (cid:13)(cid:13) q ′ L q ′ ( B ) = ˆ B (cid:12)(cid:12) Dρ − ( y ) (cid:12)(cid:12) q ′ d y = ˆ B (cid:12)(cid:12) Dρ − ( ρ ( x )) (cid:12)(cid:12) q ′ det Dρ ( x ) d x = ˆ B | cof Dρ ( x ) | q ′ det Dρ ( x ) − q ′ d x . ˆ B det Dρ ( x ) − q ′ d x < ∞ . Equality E B ( u ◦ ρ, φ, g ) = E B ( u, φ ◦ ρ − , g ) is clear in view of (4.2). Define˜ φ := ( φ in Ω \ B,φ ◦ ρ − in B. As ρ | ∂B = id | ∂B , we have that ˜ φ is Sobolev; in fact, ˜ φ ∈ W ,q ′ (Ω). Thanks to Lemma 4.9 wehave E Ω ( u, ˜ φ, g ) = 0, so0 = E Ω ( u, ˜ φ, g ) = E Ω \ ¯ B ( u, φ, g ) + E B ( u, φ ◦ ρ − , g ) = E Ω \ ¯ B ( u, φ, g ) + E B ( u ◦ ρ, φ, g ) = E Ω ( z, φ, g )and, hence, z ∈ A p,q (Ω).If, in addition, u ∈ A p (Ω) and det Dρ = 1 a.e. then det Dz ( x ) = det Du ( ρ ( x )) det Dρ ( x ) =1 for a.e. x ∈ B and det Dz ( x ) = det Du ( x ) = 1 for a.e. x ∈ Ω \ B . Therefore, z ∈ A p (Ω). In this section we prove existence of minimizers of I under the assumptions that W is poly-convex in the first variable and V is tangentially quasiconvex.We first define the set of admissible functions. We will distinguish two cases, accordingto whether the material is compressible (admissible set B and energy functional I ) or incom-pressible (admissible set B and energy functional I ). The energy functional is, in principle,defined in the whole L (Ω , R n ) × L ( R n , R n ) but it will be infinity outside the set of admissiblefunctions.Fix p > n − s >
1. Let Ω ⊂ R n be a bounded Lipschitz domain. Let Γ be an( n − ∂ Ω, and let u : Γ → R n . We define B as the set of ( u, ~n ) ∈ L (Ω , R n ) × L ( R n , R n ) such that u ∈ A p (Ω), u | Γ = u in the sense of traces, Du ( x ) ∈ R n × n + for a.e. x ∈ Ω, ~n | im T ( u, Ω) ∈ W ,s (im T ( u, Ω) , S n − ) and ~n | R n \ im T ( u, Ω) = 0 . Note that no boundary conditions are prescribed for ~n . As for the incompressible case, wedefine B as the set of ( u, ~n ) ∈ B such that Du ( x ) ∈ SL ( n ) for a.e. x ∈ Ω.We define the energy functionals(5.1)
I, I mec , I nem , I , I , mec , I , nem : L (Ω , R n ) × L ( R n , R n ) → [0 , ∞ ]15escribing the nematic elastomer as follows: I mec ( u, ~n ) = ˆ Ω W ( Du ( x ) , ~n ( u ( x ))) d x, if ( u, ~n ) ∈ B , ∞ , otherwise, I nem ( u, ~n ) = ˆ im T ( u, Ω) V ( ~n ( y ) , D~n ( y )) d y, if ( u, ~n ) ∈ B , ∞ , otherwise, I , mec ( u, ~n ) = ( I mec ( u, ~n ) , if ( u, ~n ) ∈ B , ∞ , otherwise, I , nem ( u, ~n ) = ( I nem ( u, ~n ) , if ( u, ~n ) ∈ B , ∞ , otherwise.Finally, I := I mec + I nem and I := I , mec + I , nem .The following result establishes the lower semicontinuity of I in B with respect to the L topology. Its proof is essentially a rewriting of the proofs of [12, Props. 7.1, 7.8 and Th. 8.2],and will only be sketched. Proposition 5.1.
Let s > and p > n − . Let (5.2) ( u j , ~n j ) → ( u, ~n ) in L (Ω , R n ) × L ( R n , R n ) as j → ∞ . Let W : R n × n + × S n − → [0 , ∞ ) be continuous, polyconvex and such that (5.3) W ( F, ~n ) > c | F | p + θ (det F ) , F ∈ R n × n + , ~n ∈ S n − for a constant c > and a Borel function θ : (0 , ∞ ) → [0 , ∞ ) with (5.4) lim t ց θ ( t ) = lim t →∞ θ ( t ) t = ∞ . Let V : T n S n − → [0 , ∞ ) be continuous and tangentially quasiconvex such that (5.5) c | ζ | s − c V ( z, ζ ) c (1 + | ζ | s ) , ( z, ζ ) ∈ T n S n − . Then (5.6) I ( u, ~n ) lim inf j →∞ I ( u j , ~n j ) . Proof.
By taking a subsequence, we can assume that the lim inf of the right-hand side of (5.6)is a limit, and that, in fact, it is finite. The proof of [12, Th. 8.2] shows that u j ⇀ u in W ,p (Ω , R n ) , det Du j ⇀ det Du in L (Ω) ,χ im T ( u j , Ω) D~n j ⇀ χ im T ( u, Ω) D~n in L s ( R n , R n × n ) as j → ∞ χ im T ( u, Ω) D~n stands for the extension of
D~n by zero outside im T ( u, Ω), and analogouslyfor χ im T ( u j , Ω) D~n j , and, by [12, Prop. 7.8], I mec ( u, ~n ) lim inf j →∞ I mec ( u j , ~n j ) . Now let G ⊂⊂ im T ( u, Ω) be open. Then, by [12, Lemma 3.6], there exists j ∈ N suchthat for all j > j we have G ⊂ im T ( u j , Ω). Therefore, ~n j ⇀ ~n in W ,s ( G, R n ) as j → ∞ , soby Theorem 3.8, ˆ G V ( ~n ( y ) , D~n ( y )) d y lim inf j →∞ ˆ G V ( ~n j ( y ) , D~n j ( y )) d y lim inf j →∞ I nem ( u j , ~n j ) . As this is true for all open G ⊂⊂ im T ( u, Ω) we obtain I nem ( u, ~n ) lim inf j →∞ I nem ( u j , ~n j ) , which concludes the proof.The compactness for sequences bounded in energy is as follows. Its proof, again, is arewriting of that of [12, Prop. 7.1 and Th. 8.2] and will be omitted. Proposition 5.2.
Let s > and p > n − . Let W : R n × n + × S n − → [0 , ∞ ) satisfy (5.3) – (5.4) for a constant c > and a Borel function θ : (0 , ∞ ) → [0 , ∞ ) . Let V : T n S n − → [0 , ∞ ) satisfy (5.7) c | ζ | s − c V ( z, ζ ) , ( z, ζ ) ∈ T n S n − . For each j ∈ N , let ( u j , ~n j ) ∈ L (Ω , R n ) × L ( R n , R n ) satisfy sup j ∈ N I ( u j , ~n j ) < ∞ . Then there exist a subsequence (not relabelled) and ( u, ~n ) ∈ B such that (5.2) holds. Proposition 5.1 and 5.2 yield, by the direct method of the calculus of variations, thefollowing result on the existence of minimizers.
Theorem 5.3.
Let s > and p > n − . Let W : R n × n + × S n − → [0 , ∞ ) be continuous,polyconvex and such that (5.3) – (5.4) hold for a constant c > and a Borel function θ :(0 , ∞ ) → [0 , ∞ ) . Let V : T n S n − → [0 , ∞ ) be continuous and tangentially quasiconvex suchthat (5.5) holds. If B 6 = ∅ and I is not identically infinity, then I attains its minimum in B . In the incompressible case, the analogue results are as follows; as commented in [12, Rks.7.9 and 8.4], the incompressibility can easily be taken into account.17 roposition 5.4.
Let s > and p > n − . Let (5.2) hold. Let W : SL ( n ) × S n − → [0 , ∞ ) be continuous, polyconvex and such that (5.8) W ( F, ~n ) > c | F | p − c , F ∈ SL ( n ) , ~n ∈ S n − for a constant c > . Let V : T n S n − → [0 , ∞ ) be continuous and tangentially quasiconvexsuch that bound (5.5) holds. Then I ( u, ~n ) lim inf j →∞ I ( u j , ~n j ) . Proposition 5.5.
Let s > and p > n − . Let W : SL ( n ) × S n − → [0 , ∞ ) satisfy (5.8) for a constant c > . Let V : T n S n − → [0 , ∞ ) satisfy (5.7) . For each j ∈ N , let ( u j , ~n j ) ∈ L (Ω , R n ) × L ( R n , R n ) satisfy sup j ∈ N I ( u j , ~n j ) < ∞ . Then there exist a subsequence (not relabelled) and ( u, ~n ) ∈ B such that (5.2) holds. Theorem 5.6.
Let s > and p > n − . Let W : SL ( n ) × S n − → [0 , ∞ ) be continuous,polyconvex and such that (5.8) holds for a constant c > . Let V : T n S n − → [0 , ∞ ) becontinuous and tangentially quasiconvex such that (5.5) holds. If B = ∅ and I is notidentically infinity, then I attains its minimum in B . In this section we state three results proved in [18] about the product of L functions and thecomposition of a Lipschitz function with a Sobolev function.The next lemma, taken from [18, Lemma 3.1], states that there are many translationssuch that the product of the translated L functions is in L . Lemma 6.1.
Let x ∈ R n and r > . Let ψ ∈ W , ∞ ( B (0 , r ) , ¯ B (0 , r )) , g ∈ L ( B (0 , r )) and f ∈ L ( B ( x , r )) . Then, there exists a measurable set E ⊂ B ( x , r ) of positive measure suchthat for any a ∈ E , the function ˜ f ( x ) := f ( a + ψ ( x − a )) g ( x − a ) , x ∈ B ( a , r ) belongs to L ( B ( a , r )) and k ˜ f k L ( B ( a ,r )) | B (0 , r ) | k f k L ( B ( x , r )) k g k L ( B (0 ,r )) . The following result is a weaker version of [18, Lemma A.1].18 emma 6.2.
Let ψ ∈ W , ∞ ( B (0 , , ¯ B (0 , and f ∈ L ( B (0 , . Then the map ( x, a ) f ( a + ψ ( x − a )) is measurable and for almost all a ∈ B (0 , the function x f ( a + ψ ( x − a )) is in L ( B ( a , . The following version of the chain rule was proved in [18, Lemma A.2].
Lemma 6.3.
Let ψ ∈ W , ∞ ( B (0 , , ¯ B (0 , and u ∈ W , ( B (0 , . Then for almost all a ∈ B (0 , the function w ( x ) := u ( a + ψ ( x − a )) , x ∈ B ( a , belongs to W , ( B ( a , and Dw ( x ) = Du ( a + ψ ( x − a )) Dψ ( x − a ) . If, in addition, ψ = id on ∂B (0 , then w = u on ∂B ( a , in the sense of traces. In this section we prove the upper bound inequality by constructing a recovery sequence.We first present the coercivity, growth and continuity conditions of the energy functions W and V , which are slightly more restrictive that those of Section 5. Fix p > n − q > s >
1. In the compressible case, the conditions on W are as follows.(W) W : R n × n + × S n − → [0 , ∞ ) is continuous and there exist a convex θ : (0 , ∞ ) → [0 , ∞ ), abounded Borel h : [0 , → [0 , ∞ ) and c > θ ( t t ) . (1 + θ ( t )) (1 + θ ( t )) , t , t > , lim t →∞ θ ( t ) t = ∞ , lim inf t → t q ′ − θ ( t ) > , lim t → h ( t ) = 0 , and for all F ∈ R n × n + and ~n, ~m ∈ S n − ,1 c ( | F | p + | cof F | q + θ (det F )) − c W ( F, ~n ) c ( | F | p + θ (det F ) + 1) , | W ( F, ~n ) − W ( F, ~m ) | h ( | ~n − ~m | ) W ( F, ~n ) . The function W is extended to ( R n × n \ R n × n + ) × S n − by infinity. Observe that if ( u, ~n ) ∈ B satisfies I mec ( u, ~n ) < ∞ then u ∈ A p,q (Ω).In the incompressible case, the conditions on W are:19W ) W : SL ( n ) × S n − → [0 , ∞ ) is continuous and there exist a bounded Borel h : [0 , → [0 , ∞ ) and c > t → h ( t ) = 0 and for all F ∈ SL ( n ) and ~n, ~m ∈ S n − ,1 c | F | p − c W ( F, ~n ) c ( | F | p + 1) , | W ( F, ~n ) − W ( F, ~m ) | h ( | ~n − ~m | ) W ( F, ~n ) . The function W is extended to ( R n × n \ SL ( n )) × S n − by infinity. Note that q does not playany role in the incompressible case.The assumption for V is as follows:(V) V : T n S n − → [0 , ∞ ) is continuous and there exists c > c | ζ | s − c V ( z, ζ ) c | ζ | s + c, ( z, ζ ) ∈ T n S n − . We define the admissible spaces B and B as in Section 5, as well as the functionals (5.1).We also define the functionals I ∗ , I ∗ nem , I ∗ mec , I ∗ , I ∗ , nem , I ∗ , mec : L (Ω , R n ) × L ( R n , R n ) → [0 , ∞ ]in a similar way to their counterparts (5.1), but replacing W with W qc and V with V tqc , i.e., I ∗ mec ( u, ~n ) = ˆ Ω W qc ( Du ( x ) , ~n ( u ( x ))) d x, if ( u, ~n ) ∈ B , ∞ , otherwise, I ∗ nem ( u, ~n ) = ˆ im T ( u, Ω) V tqc ( ~n ( y ) , D~n ( y )) d y, if ( u, ~n ) ∈ B , ∞ , otherwise, I ∗ , mec ( u, ~n ) = ( I ∗ mec ( u, ~n ) , if ( u, ~n ) ∈ B , ∞ , otherwise, I ∗ , nem ( u, ~n ) = ( I ∗ nem ( u, ~n ) , if ( u, ~n ) ∈ B , ∞ , otherwise, I ∗ := I ∗ mec + I ∗ nem and I ∗ := I ∗ , mec + I ∗ , nem . Here W qc is the quasiconvexification of W withrespect to the first variable.Now we state the main result of this section: the existence of the recovery sequence. Itfollows from Lemma 7.3 below. Theorem 7.1.
Let q > , s > , p > n − , V satisfy (V) and W satisfy (W) (respec-tively, (W )). Let Ω ⊂ R n open bounded and Lipschitz. Then, for any ( u, ~n ) ∈ L (Ω , R n ) × L ( R n , R n ) there is a sequence { ( u j , ~n j ) } j ∈ N ⊂ L (Ω , R n ) × L ( R n , R n ) such that ( u j , ~n j ) → ( u, ~n ) in L (Ω , R n ) × L ( R n , R n ) as j → ∞ and lim sup j →∞ I ( u j , ~n j ) I ∗ ( u, ~n ) (respectively, lim sup j →∞ I ( u j , ~n j ) I ∗ ( u, ~n ) ). u is modified in a ball. Lemma 7.2.
Assume one of the following:a) W satisfies (W),b) W satisfies (W ),and fix F ∈ R n × n + in case a) and F ∈ SL ( n ) in case b) , ~m ∈ S n − and η ∈ (0 , . Then thereis δ > such that for any ball B = B ( x , r ) , any ~n ∈ L ∞ (im T ( u, B ) , S n − ) and any u ∈ ( A p,q ( B ) in case a) , A p ( B ) in case b) with (7.1) B ( | Du − F | p + | θ (det Du ) − θ (det F ) | + | ~n ◦ u − ~m | p ) d x δ in case a) ,or B ( | Du − F | p + | ~n ◦ u − ~m | p ) d x δ in case b) ,there exist a ∈ B (cid:0) x , r (cid:1) and z ∈ ( A p,q ( B ) in case a) , A p ( B ) in case b) with z = u in B ( x , r ) \ B (cid:0) a , r (cid:1) , im T ( z, Ω) = im T ( u, Ω) , (7.2) ˆ B ( a , r ) W ( Dz, ~n ◦ z ) d x ˆ B ( a , r )( W qc ( Du, ~n ◦ u ) + η ) d x and (7.3) ˆ B | u − z | p d x c r p ˆ B ( W qc ( Du, ~n ◦ u ) + 1) d x for some c > depending on W , n and p . If u is Lipschitz, then so is z .Proof. This proof is partially based on that of [18, Lemma 3.2]. We will only prove the case a) , since the proof of case b) is analogous.The L p bound (7.3) follows from Poincar´e’s inequality, the growth condition of (W) and(7.2) as follows: ˆ B | u − z | p d x = ˆ B ( a , r ) | u − z | p d x . r p ˆ B ( a , r ) | Du − Dz | p d x . r p ˆ B ( a , r ) ( | Du | p + | Dz | p ) d x . r p ˆ B ( a , r ) ( W qc ( Du, ~n ◦ u ) + W ( Dz, ~n ◦ z ) + 1) d x . r p ˆ B ( a , r ) ( W qc ( Du, ~n ◦ u ) + 1) d x r p ˆ B ( W qc ( Du, ~n ◦ u ) + 1) d x,
21o the bulk of the proof consists in showing (7.2).By Definition 3.3 of quasiconvexification, there exists ϕ η ∈ W , ∞ ( B (0 , r ) , R n ) such that ϕ η ( x ) = F x on ∂B (0 , r ), det Dϕ η > B (0 , r ) W ( Dϕ η , ~m ) d x W qc ( F, ~m ) + η. The function F − ϕ η is Lipschitz and is the identity on ∂B (0 , r ), hence, by degree theory(see, if necessary, [6, Th. 1]), F − ϕ η ( B (0 , r )) ⊂ ¯ B (0 , r ). Moreover, F − ϕ η is invertible andits inverse is in W , (see [52, Th. 8] or [36, Th. 3.3]). Take a ∈ B ( x , r ) (to be chosenbelow), call B ′ = B ( a , r ) and set v ( x ) = F − ϕ η ( x − a ) + a and z = ( u ◦ v in B ′ ,u in B ( x , r ) \ B ′ . It is clear that z = u in B ( x , r ) \ B ′ , im T ( v, B ′ ) = B ′ and v − ∈ W , ( B ′ , R n ).By Lemmas 6.3 and 6.2, there exists a null set N such that for all a ∈ B ( x , r ) \ N we have that z ∈ W , ( B ′ , R n ), det Dz ∈ L ( B ) and cof Dz ∈ L q ( B, R n × n ). Moreover, since v | ∂B ′ = id | ∂B ′ we have u ◦ v | ∂B ′ = u | ∂B ′ and, hence, z ∈ W , ( B, R n ). Choose E and a ∈ E \ N using Lemma 6.1 applied to B ′ with ψ = F − ϕ η , f = | Du − F | p + | θ (det Du ) − θ (det F ) | and g = 1 + θ (det( F − Dϕ η )). Then, by (7.1),(7.5) B ′ (1 + θ (det Dv )) ( | Du − F | p + | θ (det Du ) − θ (det F ) | ) ◦ v d x c η δ, with c η depending on η and F .By (W) and (7.4) we have θ (det Dϕ η ) ∈ L ( B (0 , r )), θ (det( F − Dϕ η )) ∈ L ( B (0 , r )) and(det Dϕ η ) − q ′ ∈ L ( B (0 , r )). Therefore, there exists γ > F , ~m and η ) suchthat(7.6) ˆ B ( , r ) ∩{ det Dϕ η <γ } (1 + θ (det( F − Dϕ η ))) d x (cid:12)(cid:12) B (cid:0) , r (cid:1)(cid:12)(cid:12) η (cid:0) k F − Dϕ η k pL ∞ (cid:1) (1 + | F | p + θ (det F ))and(7.7) ˆ B ( , r ) ∩{ det Dϕ η <γ } (1 + | Dϕ η | p + θ (det Dϕ η )) d x c (cid:12)(cid:12)(cid:12) B (cid:16) , r (cid:17)(cid:12)(cid:12)(cid:12) η, where c is the constant of (W).Let R η = k Dv k L ∞ and M η = k Dϕ η k L ∞ . Since W is continuous in R n × n + there is ε > u , ~n or δ with εR η ε | W ( σ, ~ℓ ) − W ( ζ, ~k ) | η σ, ζ ∈ R n × n + and ~ℓ, ~k ∈ S n − with | ζ | M η , det ζ > γ and | σ − ζ | + | ~ℓ − ~k | εR η .Moreover, by Proposition 3.4 and the continuity of θ , the number ε can be chosen so that(7.9) | W qc ( ζ, ~ℓ ) − W qc ( F, ~m ) | + | θ (det ζ ) − θ (det F ) | η for all ζ ∈ R n × n + and ~ℓ ∈ S n − satisfying | ζ − F | + | ~m − ~ℓ | ε .Set ˆ ϕ η ( x ) = ϕ η ( x − a ), and write ˆ B ′ ( W ( Dz, ~n ◦ z ) − W qc ( Du, ~n ◦ u )) d x = I + I + I + I , with I = ˆ B ′ ( W ( Dz, ~n ◦ z ) − W ( D ˆ ϕ η , ~n ◦ z )) d x, I = ˆ B ′ ( W ( D ˆ ϕ η , ~n ◦ z ) − W ( D ˆ ϕ η , ~m )) d x,I = ˆ B ′ ( W ( D ˆ ϕ η , ~m ) − W qc ( F, ~m )) d x and I = ˆ B ′ ( W qc ( F, ~m ) − W qc ( Du, ~n ◦ u )) d x. We will estimate these four integrals separately. Thanks to (7.4) we have I η | B ′ | . Toestimate I we use (7.9) to get W qc ( F, ~m ) W qc ( Du, ~n ◦ u ) + η on the set where | Du − F | + | ~m − ~n ◦ u | ε. In { x ∈ B ′ : | Du ( x ) − F | + | ~m − ~n ◦ u ( x ) | > ε } we use (7.1) and Chebyshev’s inequality to get I η | B ′ | + W qc ( F, ~m ) (cid:12)(cid:12) { x ∈ B ′ : | Du ( x ) − F | + | ~m − ~n ◦ u ( x ) | > ε } (cid:12)(cid:12) η | B ′ | + W qc ( F, ~m ) 2 p − ε p | B | δ. To estimate I we need to define the sets ω = { x ∈ B ′ : | ~n ◦ u ( x ) − ~m | > εR η } and ω d = { x ∈ B ′ : det D ˆ ϕ η ( x ) > γ } , where ε and γ are those of (7.8). Doing the change of variables z ( x ) = u ( x ′ ), i.e., x = v − ( x ′ ),we obtain I = ˆ B ′ (cid:0) W (( D ˆ ϕ η ) ◦ v − ( x ′ ) , ~n ◦ u ( x ′ )) − W (( D ˆ ϕ η ) ◦ v − ( x ′ ) , ~m ) (cid:1) det Dv − ( x ′ ) d x ′ , and ˆ B ′ det Dv − ( x ′ ) d x ′ = | B ′ | . Using (W) and (7.8) we get I ˆ v ( ω d ) \ ω η det Dv − ( x ′ ) d x ′ + ˆ B ′ \ ( v ( ω d ) \ ω ) W (( D ˆ ϕ η ) ◦ v − ( x ′ ) , ~n ◦ u ( x ′ )) det Dv − ( x ′ ) d x ′ η | B ′ | + c ˆ B ′ \ ( v ( ω d ) \ ω ) (cid:0) | ( D ˆ ϕ η ) ◦ v − ( x ′ ) | p + θ (det D ˆ ϕ η ) ◦ v − ( x ′ ) (cid:1) det Dv − ( x ′ ) d x ′ . x = v − ( x ′ ) and using (7.7) we obtain c ˆ B ′ \ v ( ω d ) (cid:0) | ( D ˆ ϕ η ) ◦ v − ( x ′ ) | p + θ (det D ˆ ϕ η ) ◦ v − ( x ′ ) (cid:1) det Dv − ( x ′ ) d x ′ c ˆ B ′ \ ω d (1 + | D ˆ ϕ η ( x ) | p + θ (det D ˆ ϕ η ( x ))) d x η | B ′ | . On the other hand, for x ∈ ω d we have that det Dv ( x ) > γ det F − , so det Dv − ∈ L ∞ ( v ( ω d )).Then, using (7.1), θ (det D ˆ ϕ η ) ∈ L ∞ ( ω d ) and Chebyshev’s inequality we get c ˆ ω ∩ v ( ω d ) (cid:0) | ( D ˆ ϕ η ) ◦ v − ( x ′ ) | p + θ (det( D ˆ ϕ η )) ◦ v − ( x ′ ) (cid:1) det Dv − ( x ′ ) d x ′ . | ω | . ε − p δ | B ′ | , with the constant under . depends on W , γ and η but not on δ , ~n , u or z .Hence, we have that there exists a constant ˜ c depending on η and W but not on δ suchthat I (2 η + ˜ cε − p δ ) | B ′ | . Next, we estimate I . Let ω ′ = { x ∈ B ′ : | Du ( x ) − F | ◦ v > ε } . Using that, in B ′ , Dz = ( Du ◦ v ) Dv = [( Du − F ) ◦ v ] Dv + D ˆ ϕ η and that in ω d \ ω ′ we have det D ˆ ϕ η > γ and | Du ( x ) − F | ◦ v ε we get | Dz − D ˆ ϕ η | [ | Du − F | ◦ v ] | Dv | εR η . By (7.8) we have ˆ ω d \ ω ′ ( W ( Dz, ~n ◦ z ) − W ( D ˆ ϕ η , ~n ◦ z )) d x η | B ′ | . Using the growth estimate (W) we obtain W ( Dz, ~n ◦ z ) c (1 + [ | Du | p ◦ v ] | Dv | p + θ ((det Du ) ◦ v det Dv )) . Hence using | Dv | R η and (W) we get that, in B ′ ,(7.10) W ( Dz, ~n ◦ z ) c (cid:0) R pη | Du | p ◦ v + 1 + θ ((det Du ) ◦ v ) (cid:1) (1 + θ (det Dv )) . To estimate the integral in ω ′ we observe that | Du − F | ◦ v > ε implies | Du | ◦ v + 1 | Du − F | ◦ v + | F | + 1 (cid:18) | F | + 1 ε + 1 (cid:19) | Du − F | ◦ v θ (det Du ) ◦ v | θ (det Du ) ◦ v − θ (det F ) | + θ (det F ) ε p | Du − F | p ◦ v. Therefore, from (7.10) and (7.5) we obtain ˆ ω ′ W ( Dz, ~n ◦ z ) c ˆ ω ′ (1 + θ (det Dv ( x ))) (cid:0) R pη | Du | p + θ (det Du ) (cid:1) ◦ v ( x ) d x c ′ ˆ ω ′ (1 + θ (det Dv ( x ))) ( | Du − F | p + | θ (det Du ) − θ (det F ) | ) ◦ v ( x ) d x c ′ η δ | B ′ | . The constant c ′ η depends on W , η and F but not on δ . In B ′ \ ( ω d ∪ ω ′ ) we have | Du − F | ◦ v ε D ˆ ϕ η < γ . Then we have | Du | ◦ v | F | + 1 and thanks to (7.9) we also obtain θ (det Du ) ◦ v θ (det F ) + 1. Therefore (7.10) implies W ( Dz, ~n ◦ z ) c (cid:0) R pη (1 + | F | ) p + θ (det F ) (cid:1) (1 + θ (det Dv )) c ∗ (cid:0) k F − Dϕ η k pL ∞ (cid:1) (1 + | F | p + θ (det F )) (1 + θ (det Dv )) , with c ∗ depending only on W . Hence, thanks to (7.6) we get ˆ B ′ \ ( ω ′ ∪ ω d ) W ( Dz, ~n ◦ z ) d x c ∗ η | B ′ | . Consequently, I ( η + c ′ η δ + c ∗ η ) | B ′ | . Adding the estimates for I , I , I and I we obtain ˆ B ′ ( W ( Dz, ~n ◦ z ) − W qc ( Du, ~n ◦ u )) d x (cid:18) η + c ′ η δ + c ∗ η + 2 η + ˜ cε p δ + η + η + W qc ( F, ~m ) 2 p − ε p δ (cid:19) | B ′ | . Recall that η , c ′ η , c ∗ , ˜ c and ε do not depend on δ . Then, choosing δ small enough, we have(7.2). Using the growth condition (W) we obtain Dz ∈ L p ( B ), so z ∈ W ,p ( B ).Recall that a was chosen so that det Dz ∈ L ( B ) and cof Dz ∈ L q ( B ). Then Lemma4.10 gives z ∈ A p,q ( B ) and the proof is completed.In the following lemma we apply Lemma 7.2 in the Lebesgue points of Du and ~n ◦ u . Theproof is based on that of [18, Lemma 3.3]. Lemma 7.3.
Let Ω ⊂ R n open, Lipschitz and bounded, and assume a) or b) of Lemma 7.2.Then for any u ∈ ( A p,q ( B ) in case a) , A p ( B ) in case b)25 nd any ~n ∈ W ,s (im T ( u, Ω) , S n − ) , there are two sequences u j ∈ ( A p,q (Ω) in case a) , A p (Ω) in case b) and ~n j ∈ W ,s (im T ( u, Ω) , S n − ) such that u j ⇀ u in W ,p (Ω , R n ) , u j = u on ∂ Ω , im T ( u j , Ω) = im T ( u, Ω) for all j ∈ N , ~n j ⇀ ~n in W ,s (im T ( u, Ω) , S n − ) , lim sup j →∞ ˆ im T ( u j , Ω) V ( ~n j ( y ) , D~n j ( y )) d y ˆ im T ( u, Ω) V tqc ( ~n ( y ) , D~n ( y )) d y and lim sup j →∞ ˆ Ω W ( Du j , ~n j ◦ u j ) d x ˆ Ω W qc ( Du, ~n ◦ u ) d x. If, additionally, u ∈ W , ∞ (Ω , R n ) , then we can take u j ∈ W , ∞ (Ω , R n ) .Proof. We will only prove the case a) , the proof of case b) being completely analogous. Thanksto Theorem 3.8, there exists a sequence { ~n k } k ∈ N in W ,s (im T ( u, Ω) , S n − ) such that ~n k ⇀ ~n in W ,s (im T ( u, Ω) , S n − ) andlim k →∞ ˆ im T ( u, Ω) V ( ~n k ( y ) , D~n k ( y )) d y = ˆ im T ( u, Ω) V tqc ( ~n ( y ) , D~n ( y )) d y. Fix η ∈ (0 , { u j } j ∈ N it is enough to construct w ∈ ( A p,q (Ω) in case a) , A p (Ω) in case b) such that k u − w k L p η , w = u on ∂ Ω, im T ( w, Ω) = im T ( u, Ω) and(7.11) ˆ Ω W ( Dw, ~n ◦ w ) d x ˆ Ω W qc ( Du, ~n ◦ u ) d x + η. Indeed for each j ∈ N , we can construct u j as the w of the claim above corresponding to η = 1 /j . Then u j → u in L p and, thanks to (7.11) and (W), we will have sup j ∈ N k u j k W ,p < ∞ ,so u j ⇀ u in W ,p (Ω , R n ). On the other hand, since ~n k → ~n a.e. in im T ( u, Ω) and u j satisfiesLuzin’s N − condition (i.e., the preimage of a set of measure zero has measure zero: this is aconsequence of the fact that det Du j > j ∈ N we have ~n k ◦ u j → ~n ◦ u j a.e.in Ω as k → ∞ , hence using (W) we obtain ˆ Ω W ( Du j , ~n k ◦ u j ) d x ˆ Ω ( h ( | ~n k ◦ u j − ~n ◦ u j | ) + 1) W ( Du j , ~n ◦ u j ) d x. By dominated convergence, we havelim sup k →∞ ˆ Ω W ( Du j , ~n k ◦ u j ) d x ˆ Ω W ( Du j , ~n ◦ u j ) d x, j ∈ N we can take k j ∈ N big enough to have ˆ Ω W ( Du j , ~n k j ◦ u j ) d x ˆ Ω W qc ( Du, ~n ◦ u ) d x + 2 j − . Therefore, relabelling the sequence { ~n j } j ∈ N we havelim sup j →∞ ˆ Ω W ( Du j , ~n j ◦ u j ) d x ˆ Ω W qc ( Du, ~n ◦ u ) d x. If ´ Ω W qc ( Du, ~n ◦ u ) d x = ∞ , we can take w = u , so we will assume W qc ( Du, ~n ◦ u ) ∈ L (Ω).Using (W) we have1 c | F | p + 1 c θ (det F ) − c W qc ( F, ~m ) for all F ∈ R n × n + and ~m ∈ S n − . This is because the left-hand side of the inequality above is polyconvex, hence quasiconvex.Hence, | Du | p and θ (det Du ) are integrable. On the other hand, we have ~n ◦ u ∈ L ∞ (Ω , S n − ),because thanks to [12, Lemma 7.7], ~n ◦ u is measurable. Denote by E the intersection of theset of p -Lebesgue points of Du and ~n ◦ u and Lebesgue points of θ (det Du ). Given x ∈ E ,let F x = Du ( x ) and ~m x = ~n ◦ u ( x ), and choose δ x as in Lemma 7.2 for this F x , ~m x and η asabove.We will construct a sequence of { ( w j , Ω j ) } j ∈ N such that w j ∈ A p,q (Ω), { Ω j } j ∈ N is adecreasing sequence of open subsets of Ω, w j = u on Ω j and im T ( w j , Ω) = im T ( u, Ω). Set w = u and Ω = Ω. The passage from ( w j , Ω j ) to ( w j +1 , Ω j +1 ) is as follows. For all x ∈ E ∩ Ω j we choose r j ( x ) ∈ (0 , η ) such that B ( x, r j ( x )) ⊂ Ω j , u ∗ ∈ W ,p ( ∂B ( x, r j ( x )) , R n ) (recall fromSubsection 4.1 the definition of precise representative) and B ( x,r ) (cid:0) | Dw j ( x ′ ) − F x | p + | θ (det Dw j ( x ′ )) − θ (det F x ) | + | ~n ◦ w j ( x ′ ) − ~m x | p (cid:1) d x ′ δ x for all r < r j ( x ). The union of this collection of balls B ( x, r j ( x )) covers Ω j up to a set ofmeasure zero. Extract a finite disjoint subset { B ( x k , r k ) } Mk =0 such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M [ k =0 B ( x k , r k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > | Ω j | . Define w j +1 as w j on Ω \ S Mk =0 B ( x k , r k ) and as the function z of Lemma 7.2 in each of theballs B ( x k , r k ). Then w j +1 = w j = u on ∂ Ω and thanks to Lemma 4.8, we get w j +1 ∈ ( A p,q (Ω) if W satisfies a) , A p (Ω) if W satisfies b) .Let B ( x ′ k , r k ) ⊂ B ( x k , r k ) be the ball given by Lemma 7.2. Take an increasing sequence { U i } i ∈ N of open subsets compactly contained in Ω such that S i ∈ N U i = Ω, S Mk =0 B ( x k , r k ) ⊂ U w ∗ j ∈ W ,p ( ∂U i , R n ) for all i ∈ N . Then, w j and w j +1 coincide in a neighbourhood of each ∂U i , so w ∗ j +1 ∈ W ,p ( ∂U i , R n ) and im T ( w j , U i ) = im T ( w j +1 , U i ) (recall Definition 4.3), sincethe degree only depends on the boundary values. Therefore, im T ( w j , Ω) = im T ( w j +1 , Ω), and,by induction, im T ( w j +1 , Ω) = im T ( u, Ω).By Lemma 7.2,(7.12) ˆ B ( x ′ k , rk ) W ( Dw j +1 , ~n ◦ w j +1 ) d x ˆ B ( x ′ k , rk ) ( W qc ( Du, ~n ◦ u ) + η ) d x and(7.13) ˆ B ( x k ,r k ) | w j +1 − u | p d x c η p ˆ B ( x k ,r k ) ( W qc ( Du, ~n ◦ u ) + 1) d x. Set Ω j +1 = Ω j \ S Mk =0 ¯ B (cid:0) x ′ k , r k (cid:1) . It is clear that w j +1 = w j = u on Ω j +1 and that | Ω j +1 | (1 − − n − ) | Ω j | . The construction of w j +1 is completed and, hence, so is the sequence { w j } j ∈ N .Thus, we only have to show that for j big enough, w j has the desired properties, namely, (7.11)and that k u − w j k L p is small.Thanks to (7.13) we have ˆ Ω | w j − u | p d x c η p ˆ Ω ( W qc ( Du, ~n ◦ u ) + 1) d x, so w j is close to u in L p , independently of j . On the other hand, from (7.12) we obtain ˆ Ω \ Ω j W ( Dw j +1 , ~n ◦ w j +1 ) d x ˆ Ω \ Ω j ( W qc ( Du, ~n ◦ u ) + η ) d x, which implies ˆ Ω W ( Dw j +1 , ~n ◦ w j +1 ) d x ˆ Ω \ Ω j ( W qc ( Du, ~n ◦ u ) + η ) d x + ˆ Ω j W ( Du, ~n ◦ u ) d x. Using | Ω j | (1 − − n − ) j | Ω | → W ( Du, ~n ◦ u ) ∈ L (Ω)(since | Du | p and θ (det Du ) are integrable), for j large enough we get ˆ Ω W ( Dw j +1 , ~n ◦ w j +1 ) d x ˆ Ω ( W qc ( Du, ~n ◦ u ) + 2 η ) d x and the proof is concluded. Once the recovery sequence has been constructed in Theorem 7.1 and the lower semicontinuityand compactness results have been established in Section 5, the general theory of relaxation(see, e.g., [4, Th. 11.1.1 and 11.1.2]) provides the following result. We recall that the lowersemicontinuous envelope is the largest lower semicontinuous function below a given one.28 heorem 8.1.
Let W satisfy (W) and let V satisfy (V). Assume W qc is polyconvex. Then I ∗ is the lower semicontinuous envelope of I with respect to the L (Ω , R ) × L ( R n , R n ) topologyand, for each ( u, ~n ) ∈ L (Ω , R ) × L ( R n , R n ) , I ∗ ( u, ~n ) = inf (cid:26) lim inf j →∞ I ( u j , ~n j ) : ( u j , ~n j ) → ( u, ~n ) as j → ∞ in L (Ω , R ) × L ( R n , R n ) (cid:27) . If, in addition, I is not identically infinity thena) There exists a minimizer of I ∗ .b) Every minimizer of I ∗ is the limit in L (Ω , R ) × L ( R n , R n ) of a minimizing sequence for I .c) Every minimizing sequence of I converges in L (Ω , R ) × L ( R n , R n ) , up to a subsequence,to a minimizer of I ∗ . The analogue of Theorem 8.1 remains true in the incompressible case, i.e., when W isassumed to satisfy (W ) and every instance of I is replaced by I , and every instance of I ∗ by I ∗ .As usual in relaxation and Γ-convergence problems (see, e.g., [13, Rk. 2.2]), if F is afunctional continuous with respect to the topology L (Ω , R ) × L ( R n , R n ), then the relaxationof I + F is I ∗ + F . An example of such an F is given by F ( u ) = ´ Ω f ( x, u ( x )) d x with f : Ω × R n → R measurable in the first variable and continuous in the second such that | f ( x, y ) | C | y | r + γ ( x ) for a.e. x ∈ Ω and all y ∈ R n , for some C > γ ∈ L (Ω) and0 r < p ∗ , where p ∗ is the conjugate Sobolev exponent of p (see, e.g., [28, Cor. 6.51]). Thisis because, as shown in Proposition 5.1 (in truth, [12, Th. 8.2]), if u j → u in L (Ω , R n ) andsup j ∈ N I mec ( u j , ~n j ) < ∞ then, for a subsequence, u j ⇀ u in W ,p (Ω , R n ) and, by the compactSobolev embedding, u j → u in L r (Ω , R n ). Acknowledgements
We thank G. Leoni for bringing to our notice the concept of tangential quasiconvexity. C.M.-C.has been supported by Project MTM2014-57769-C3-1-P and the “Ram´on y Cajal” programmeRYC-2010-06125 of the Spanish Ministry of Economy and Competitivity. Both authors havebeen supported by the ERC Starting grant no. 307179.
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