On the passage from atomistic systems to nonlinear elasticity theory
aa r X i v : . [ m a t h . A P ] M a y On the passage from atomistic systemsto nonlinear elasticity theory
Julian Braun and Bernd Schmidt November 8, 2018
Abstract
We derive continuum limits of atomistic models in the realm of nonlinear elasticitytheory rigorously as the interatomic distances tend to zero. In particular we obtain anintegral functional acting on the deformation gradient in the continuum theory whichdepends on the underlying atomistic interaction potentials and the lattice geometry.The interaction potentials to which our theory applies are general finite range mod-els on multilattices which in particular can also account for multi-pole interactionsand bond-angle dependent contributions. Furthermore, we discuss the applicabilityof the Cauchy-Born rule. Our class of limiting energy densities consists of generalquasiconvex functions and the class of linearized limiting energies consistent with theCauchy-Born rule consists of general quadratic forms not restricted by the Cauchyrelations.
The main aim of this work is to provide a rigorous derivation of nonlinear elasticity func-tionals from atomistic models. The investigation of such discrete-to-continuum limits hasbeen an active area of research in continuum mechanics over the last years in particularfor, but not limited to, elastic interactions. For a recent account on this line of researchand a summary of the related literature we refer to the survey article [BLL07] by Blanc,LeBris and Lions.Classically, the stored energy density in elasticity theory is derived from atomisticmodels by applying the Cauchy-Born rule: Given a macroscopic deformation y of theelastic body, one assumes that microscopically near every material point x , all the atomsdeform by just following the macroscopic deformation gradient F = ∇ y ( x ) . Inserting thisansatz into the atomistic potentials then leads to a continuum stored energy density W asa function of F ∈ R × . Assuming validity of the Cauchy-Born rule, very general and evenquantum mechanical interactions have, e.g., been investigated by Blanc, LeBris and Lionsin [BLL02]. A priori, however, it is not clear if the Cauchy-Born hypothesis does hold true.For a two-dimensional mass-spring model, it has been shown by Friesecke and Theil in[FT02] that the Cauchy-Born rule does indeed hold true for small strains, while it in generalfails for large strains. This result has then be generalized to a wider class of discrete modelsand more than two dimensions by Conti, Dolzmann, Kirchheim and Müller in [CDKM06].A fundamental contribution towards a rigorous derivation of continuum limits in elas-ticity has been made by Alicandro and Cicalese in [AC04], where they prove a generalintegral representation result for continuum limits of atomistic pair interaction potentials.It is our main aim, departing from this result, to derive a continuum theory for more gen-eral interaction potentials which, in particular, can also incorporate bond-angle dependentpotentials. Such an extension is desirable in applications, as many atomistic models such Institut für Mathematik, Universität Augsburg, Germany, [email protected] Institut für Mathematik, Universität Augsburg, Germany, [email protected] Γ -limits and integral representation results for functionals on Sobolev spacesand thus follow the scheme set forth in [AC04], which is dictated by verifying the hypothesesof that abstract approach by the localization method. A few of the arguments in this proofcan be used with only minor adjustments. There are, however, some major differencesas compared to the pair interaction case treated by Alicandro and Cicalese. While theseauthors use slicing arguments in order to obtain energy estimates on the usual d × d deformation gradients in the direction of interacting pairs, we will have to estimate themuch higher dimensional d × d discrete deformation gradients. In fact, as in general ourdiscrete energies cannot be recovered by slicing techniques, we will instead work with verycarefully chosen interpolations of the discrete deformations which encode the full discretegradient on lattice cells.More specifically, if L = A Z d is some Bravais lattice, Ω ⊂ R d a bounded open set withLipschitz boundary that will be viewed as the ‘macroscopic’ domain occupied by the elasticbody, whose atoms are at positions ε L ∩ Ω , we assume that the energy of a deformation y : ε L ∩ Ω → R d can be written in the form F ε ( y, Ω) = ε d X x ∈ ( L ′ ε (Ω)) ◦ W cell ( ¯ ∇ y ( x )) , where x runs over all midpoints of elementary lattice cells of ε L inside Ω . Here ¯ ∇ y ( x ) is thediscrete gradient of y on the corresponding cell Q which encodes all relative displacementsof atoms lying on the corners of Q . (See Section 2 for precise definitions.) ε is the smallparameter in the system measuring the typical interatomic distance and tending to zeroeventually in the continuum limit. The rescaling by ε d is introduced in order to pass fromunits of finite energy per atom to units of finite energy per unit volume.In fact, our analysis is not restricted to interactions within unit lattice cells, but alsoapplies to general finite range interactions. In such models, the energy is still given as2he sum over unit lattice cells ( L ′ ε (Ω)) ◦ , but the cell energy now depends on the discretedeformation gradient ¯ ∇ y ( x ) ∈ R d × N on a larger ‘super-cell’: F ε ( y, Ω) = ε d X x ∈ ( L ′ ε (Ω)) ◦ W super − cell ( ¯ ∇ y ( x )) , where the definition of the lattice interior ( L ′ ε (Ω)) ◦ is suitably adjusted so that only latticepoints within Ω may interact.Another complication arises when extending our results to general finite-range inter-actions on multi-lattices. For such systems, the discrete gradients are augmented withadditional internal variables describing the relative shifts of the underlying single lattice.With the help of a mixed Sobolev/Lebesgue space representation theorem we are thenled to a boundary value/mean value cell formula for the limiting energy density. Thiscell formula is in fact related to the cell formula derived in [Sch08] for thin membraneswhere internal variables measure relative shifts of the thin film’s layers. On multi-lattices ε ( { , s , . . . , s m } + L ) with m shift vectors s , . . . , s m ∈ R d the general discrete energyfunctional then reads F ε ( y, s, Ω) = ε d X x ∈ ( L ′ ε (Ω)) ◦ W super − cell ( ¯ ∇ y ( x ) , s ( x )) , where y : ε L ∩ Ω → R d , s : ε L ∩ Ω → R d × m . For notational convenience we will restrict tosimple unit cell interactions on a Bravais lattice for the largest part of the paper and onlycomment on the necessary modifications in the more general case at the end of Section 5.Our main results are summarized in the following theorems. The necessary assumptionsAssumptions 1, 2 and 3 on the cell energy are specified in Section 2. (Assumptions 1 and2 are nothing but standard p -growth assumptions on W cell .) Theorem 1.1 ( Γ -convergence) . Suppose Assumptions 1 and 2 are true. Then F ε ( · , Ω)Γ( L p (Ω; R d )) - and Γ( L ploc (Ω; R d ) / R ) -converges to the functional F , defined by F ( y ) = ˆ Ω W cont ( ∇ y ( x )) dx, if y ∈ W ,p (Ω; R d ) , ∞ otherwise,where the continuum density W cont : R d × d → [0 , ∞ ) is given in terms of W cell by W cont ( M ) = 1 | det A | lim N →∞ N d inf X x ∈ ( L ′ ( A (0 ,N ) d )) ◦ W cell ( ¯ ∇ y ( x )) : y ∈ B ( A (0 , N ) d , y M ) . Here B ( A (0 , N ) d , y M ) is the space of lattice deformations of L ∩ A (0 , N ) d with linearboundary conditions M on ∂ L ( A (0 , d ) , cf. Section 4.As in non-convex homogenization (see [Mül87]), in the representation result for W cont it is nesessary to minimize W cell over larger and larger cubes and the limit is in gen-eral not obtained for finite N . A simple 2d example for this effect is given by a squarelattice where nearest neighbor atoms interact via a harmonic spring potential: F ε ( y ) = P | x − x ′ | = ε ( | y ( x ) − y ( x ′ ) | − ε ) (which can be written in the above form). This is a simpli-fied version of the example discussed in [Sch08, Sect. 4.4]. The arguments sketched there,which amount to considering deformations y ( x , x ) = M ( x , x ) + σ ( x ) + σ ( x ) -periodic functions σ and σ like σ i ( z ) = 12 ( − z s m i + m i − (cid:18) − m i m i (cid:19) ,i = 1 , , and suitably modified on the boundary, show that N inf X x ∈ ( L ′ ((0 ,N ) d )) ◦ W cell ( ¯ ∇ y ( x )) : y ∈ B ((0 , N ) d , y M ) converges to (max { , | m · | − } ) + (max { , | m · | − } ) with error bound O ( N − ) , where m · j denotes the j th column of M . Evaluating, however, the energy of the bonds on andclose to the boundary it is easily seen that, for compressive boundary conditions | m · | < or | m · | < , this limiting energy is always over-estimated by a constant times N − . Theorem 1.2 (Compactness) . Under the assumptions of Theorem 1.1, if y ε is a sequencewith equibounded energies F ε ( y ε , Ω) and Ω is connected, then there exist a sequence ε k → and y ∈ W ,p (Ω; R d ) such that y ε k → y in L ploc (Ω; R d ) / R . Of course, if Ω is not connected, one has compactness in L ploc up to translation on everyconnected component.Analogous results hold true under boundary conditions g ∈ W , ∞ ( R d ; R d ) . Let F g and F gε denote the functionals obtained from F and F ε , respectively, with values set to infinityif the boundary conditions are not met, cf. Section 4. Theorem 1.3 ( Γ -convergence) . If Assumptions 1 and 2 are true, then F gε ( · , Ω) Γ( L p (Ω; R d )) -converges to the functional F g . Theorem 1.4 (Compactness) . Under the assumptions of Theorem 1.3, if y ε is a se-quence with equibounded energies F gε ( y ε , Ω) , then there exist a sequence ε k → and y ∈ W ,p (Ω; R d ) with y = g on ∂ Ω such that y ε k → y in L p (Ω; R d ) . A standard argument then yields that almost minimizers of F gε ( · , Ω) converge to min-imizers of F g ( · , Ω) and almost minimizers of F ε ( · , Ω) up to translation converge to mini-mizers of F ( · , Ω) , more precisely: Corollary 1.5 (Convergence of almost minimizers) . Suppose Assumptions 1 and 2 aretrue. Then every sequence of almost minimizers of F ε ( · , Ω) for connected Ω is compactin L ploc (Ω; R d ) / R and every limit is a minimizer of F , while every sequence of almostminimizers of F gε ( · , Ω) is compact in L p (Ω; R d ) and every limit is a minimizer of F g . It is not hard to include body forces in the energy expression as these will only becontinuous perturbations of the energy functional which converge uniformly on boundedsets and thus preserve Γ -convergence by general theory.We also remark that the point why the theory can be adapted to the case of generalfinite range interactions, is that in this case W cell , while naturally still bounded from aboveby the discrete gradient through Assumption 2, from below has to be bounded only interms of the discrete gradient on the unit cell. See Section 5 for details. For general finiterange interactions on multi-lattices we state here only the analogue of Theroem 1.1, asin the cell formula there are now additional internal variables that need to be taken intoaccount. Theorems 1.2, 1.3 and 1.4 and Corollary 1.5 extend in a straightforward manner.4 heorem 1.6 ( Γ -convergence) . Suppose W super − cell satisfies the growth assumptions (asstated in Section 5). Then F ε ( · , · , Ω) Γ( L p (Ω; R d ) × w- L q (Ω; R d × m )) -converges to the func-tional F , defined by F ( y, s ) = ˆ Ω W cont ( ∇ y ( x ) , s ( x )) dx, if y ∈ W ,p (Ω; R d ) , ∞ otherwise,where the continuum density W cont : R d × d × R d × m → [0 , ∞ ) is given in terms of W super − cell by W cont ( M, s ) = 1 | det A | lim N →∞ N d inf ( X x ∈ ( L ′ ( A (0 ,N ) d )) ◦ W super − cell ( ¯ ∇ y ( x ) , s ( x )) :( y, s ) ∈ B ( A (0 , N ) d , y M , s ) ) . Here B ( A (0 , N ) d , y M , s ) is the space of lattice deformations of L ∩ A (0 , N ) d withlinear boundary conditions M on ∂ L ( A (0 , d ) for y and average s for s , cf. Section 5.As macroscopic deformations are solely given in terms of a deformation mapping y ∈ W ,p (Ω; R d ) , we are also interested in the effective macroscopic energy density obtained byminimizing out the internal variables s : Theorem 1.7.
For every y ∈ W ,p (Ω; R d ) we have min s ∈ L q (Ω; R d × m ) ˆ Ω W cont ( ∇ y ( x ) , s ( x )) dx = ˆ Ω min s ∈ R d × m W cont ( ∇ y ( x ) , s ) dx. Moreover, F s − min ε ( · , Ω) := inf s ∈ L q F ε ( · , s, Ω) Γ( L p (Ω; R d )) -converges to the functional F s − min ,defined by F s − min ( y ) = ˆ Ω min s ∈ R d × m W cont ( ∇ y ( x ) , s ) dx, if y ∈ W ,p (Ω; R d ) , ∞ otherwise, Returning to our basic setting on a Bravais lattice, under an additional assumption, wecan calculate the limiting density for small strains explicitly by the Cauchy-Born rule:
Theorem 1.8.
In addition to Assumptions 1 and 2 suppose that W cell satisfies Assumption3. Then there is a neighborhood U of SO ( d ) such that W cont is given by W cont ( M ) = W CB ( M ) := 1 | det A | W cell ( M Z ) . for all M ∈ U . Here Z ∈ R d × d is a ‘discrete identity matrix’, see Section 2 for details.As W cont arises as the energy density of a Γ -limit it has to be quasiconvex (cf. Section 2for the definition of these concepts). The next proposition shows that our class of atomisticinteractions is rich enough to model any quasiconvex energy density in the continuum limit.5 roposition 1.9. Suppose V : R d × d → R is quasiconvex with standard p -growth c | M | p − c ′ ≤ V ( M ) ≤ c ′′ ( | M | p + 1) for some constants c, c ′ , c ′′ > and all M ∈ R d × d . Then there exists a cell energy W cell satisfying Assumptions 1 and 2 such that W cont = V . We remark that, by way of contrast, a restriction to pair interaction models will onlylead to a restricted class of limiting continuum energies, as can be quantified in terms ofthe so-called Cauchy relations: If the Cauchy-Born rule applies (e.g., due to Assumption3), an atomistic interaction energy E ( y ) = ε d X x,x ′∈L ε ∩ Ω x = x ′ V | x − x ′| ε (cid:18) | y ( x ) − y ( x ′ ) | ε (cid:19) yields the continuum density W CB ( M ) = 1 | det A | X x ∈L x =0 V | x | ( | M x | ) . Assuming V | x | is smooth and, for large | x | , sufficiently rapidly decreasing a direct calculationyields D W CB (Id)( M, M ) = d X i,j,k,l =1 c ijkl m ij m kl , where the elastic constants c ijkl are given by c ijkl = 1 | det A | X x ∈L x =0 V ′′| x | ( | x | ) x i x j x k x l | x | + V ′| x | ( | x | ) (cid:18) x j x l δ ki | x | − x i x j x k x l | x | (cid:19) . While the symmetry relations c ijkl = c klij and c ijkl = c jikl naturally follow from thesymmetry of the Hessian D W CB (Id) and frame indifference of W CB , the particular formof W CB in addition gives c ijkl = c ilkj = c kjil for every i, j, k, l .In the 3-dimensional setting of elasticity theory these additional relations lower thedimension of admissible elasticity tensors from to (symmetric in all indices) and socan be written as equations, the Cauchy-relations, namely c = c , c = c , c = c ,c = c , c = c , c = c . The question whether in elasticity theory the Cauchy-relations hold true (rari-constanttheory) or fail (multi-constant theory) had been under discussion for quite some time inphysics and was finally decided by experimental data in favour of the multi-constant theory(for some experimental data and further physical considerations cf. [Hau67]). This means,in particular, that the interaction in a lattice is a complex multibody interaction whichcannot be reduced to pair-potentials. Our model in this paper using general cell energiesis not limited by the Cauchy-relations:
Proposition 1.10.
Suppose Q : R d × d → R is a positive semidefinite quadratic form whichis positive definite on the symmetric d × d matrices and vanishes on antisymmetric matrices.Then there exists W cell satisfying Assumptions 1, 2 and 3 such that D W CB (Id)( M, M ) = Q ( M ) . Γ -convergence and integral representations of functionalson Sobolev spaces. In Section 3 we then proceed to state precisely and prove a general Γ -compactness and representation theorem. This in particular requires a number of technicalpreliminaries in order to investigate discrete deformations. A version of this representationresult for boundary value problems is then provided in Section 4. Finally, the limitingstored energy function is identified in Section 5 through minimizing a sequence of cellproblems, leading to the main discrete-to-continuum convergence result and the proofs ofthe results stated above. In this section we introduce the atomistic model and recall some general facts on Γ -convergence and integral representation results required by the localization method. Let
L ⊂ R d be a Bravais lattice, i.e., there are linearly independent vectors v , . . . , v d suchthat L = { n v + · · · + n d v d | n , . . . , n d ∈ Z } = A Z d , if we set A = ( v , . . . , v d ) . The scaled lattices L ε = ε L partition R d into the ε -cells z + A [0 , ε ) d ( z ∈ L ε ). Let Q ε ( x ) denote the ε -cell containing x . The centers of the cellsare L ′ ε = L ε + A ( , . . . , ) and we denote by ¯ x the center of the cell containing x . Thesecenters give a convenient labeling of the cells. Furthermore let z , . . . , z d be the points in A (cid:8) − , (cid:9) d and Z := ( z , . . . , z d ) ∈ R d × d .For a set U ⊂ R d we define the following lattice subsets in the spirit of its closed hull,interior or boundary with respect to ε L ′ or its corners ε L by L ′ ε ( U ) = { x ∈ L ′ ε | Q ε ( x ) ∩ U = ∅} , L ε ( U ) = L ′ ε ( U ) + ε { z , . . . , z d } , ( L ′ ε ( U )) ◦ = { x ∈ L ′ ε | Q ε ( x ) ⊂ U } , ( L ε ( U )) ◦ = ( L ′ ε ( U )) ◦ + ε { z , . . . , z d } ,∂ L ′ ε ( U ) = L ′ ε ( U ) \ ( L ′ ε ( U )) ◦ , ∂ L ε ( U ) = ∂ L ′ ε ( U ) + ε { z , . . . , z d } . Furthermore let U ε = [ ¯ x ∈L ′ ε ( U ) Q ε (¯ x ) , U ε = [ ¯ x ∈ ( L ′ ε ( U )) ◦ Q ε (¯ x ) . A lattice deformation should be thought of as a mapping L ε ∩ Ω → R d . Choosing asuitable extension (e.g., by ) and piecewise constant interpolation, we can and will assumethat the lattice deformations B ε (Ω) are the functions Ω → R d , which are constant on everycell Q ε ( x ) , x ∈ L ′ ε (Ω) . (This will not change the energy, see below.)If we have a deformation y ∈ B ε (Ω) and x ∈ Ω ε , we set y i ( x ) = y (¯ x + εz i ) , ¯ y ( x ) = 12 d d X i =1 y i ( x ) and ¯ ∇ y ( x ) = 1 ε ( y ( x ) − ¯ y ( x ) , . . . , y d ( x ) − ¯ y ( x )) . Let A ( U ) be the set of all bounded open subsets of U ⊂ R d and A L ( U ) the set of allthose, that have a Lipschitz boundary. In the following, we will consider a set Ω ∈ A L ( R d ) F ε : L p (Ω; R d ) × A (Ω) → [0 , ∞ ] for some fixed < p < ∞ , defined by F ε ( y, U ) = ε d X x ∈ ( L ′ ε ( U )) ◦ W cell ( ¯ ∇ y ( x )) if y ∈ B ε ( U ) , ∞ otherwise. (2.1)In this definition the energy only depends on the values of y in ( L ε ( U )) ◦ ⊂ L ε ∩ U . Ofcourse, there can be some points in L ε ∩ U which we do not use at all, but this is negligibleif we impose Dirichlet boundary conditions as we will do later on.We make some assumptions on the cell energy W cell : R d × d → [0 , ∞ ) . Note, that adiscrete gradient can take values precisely in the space V = F ∈ R d × d : d X j =1 a ij = 0 , for every i = 1 , . . . , d . Therefore, we are only interested in the values of W cell on V . Assumption 1.
There are c, c ′ > such that for every F ∈ V W cell ( F ) ≥ c | F | p − c ′ . Assumption 2.
There is a c > such that for every F ∈ V W cell ( F ) ≤ c ( | F | p + 1) . While these conditions are supposed to hold true for all our results, we also state athird assumption, which, if satisfied, allows for an application of the Cauchy-Born rulelocally near SO ( d ) . The so-called Cauchy-Born energy density is defined by letting eachatom follow the macroscopic gradient: W CB ( M ) := 1 | det A | W cell ( M Z ) for M ∈ R d × d . Assumption 3. (i) W cell : R d × d → R is invariant under translations and rotations, i.e.for F ∈ R d × d , W cell ( RF + ( c, . . . , c )) = W cell ( F ) for all R ∈ SO ( d ) , c ∈ R d .(ii) W cell ( F ) is minimal ( = 0 ) if and only if there exists R ∈ SO ( d ) and c ∈ R d such that F = RZ + ( c, . . . , c ) . (iii) W cell is C in a neighborhood of ¯ SO ( d ) := SO ( d ) Z and the Hessian D W cell ( Z ) at theidentity is positive definite on the orthogonal complement of the subspace spannedby translations ( c, . . . , c ) and infinitesimal rotations F Z , with F T = − F .(iv) p ≥ d , which together with Assumption 1 implies in particular that W cell satisfies thegrowth assumption lim inf | F |→∞ F ∈ V W cell ( F ) | F | d > . .2 Γ -convergence and integral representations In our analysis, we consider energies on discrete systems depending on a small parameter ε ,the scale of the lattice spacing. To make the limit for ε → precise and gain some knowl-edge about the behavior of associated minimizers, we will use De Giorgi’s Γ -convergence.We recall the definition and some basic properties that will be needed in the sequel. Definition 2.1.
Let X be a metric space and F n , F : X → ¯ R = R ∪ {−∞ , ∞} . We say F n Γ( X ) -converges to F ( F n Γ −→ F ), if(i) (liminf-inequality) For every y, y n ∈ X with y n → y , we have F ( y ) ≤ lim inf n →∞ F n ( y n ) , (ii) (recovery sequence) For every y ∈ X , there is a sequence y n ∈ X such that F ( y ) ≥ lim sup n →∞ F n ( y n ) . If ( F ε ) ε> is a family of functionals depending on a positive real parameter ε , we say F ε Γ( X ) -converges to F , if for every sequence ε n > converging to , we have F ε n Γ −→ F .We will also use the Γ - lim sup and the Γ - lim inf , given by F ′ ( y ) = Γ( X ) - lim inf n →∞ F n ( y ) = inf { lim inf n →∞ F n ( y n ) : y n → y in X } ,F ′′ ( y ) = Γ( X ) - lim sup n →∞ F n ( y ) = inf { lim sup n →∞ F n ( y n ) : y n → y in X } . Note that (i) is equivalent to F ≤ F ′ and (ii) is equivalent to F ≥ F ′′ . Hence, F n Γ −→ F ifand only if F ′ = F ′′ = F . Furthermore, we see that Γ -convergence is a pointwise property,so we can speak about Γ -convergence at a specific point.In the following proposition we assemble some basic properties of Γ -convergence thatwe will not prove here. Proposition 2.2. (i) The infima in the definitions of F ′ and F ′′ are actually attainedminima in ¯ R ;(ii) every sequence of functionals on a separable metric space, like L p ( U ; R d ) , has a Γ -convergent subsequence;(iii) F ′ , F ′′ and F are lower semicontinuous with respect to convergence in X .(iv) Γ -convergence satisfies the Urysohn property, i.e., F n Γ -converges to F , if and onlyif every subsequence of F n has a further subsequence, that Γ -converges to F ;(v) if F n Γ -converges to F and G n converges uniformly on bounded sets to a continuousfunctional G , then F n + G n Γ -converges to F + G . In view of applications, the most interesting property of Γ -convergence is the followingtheorem. Theorem 2.3. If F n Γ -converges to F and sequences y n in X with equibounded F n ( y n ) are pre-compact then F attains its minimum on X and we have min x ∈ X F ( x ) = lim n →∞ inf x ∈ X F n ( x ) . urthermore, let y n ∈ X be a sequence with F n ( y n ) → lim n →∞ inf x ∈ X F n ( x ) , then the limit of every converging subsequence of y n is a minimizer of F . For proofs of Proposition 2.2 and Theorem 2.3 see, e.g., [DM93].Returning to our specific setting, for a sequence ε n > such that ε n → , we define F ′ ( y, U ) := Γ( L p (Ω; R d )) - lim inf n →∞ F ε n ( y, U )= min { lim inf n →∞ F ε n ( y n , U ) : y n → y in L p (Ω; R d ) } ,F ′′ ( y, U ) := Γ( L p (Ω; R d )) - lim sup n →∞ F ε n ( y, U ) . = min { lim sup n →∞ F ε n ( y n , U ) : y n → y in L p (Ω; R d ) } . The limiting functionals we will encounter in the next section are integral functionalsof the form I : W ,p ( U ; R k ) → [0 , ∞ ] , I ( y ) = ˆ U f ( ∇ y ( x )) dx with < p < ∞ , U ∈ A ( R d ) , f : R k × d → [0 , ∞ ) continuous. Recall that a Borel measurableand locally bounded function f : R k × d → R is quasiconvex, if f ( M ) ≤ U f ( M + ∇ ϕ ( x )) dx, for every nonempty U ∈ A ( R d ) , M ∈ R k × d and ϕ ∈ W , ∞ ( U ; R k ) . In our analysis, thequasiconvexity of f will be due to the following result. Theorem 2.4. If I is sequentially weakly lower semicontinuous in W ,p ( U ; R k ) , then f isquasiconvex. A detailed discussion of quasiconvexity and related properties, including proofs of theabove statements, is given, e.g., in [Dac08].In order to guarantee that indeed our limiting functional is an integral functional, wewill resort to the following general integral representation result on Sobolev spaces.
Theorem 2.5.
Let ≤ p < ∞ and let F : W ,p (Ω; R d ) × A (Ω) → [0 , ∞ ] satisfy thefollowing conditions:(i) (locality) F ( y, U ) = F ( v, U ) , if y ( x ) = v ( x ) for a.e. x ∈ U ;(ii) (measure property) F ( y, · ) is the restriction of a Borel measure to A (Ω) ;(iii) (growth condition) there exists c > such that F ( y, U ) ≤ c ˆ U |∇ y ( x ) | p + 1 dx ; (iv) (translation invariance in y) F ( y, U ) = F ( y + a, U ) for every a ∈ R d ; v) (lower semicontinuity) F ( · , U ) is sequentially lower semicontinuous with respect toweak convergence in W ,p (Ω; R d ) ;(vi) (translation invariance in x) With y M ( x ) = M x we have F ( y M , B r ( x )) = F ( y M , B r ( x ′ )) for every M ∈ R d × d , x, x ′ ∈ Ω and r > such that B r ( x ) , B r ( x ′ ) ⊂ Ω .Then there exists a continuous f : R d × d → [0 , ∞ ) such that ≤ f ( M ) ≤ C (1 + | M | p ) for every M ∈ R d × d and F ( y, U ) = ˆ U f ( ∇ y ( x )) dx. A proof can be found in [BD98, pp.77-81] or in the scalar-valued setting, which isessentially the same, in [DM93, pp.215-220].To show the measure property in the previous theorem, we will use the following lemma.
Lemma 2.6 (De Giorgi-Letta) . Let X be a metric space with open sets τ . Assume that ρ : τ → [0 , ∞ ] is an increasing set function such that(i) ρ ( ∅ ) = 0 ,(ii) (subadditivity) ρ ( U ∪ V ) ≤ ρ ( U ) + ρ ( V ) for all U, V ∈ τ ,(iii) (inner regularity) ρ ( U ) = sup { ρ ( V ) : V ∈ τ, V ⊂⊂ U } for all U ∈ τ ,(iv) (superadditivity) ρ ( U ∪ V ) ≥ ρ ( U ) + ρ ( V ) for all U, V ∈ τ with U ∩ V = ∅ .Then the extension µ of ρ to all subsets of X , defined by µ ( E ) = inf { ρ ( U ) : U ∈ τ, E ⊂ U } , is an outer measure and every Borel set is µ -measurable. For a proof see, e.g., [FL07, pp.32-34].
In this section we will prove a general compactness and representation result for sequencesof discrete deformations. For pair interactions, the following theorem has first been estab-lished by Alicandro and Cicalese in [AC04].
Theorem 3.1 (compactness and integral representation) . Suppose Assumptions 1 and 2are true. For every sequence ε n > such that ε n → , there exists a subsequence ε n k and a functional F : L p (Ω; R d ) × A (Ω) → [0 , ∞ ] such that for every U ∈ A (Ω) and y ∈ W ,p (Ω; R d ) the functionals F ε nk ( · , U ) Γ( L p (Ω; R d )) -converge to F ( · , U ) at y . Furthermorethere exists a quasiconvex function f : R d × d → [0 , ∞ ) satisfying c | M | p − c ′ ≤ f ( M ) ≤ c ′ ( | M | p + 1) or some c, c ′ > such that F ( y, U ) = ˆ U f ( ∇ y ( x )) dx if y ∈ W ,p (Ω; R d ) . In addition, if U ∈ A L (Ω) (in particular, if U = Ω ), we have F ( y, U ) = ˆ U f ( ∇ y ( x )) dx if y | U ∈ W ,p ( U ; R d ) , ∞ otherwise,and the functionals F ε nk ( · , U ) Γ -converge to F ( · , U ) . We now define the continuous and piecewise affine interpolation ˜ y of y , similar to [Sch09]:First consider the cell A (cid:2) − , (cid:3) d and y : A (cid:8) − , (cid:9) d → R d . On every -dimensional face ofthe cell just take ˜ y = y . Now assume we already have chosen a simplicial decomposition onevery ( k − -dimensional face and have interpolated affine there. Let F = co { z i , . . . , z i k } be a k -dimensional face. Set ¯ z = 12 k k X m =1 z i m , ˜ y (¯ z ) = 12 k k X m =1 y ( z i m ) . To complete the induction, we decompose F into the simplices co { w , . . . , w k , ¯ z } , where co { w , . . . , w k } is a simplex belonging to a simplicial decomposition of an ( n − -dimensionalface. Define ˜ y to be the interpolation affine on every constructed simplex. If y ∈ B ε (Ω) ,we get ˜ y on Ω ε by interpolating as above on every cell.The following proposition is about the relation of ¯ ∇ y and ∇ ˜ y . Proposition 3.2.
There are
C, c > such that for every x ∈ Ω ε and y ∈ B ε (Ω) c (cid:12)(cid:12) ¯ ∇ y ( x ) (cid:12)(cid:12) p ≤ Q ε ( x ) (cid:12)(cid:12) ∇ ˜ y ( x ′ ) (cid:12)(cid:12) p dx ′ ≤ C (cid:12)(cid:12) ¯ ∇ y ( x ) (cid:12)(cid:12) p . (3.1) Proof.
Every z i belongs to some simplex K of the construction. Choose a ∈ K ◦ , wherethe gradient is well-defined. Since the interpolation is linear on K , we see that (cid:12)(cid:12)(cid:12)(cid:12) y (¯ x + εz i ) − ¯ yε (cid:12)(cid:12)(cid:12)(cid:12) p = |∇ ˜ y ( a ) z i | p ≤ C |∇ ˜ y ( a ) | p = C | K | ˆ K (cid:12)(cid:12) ∇ ˜ y ( x ′ ) (cid:12)(cid:12) p dx ′ ≤ C | Q ε ( x ) || K | Q ε ( x ) (cid:12)(cid:12) ∇ ˜ y ( x ′ ) (cid:12)(cid:12) p dx ′ ≤ C Q ε ( x ) (cid:12)(cid:12) ∇ ˜ y ( x ′ ) (cid:12)(cid:12) p dx ′ , C is independent of x, ε and y . We immediately get the first inequality. For thesecond inequality we prove by induction over k that for every k -dimensional simplex S =co { ¯ z, z i , w , . . . , w k − } in the construction regarding Q ε ( x ) we have |∇ ˜ y ( a ) P V | p ≤ C (cid:12)(cid:12) ¯ ∇ y ( x ) (cid:12)(cid:12) p (3.2)for every a ∈ S , where P V is the projection on V = span { ¯ z − z i , w − z i , . . . , w k − − z i } .The case k = 1 is clear since then for some j we have V = span { z j − z i } and ∇ ˜ y ( a )( z j − z i ) = ¯ ∇ y ( x )( e j − e i ) . If the statement is true for k − , we immediately have (3.2) for V ′ = span { w − z i , . . . , w k − − z i } . But as in the k = 1 case we also have (3.2) for V ′′ = span { ¯ z − z i } = span { z j − z i } . Letus define k v k V = (cid:12)(cid:12) v ′ (cid:12)(cid:12) + (cid:12)(cid:12) v ′′ (cid:12)(cid:12) , if v ∈ V , v ′ ∈ V ′ and v ′′ ∈ V ′′ such that v = v ′ + v ′′ . This is a norm on V and hence wecan calculate using the equivalence of all norms on finite dimensional spaces |∇ ˜ y ( a ) P V | p ≤ C sup {|∇ ˜ y ( a ) v | p : v ∈ V, k v k V ≤ }≤ C (sup { (cid:12)(cid:12) ∇ ˜ y ( a ) v ′ (cid:12)(cid:12) p : v ′ ∈ V ′ , (cid:12)(cid:12) v ′ (cid:12)(cid:12) ≤ } + sup { (cid:12)(cid:12) ∇ ˜ y ( a ) v ′ (cid:12)(cid:12) p : v ′ ∈ V ′ , (cid:12)(cid:12) v ′ (cid:12)(cid:12) ≤ } ) ≤ C (cid:12)(cid:12) ¯ ∇ y ( x ) (cid:12)(cid:12) p . Since we have only finite many possibilities for
V, V ′ , V ′′ , this C can be chosen independentof them, which concludes the induction. Take k = d and integrate to get the result. Proposition 3.3.
Let ε n > , with ε n → , y n ∈ B ε n (Ω) and y ∈ L p (Ω; R d ) such that y n → y in L p (Ω; R d ) . For every V ⊂⊂ Ω , we then have ˜ y n → y in L p ( V ; R d ) .Proof. It is enough to show k y n − ˜ y n k L p ( V ; R d ) → . Let λ i : R d → [0 , denote the cell-periodic functions such that ˜ y n ( x ) = d X i =1 λ i (cid:18) xε n (cid:19) y n (¯ x + ε n z i )= d X i =1 λ i (cid:18) xε n (cid:19) y n ( x + ε n ( z i − z )) , where without loss of generality we have chosen a numbering of A {− , } d such that z = A ( − , . . . , − ) . Of course, λ i ≥ and the λ i add up to in any point and so for n large enough ˆ V | y n ( x ) − ˜ y n ( x ) | p dx ≤ ˆ V d X i =1 λ i (cid:18) xε n (cid:19) | y n ( x ) − y n ( x + ε n ( z i − z )) | p dx ≤ ˆ V max i =1 ,..., d | y n ( x ) − y n ( x + ε n ( z i − z )) | p dx ≤ d X i =1 ˆ V | y n ( x ) − y n ( x + ε n ( z i − z )) | p dx. since for every i ∈ { , . . . , d }k y n − y n ( · + ε n ( z i − z )) k L p ( V ; R d ) ≤ k y n − y k L p (Ω; R d ) + k y − y ( · + ε n ( z i − z )) k L p ( V ; R d ) → . We proceed to collect further lemmata. We will use them later to prove the requirementsof Theorem 2.5. In the following, fix some sequence of positive real numbers ε n → . Lemma 3.4.
Suppose Assumption 1 is true. If y ∈ L p (Ω; R d ) and U ∈ A (Ω) are such that F ′ ( y, U ) < ∞ , then y ∈ W ,p ( U ; R d ) and F ′ ( y, U ) ≥ c k∇ y k pL p ( U ; R d × d ) − c ′ | U | , (3.3) for some c, c ′ > independent of y and U .Proof. Let y n → y in L p (Ω; R d ) such that lim inf n →∞ F ε n ( y n , U ) < ∞ . For some subsequence n k , we have lim k →∞ F ε nk ( y n k , U ) = lim inf n →∞ F ε n ( y n , U ) ,y n k ∈ B ε nk ( U ) and F ε nk ( y n k , U ) ≤ M < ∞ for some fixed M > . By Proposition 3.3we have ˜ y n k → y in L p ( V, R d ) for every V ⊂⊂ U . Furthermore, by Assumption 1 andProposition 3.2, we get M ≥ F ε nk ( y n k , U ) = ε dn k X x ∈ ( L ′ εnk ( U )) ◦ W cell ( ¯ ∇ y n k ( x )) ≥ ε dn k X x ∈ ( L ′ εnk ( U )) ◦ (cid:0) c (cid:12)(cid:12) ¯ ∇ y n k ( x ) (cid:12)(cid:12) p − c ′ (cid:1) ≥ ε dn k X x ∈ ( L ′ εnk ( U )) ◦ c Q εnk ( x ) (cid:12)(cid:12) ∇ ˜ y n k ( x ′ ) (cid:12)(cid:12) p dx ′ − c ′ . We thus obtain c ˆ U εnk (cid:12)(cid:12) ∇ ˜ y n k ( x ′ ) (cid:12)(cid:12) p dx ′ ≤ M + c ′ | U | , hence the gradients are bounded in L p ( V ; R d ) . By the properties of weak convergenceon Sobolev spaces this means y ∈ W ,p ( V, R d ) and ∇ ˜ y n k ⇀ ∇ y in L p ( V ; R d ) . Weaksequentially lower semicontinuity of the norm yields c k∇ y k pL p ( V ; R d × d ) ≤ lim inf n →∞ F ε n ( y n , U ) + c ′ | U | , but the right hand side is independent of V , thus y ∈ W ,p ( U, R d ) and c k∇ y k pL p ( U ; R d × d ) ≤ lim inf n →∞ F ε n ( y n , U ) + c ′ | U | . The definition of the Γ - lim inf now yields the lemma.14 emma 3.5. Suppose Assumption 2 is true. Then there is a
C > such that for every V ∈ A L (Ω) , U ∈ A ( V ) and y ∈ L p (Ω; R d ) ∩ W ,p ( V ; R d ) we have F ′′ ( y, U ) ≤ C (cid:16) k∇ y k pL p ( U ; R d × d ) + | U | (cid:17) . (3.4) Proof.
We first prove (3.4) for every y ∈ C ∞ c ( R d ; R d ) . For x ∈ L ′ ε n and a ∈ Q ε n ( x ) define y n ( a ) = y ( x ) . Thus y n ∈ B ε n ( U ) and since y is uniformly continuous, we have y n → y uniformly andhence in L p (Ω; R d ) . By Taylor expansion we have (cid:12)(cid:12)(cid:12)(cid:12) y n (¯ x ) − y n (¯ x + ε n z i ) ε n (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) y (¯ x ) − y (¯ x + ε n z i ) ε n (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( |∇ y (¯ x ) | + ε n (cid:13)(cid:13) ∇ y (cid:13)(cid:13) ∞ ) . With Assumption 2 we can calculate F ε n ( y n , U ) = ε dn X x ∈ ( L ′ εn ( U )) ◦ W cell ( ¯ ∇ y n ( x )) ≤ Cε dn X x ∈ ( L ′ εn ( U )) ◦ ( (cid:12)(cid:12) ¯ ∇ y n ( x ) (cid:12)(cid:12) p + 1) ≤ C ′ | U | + Cε dn X x ∈ ( L ′ εn ( U )) ◦ ( |∇ y ( x ) | p + ε pn (cid:13)(cid:13) ∇ y (cid:13)(cid:13) p ∞ ) ≤ C ′ | U | + C ′′ | U | ε pn (cid:13)(cid:13) ∇ y (cid:13)(cid:13) p ∞ + Cε dn X x ∈ ( L ′ εn ( U )) ◦ |∇ y ( x ) | p . Furthermore for every x ′ ∈ Q ε n ( x ) , x ∈ ( L ′ ε n ( U )) ◦ |∇ y ( x ) | p ≤ C ( (cid:12)(cid:12) ∇ y ( x ′ ) (cid:12)(cid:12) p + (cid:12)(cid:12) ∇ y ( x ) − ∇ y ( x ′ ) (cid:12)(cid:12) p ) ≤ C ( (cid:12)(cid:12) ∇ y ( x ′ ) (cid:12)(cid:12) p + ε pn (cid:13)(cid:13) ∇ y (cid:13)(cid:13) p ∞ ) , and, by integrating over x ′ and summing over x , we get ε dn X x ∈ ( L ′ εn ( U )) ◦ |∇ y ( x ) | p ≤ C ˆ U εn (cid:12)(cid:12) ∇ y ( x ′ ) (cid:12)(cid:12) p dx ′ + | U ε n | ε pn (cid:13)(cid:13) ∇ y (cid:13)(cid:13) p ∞ ≤ C ˆ U (cid:12)(cid:12) ∇ y ( x ′ ) (cid:12)(cid:12) p dx ′ + | U | ε pn (cid:13)(cid:13) ∇ y (cid:13)(cid:13) p ∞ . Putting the two inequalities together and letting n → ∞ , we obtain lim sup n →∞ F ε n ( y n , U ) ≤ C (cid:16) k∇ y k pL p ( U ; R d × d ) + | U | (cid:17) . So by the definition of the Γ - lim sup we have (3.4).The general case follows easily: Since V has Lipschitz boundary, we can take y k ∈ C ∞ c ( R d ; R d ) such that y k → y in W ,p ( V ; R d ) . Then we have by lower semicontinuity of15 ′′ ( · , U ) F ′′ ( y, U ) ≤ lim inf k →∞ F ′′ ( y k , U ) ≤ lim inf k →∞ C (cid:16) k∇ y k k pL p ( U ; R d × d ) + | U | (cid:17) = C (cid:16) k∇ y k pL p ( U ; R d × d ) + | U | (cid:17) . Lemma 3.6.
Suppose Assumptions 1 and 2 are true. Let
U, V, U ′ ∈ A (Ω) be such that U ′ ⊂⊂ U . Then for every y ∈ W ,p (Ω; R d ) F ′′ ( y, U ′ ∪ V ) ≤ F ′′ ( y, U ) + F ′′ ( y, V ) . Proof.
Without loss of generality, we can assume the terms on the right hand side to befinite. According to the properties of the Γ - lim sup it is possible to find sequences u n , v n such that lim sup n →∞ F ε n ( u n , U ) = F ′′ ( y, U )lim sup n →∞ F ε n ( v n , V ) = F ′′ ( y, V ) u n → y in L p (Ω; R d ) v n → y in L p (Ω; R d ) For n large enough F ε n ( u n , U ) and F ε n ( v n , V ) are bounded and u n ∈ B ε n ( U ) , v n ∈ B ε n ( V ) .Fix N ∈ N , N ≥ and then define D = dist( U ′ , U c ) and U j = { x ∈ U : dist( x, U ′ ) < jDN } . Choose cut-off functions ϕ j such that ϕ j ( x ) = 1 ∀ x ∈ U j ,ϕ j ∈ C ∞ c ( U j +1 ; [0 , , k∇ ϕ j k ∞ ≤ ND .
Next we define w n,j ( x ) = ϕ j (¯ x ) u n ( x ) + (1 − ϕ j (¯ x )) v n ( x ) and calculate w n,j ( x + ε n z i ) − w n,j ( x ) ε n = ϕ j ( x + ε n z i ) u n ( x + ε n z i ) − u n ( x ) ε n + (1 − ϕ j ( x + ε n z i )) v n ( x + ε n z i ) − v n ( x ) ε n (3.5) + ( u n ( x ) − v n ( x )) ϕ j ( x + ε n z i ) − ϕ j (¯ x ) ε n . To estimate F ε n ( w n,j , U ′ ∪ V ) , we have to look at ( L ′ ε n ( U ′ ∪ V )) ◦ . Clearly, if x is in ( L ′ ε n ( U j )) ◦ , then ¯ ∇ w n,j ( x ) = ¯ ∇ u n ( x ) and if x is in ( L ′ ε n ( V \ U j +1 )) ◦ , then ¯ ∇ w n,j ( x ) =¯ ∇ v n ( x ) . To control the other cases, observe that for n large enough diam( Q ε n ) ≤ D N andthus ( L ′ ε n ( U ′ ∪ V )) ◦ ⊂ ( L ′ ε n ( U j )) ◦ ∪ ( L ′ ε n ( V \ U j +1 )) ◦ ∪ ( L ′ ε n ( V ∩ ( U j +2 \ U j − ))) ◦ j ∈ { , . . . , N − } and n large enough. With W j = V ∩ ( U j +2 \ U j − ) , we thenhave F ε n ( w n,j , U ′ ∪ V ) = ε dn X x ∈ ( L ′ εn ( U ′ ∪ V )) ◦ W cell ( ¯ ∇ w n,j ( x )) ≤ F ε n ( u n , U ) + F ε n ( v n , V ) + ε dn X x ∈ ( L ′ εn ( W j )) ◦ W cell ( ¯ ∇ w n,j ( x )) | {z } := S j,n . We now have to estimate S j,n . For all n large enough, use first Assumption 2 and then(3.5) to get S j,n ≤ Cε dn X x ∈ ( L ′ εn ( W j )) ◦ ( (cid:12)(cid:12) ¯ ∇ w n,j ( x ) (cid:12)(cid:12) p + 1) ≤ Cε dn X x ∈ ( L ′ εn ( W j )) ◦ ( (cid:12)(cid:12) ¯ ∇ u n ( x ) (cid:12)(cid:12) p + (cid:12)(cid:12) ¯ ∇ v n ( x ) (cid:12)(cid:12) p + | u n ( x ) − v n ( x ) | p k∇ ϕ j k p ∞ + 1) ≤ Cε dn X x ∈ ( L ′ εn ( W j )) ◦ ( (cid:12)(cid:12) ¯ ∇ u n ( x ) (cid:12)(cid:12) p + (cid:12)(cid:12) ¯ ∇ v n ( x ) (cid:12)(cid:12) p + | u n ( x ) − v n ( x ) | p N p + 1) ≤ C ˆ ( W j ) εn |∇ ˜ u n ( x ) | p + |∇ ˜ v n ( x ) | p + N p | u n ( x ) − v n ( x ) | p + 1 dx, because of the gradient of ϕ being bounded by CN and Proposition 3.2. Averaging over j ,we get N − N − X j =2 S j,n ≤ C N − ˆ V εn |∇ ˜ u n ( x ) | p + |∇ ˜ v n ( x ) | p + 1 dx + N p ˆ V εn | u n ( x ) − v n ( x ) | p dx. (3.6)Of course we can always find a number j ( n ) such that S j ( n ) ,n ≤ N − N − X j =2 S j,n . By Proposition 3.2 and Assumption 1, the first integral in (3.6) is bounded, but k u n − v n k L p (Ω; R d ) → for n → ∞ , hence lim sup n →∞ S j ( n ) ,n ≤ CN − . If we define y n = w n,j ( n ) , then obviously y n ∈ B ε n ( U ′ ∪ V ) and y n → y in L p (Ω; R d ) . Wehave F ′′ ( y, U ′ ∪ V ) ≤ lim sup n →∞ F ε n ( y n , U ′ ∪ V ) ≤ lim sup n →∞ F ε n ( u n , U ) + lim sup n →∞ F ε n ( v n , V ) + lim sup n →∞ S j ( n ) ,n ≤ F ′′ ( y, U ) + F ′′ ( y, V ) + CN − . Letting N → ∞ , we get the conclusion. 17 emma 3.7. Suppose Assumptions 1 and 2 are true. Then for every V ∈ A L (Ω) , U ∈A ( V ) and y ∈ L p (Ω; R d ) ∩ W ,p ( V ; R d ) F ′′ ( y, U ) = sup U ′ ⊂⊂ U F ′′ ( y, U ′ ) . Proof.
Since F ′′ ( y, · ) is an increasing set function, we only have to show ’ ≤ ’.Let δ > . Then take a U ′′′ ⊂⊂ U such that (cid:12)(cid:12) U \ U ′′′ (cid:12)(cid:12) + k∇ y k L p ( U \ U ′′′ ; R d ) ≤ δ. Choosing U ′ , U ′′ such that U ′′′ ⊂⊂ U ′′ ⊂⊂ U ′ ⊂⊂ U, we can calculate F ′′ ( y, U ) ≤ F ′′ ( y, U ′′ ∪ U \ U ′′′ ) ≤ F ′′ ( y, U ′ ) + F ′′ ( y, U \ U ′′′ ) ≤ F ′′ ( y, U ′ ) + δC, where we used Lemma 3.6 and Lemma 3.5. Lemma 3.8.
Suppose Assumptions 1 and 2 are true. Then for every V ∈ A L (Ω) , U ∈A ( V ) and u, v ∈ L p (Ω; R d ) ∩ W ,p ( V ; R d ) such that u ( x ) = v ( x ) for almost every x ∈ U ,we have F ′′ ( u, U ) = F ′′ ( v, U ) . Proof. If u = v a.e. in U then for U ′ ⊂⊂ U we have F ′′ ( u, U ′ ) = F ′′ ( v, U ′ ) . To see this,just change any approximating discrete sequence of u outside of ( U ′ ) ε n such that the newsequence converges to v . But this is enough by Lemma 3.7. Now, we can finally prove the compactness result:
Proof of Theorem 3.1.
First we find by a suitable diagonal argument a subsequence F ε nk such that we get Γ -convergence for every U ∈ A (Ω) . For this we define A = ( U ⊂ Ω : U = N [ i =1 B r i ( x i ) , x i ∈ Q d , r i ∈ Q , r i > , N ∈ N . ) The set A is countable and we can write A = { U , U , . . . } . Now choose subsequencesas follows: F ε n ( · , U ) has a Γ -convergent subsequence F ε n k ( · , U ) ,F ε n k ( · , U ) has a Γ -convergent subsequence F ε n k ( · , U ) ,F ε n k ( · , U ) has a Γ -convergent subsequence F ε n k ( · , U ) , ... ... ...Now setting n k = n kk , we see that F ε nk ( · , U ) Γ -converges to a F ( · , U ) for every U ∈ A . Inthe following we will only consider the sequence ε n k and, in particular, define F ′ and F ′′ accordingly. Furthermore, we define F ( y, U ) := F ′ ( y, U ) for every y and U .18or W ⊂⊂ U ⊂ Ω , by compactness of W , we always find V ∈ A such that W ⊂ V ⊂⊂ U . Hence, by Lemma 3.7 we have F ′′ ( y, U ) = sup { F ′′ ( y, V ) : V ⊂⊂ U, V ∈ A } for every U ∈ A (Ω) and y ∈ W ,p (Ω , R d ) . Using, that F ′ ( y, · ) is an increasing set function,we can calculate sup { F ′ ( y, V ) : V ⊂⊂ U, V ∈ A } ≤ F ′ ( y, U ) ≤ F ′′ ( y, U )= sup { F ′′ ( y, V ) : V ⊂⊂ U, V ∈ A } . But the first and the last term are equal, thus F ′ ( y, U ) = F ′′ ( y, U ) = F ( y, U ) , whenever y ∈ W ,p (Ω , R d ) .The next step is to get an integral representation by showing that F , restricted to W ,p (Ω; R d ) , satisfies the conditions (i)-(vi) in Theorem 2.5. We immediately see thelocality (i), by Lemma 3.8, and the growth condition (iii), by Lemma 3.5. Furthermore,since the F ε nk are translation invariant in y , so is F , which yields (iv). To get the lowersemicontinuity (v), just remember that weak convergence in W ,p (Ω , R d ) implies strongconvergence in L p (Ω , R d ) and that Γ (X)-limits are sequentially lower semicontinuous withrespect to the convergence in X .To get the measure property (ii), it is enough to show that we can apply the De-Giorgi-Letta criterion (Lemma 2.6) with ρ = F ( y, · ) . Obviously F ( y, · ) is an increasing setfunction and F ( y, ∅ ) = 0 . Remark that for every W ⊂⊂ U ∪ V ( W, U, V open), there areopen sets U ′ , V ′ such that U ′ ⊂⊂ U , V ′ ⊂⊂ V and W ⊂ U ′ ∪ V ′ , which is easily seen bythe compactness of W . Hence the subadditivity follows from the Lemmata 3.6 and 3.7.The inner regularity is explicitly given by Lemma 3.7. The superadditivity we can showdirectly. Take a sequence y k ∈ B ε nk ( U ∪ V ) such that y k → y in L p (Ω; R d ) and F ( y, U ∪ V ) = lim k →∞ F ε nk ( y k , U ∪ V ) . Then, F ( y, U ∪ V ) ≥ lim inf k →∞ F ε nk ( y k , U ) + lim inf k →∞ F ε nk ( y k , V ) ≥ F ( y, U ) + F ( y, V ) , since U ∩ V = ∅ . Hence, we can apply the De-Giorgi-Letta criterion and obtain (ii).Finally, condition (vi) states that for every M ∈ R d × d , z, z ′ ∈ Ω and r > such that B r ( z ) , B r ( z ′ ) ⊂ Ω , we have F ( y M , B r ( z )) = F ( y M , B r ( z ′ )) , if we set y M ( x ) = M x . By inner regularity, it is enough to show that, for any r ′ < r , F ( y M , B r ( z )) ≥ F ( y M , B r ′ ( z ′ )) . Let y k ∈ B ε nk ( B r ( z )) such that y k → y M in L p (Ω; R d ) and lim k →∞ F ε nk ( y k , B r ( z )) = F ( y M , B r ( z )) . Denote by a k the only point in L ε nk ∩ Q ε nk ( z ′ − z ) . Then define19 k ( x ) = ( y k ( x − a k ) + M a k if x ∈ ( B r ′ ( z ′ )) ε nk M ¯ x else.If k is large enough, then x − a k ∈ ( B r ( z )) ε nk , u k ∈ B ε nk ( B r ′ ( z ′ )) and ¯ ∇ u k ( x ) = ¯ ∇ y k ( x − a k ) for all x ∈ ( B r ′ ( z ′ )) ε nk . Hence, F ε nk ( u k , B r ′ ( z ′ )) ≤ F ε nk ( y k , B r ( z )) . Furthermore, we have
M a k → M ( z ′ − z ) and y k ( · − a k ) → M ( · − ( z ′ − z )) in L p ( B r ′ ( z ′ ); R d ) and therefore u k → y M in L p (Ω; R d ) . Hence, we get F ( y M , B r ′ ( z ′ )) ≤ lim inf k →∞ F ε nk ( u k , B r ′ ( z ′ )) ≤ lim inf k →∞ F ε nk ( y k , B r ( z ))= F ( y M , B r ( z )) , and (vi) is proven.Consequently, we can apply Theorem 2.5 to the restriction of F to W ,p (Ω; R d ) × A (Ω) .In particular, there is a continuous function f : R d × d → [0 , ∞ ) such that F ( y, U ) = ˆ U f ( ∇ y ( x )) dx if y ∈ W ,p (Ω; R d ) and ≤ f ( M ) ≤ C (1 + | M | p ) for every M ∈ R d × d . (3.7)The asserted lower bound on f is instantly obtained, if we apply Lemma 3.4 to y M and usethe integral representation. And finally, f is quasiconvex by Theorem 2.4, since F ( · , Ω) issequentially lower semicontinuous with respect to weak convergence in W ,p (Ω; R d ) .Now, let U have Lipschitz boundary. Take y ∈ L p (Ω; R d ) ∩ W ,p ( U, R d ) . By Lemma3.7, we have F ′′ ( y, U ) = sup { F ′′ ( y, V ) : V ⊂⊂ U, V ∈ A } . Using that F ′ ( y, · ) is an increasing set function, we can calculate sup { F ′ ( y, V ) : V ⊂⊂ U, V ∈ A } ≤ F ′ ( y, U ) ≤ F ′′ ( y, U )= sup { F ′′ ( y, V ) : V ⊂⊂ U, V ∈ A } . But the first and the last term are equal, thus F ′ ( y, U ) = F ′′ ( y, U ) = F ( y, U ) . If y ∈ L p (Ω; R d ) \ W ,p ( U, R d ) , then ∞ = F ′ ( y, U ) = F ′′ ( y, U ) = F ( y, U ) by Lemma 3.4. Hence, F ε nk ( · , U ) Γ( L p (Ω; R d )) -converges to F ( · , U ) . To get the integral representation for y ∈ L p (Ω; R d ) ∩ W ,p ( U, R d ) , observe, that since U has Lipschitz boundary, we can find afunction v ∈ W ,p (Ω; R d ) such that y ( x ) = v ( x ) for almost every x ∈ U. Then, by Lemma 3.8, F ( y, U ) = F ( v, U ) = ˆ U f ( ∇ v ( x )) dx = ˆ U f ( ∇ y ( x )) dx. The boundary value problem
While loading terms can be included in our results so far without difficulties, the restrictionto deformations with preassigned boundary values is more subtle.
Suppose g ∈ W , ∞ ( R d ; R d ) is a boundary datum. We will then always choose the preciserepresentative for g and thus assume that g is continuous. We define the admissible latticedeformations B ε ( U, g ) as the functions in B ε ( U ) , that satisfy the boundary condition y ( x ) = g (¯ x ) , whenever x ∈ ∂ L ε ( U ) . The correspondingly restricted discrete functional is F gε ( y, U ) = ( F ε ( y, U ) if y ∈ B ε ( U, g ) , ∞ otherwise.Assume that ε n k and f are as in Theorem 3.1, let us for simplicity write just ε k in thefollowing and set F g ( y, U ) = ( F ( y, U ) if y | U ∈ g + W ,p ( U ; R d ) , ∞ otherwise.In analogy to Theorem 3.1 we then have: Theorem 4.1.
Suppose Assumptions 1 and 2 are true, g ∈ W , ∞ ( R d ; R d ) and F g , F gε k areas above. Then F gε k ( · , U ) Γ( L p (Ω; R d )) -converges to F g ( · , U ) for every U ∈ A L (Ω) . We start by improving Proposition 3.3 for sequences in B ε ( U, g ) . This is possible, becausenow we can control what happens near the boundary. Note, that now we can naturallydefine the interpolation ˜ y on all of U , namely, we just extend y by the discretization of g before we interpolate. Proposition 4.2.
Let U ∈ A L (Ω) and y k ∈ B ε k ( U, g ) . Then y k → y in L p ( U ; R d ) if andonly if ˜ y k → y in L p ( U ; R d ) .Proof. First, let y k → y in L p ( U ; R d ) . Choose some open bounded set U ′ with Lipschitzboundary and U ⊂⊂ U ′ . Extend the functions by defining y k ( x ) := g (¯ x ) and y ( x ) := g ( x ) for x ∈ U ′ \ U . So y k ∈ B ε k ( U ′ , g ) and, since g is Lipschitz, we have y k → y in L p ( U ′ ; R d ) .But then by Proposition 3.3 we get ˜ y k → y in L p ( U ; R d ) .Now, let ˜ y k → y in L p ( U ; R d ) . Let λ i : R d → [0 , again denote the cell-periodicfunctions such that, with z = A ( − , . . . , − ) , ˜ y n ( x ) = d X i =1 λ i (cid:18) xε n (cid:19) y n ( x + ε n ( z i − z )) . Of course, λ i ≥ and the λ i add up to in any point. Define W n,i = (cid:26) x ∈ U : λ i (cid:18) xε n (cid:19) ≥ and λ j (cid:18) xε n (cid:19) ≤ a for j = i (cid:27) , a will be chosen suitably later, and note that, for every x ∈ U ε n , the ratio | W n,i ∩ Q εn ( x ) || Q εn ( x ) | is independent of n and x and positive since x ∈ Q ε n , x → ¯ x + z i implies that λ i ( xε n ) → and λ j ( xε n ) → for j = i . Next, extend y and y k by g as above and define P n y ( x ) := Q εn ( x ) y ( b ) db. Of course, we have k P n y − y k L p ( U ; R d ) → . Hence, it suffices to show k P n y − y n k L p ( U ; R d ) → . For x ∈ W n,i we have | ˜ y n ( x − ε n ( z i − z )) − P n y ( x ) | ≥ | y n ( x ) − P n y ( x ) |− X j = i λ j (cid:18) xε n (cid:19) | y n ( x − ε n ( z i − z j )) − P n y ( x ) | . Since y n and P n y are constant on every cell, we thus have k y n − P n y k L p ( U εn ; R d ) = 12 | U ε n | p | U ε n ∩ W n,i | p k y n − P n y k L p ( U εn ∩ W n,i ; R d ) ≤ | U ε n | p | U ε n ∩ W n,i | p k ˜ y n ( · − ε n ( z i − z )) − P n y k L p ( U εn ∩ W n,i ; R d ) + a X j = i k y n ( · − ε n ( z i − z j )) − P n y k L p ( U εn ; R d ) . But | U εn | p | U εn ∩ W n,i | p > is independent of n and k ˜ y n ( · − ε n ( z i − z )) − P n y k L p ( U εn ∩ W n,i ; R d ) ≤ k y − P n y k L p ( U ; R d ) + k ˜ y n ( · − ε n ( z i − z )) − y k L p ( U ; R d ) converges to . To control the remaining sum, we estimate k y n ( · − ε n ( z i − z j )) − P n y k L p ( U εn ; R d ) ≤ k y n ( · − ε n ( z i − z j )) − P n y ( · − ε n ( z i − z j )) k L p ( U εn ; R d ) + k P n y ( · − ε n ( z i − z j )) − P n y k L p ( U εn ; R d ) , where the second term goes to and the first term is estimated by k y n ( · − ε n ( z i − z j )) − P n y ( · − ε n ( z i − z j )) k L p ( U εn ; R d ) ≤ k y n − P n y k L p ( U εn ; R d ) . Altogether we obtain k y n − P n y k L p ( U ; R d ) ≤ k y n − P n y k L p ( U εn \ U εn ; R d ) + k y n − P n y k L p ( U εn ; R d ) ≤ k y n − P n y k L p ( U εn \ U εn ; R d ) + 2 a (2 d − k y n − P n y k L p ( U εn ; R d ) + o (1)= (1 + 2 a (2 d − k y n − P n y k L p ( U εn \ U εn ; R d ) + 2 a (2 d − k y n − P n y k L p ( U εn ; R d ) + o (1)
22s near the boundary we can calculate k y n − P n y k L p ( U εn \ U εn ; R d ) ≤ | U ε n \ U ε n | p k g k ∞ + k P n y k L p ( U εn \ U εn ; R d ) ≤ | U ε n \ U ε n | p k g k ∞ + k y k L p ( U εn \ U εn ; R d ) → , for a = d +1 we finally get k y n − P n y k L p ( U ; R d ) → . Remark . The proof shows that without boundary conditions, i.e., for a general sequence y k ∈ B ε k ( U ) , U ∈ A (Ω) , we still have y k → y in L ploc ( U ; R d ) if and only if ˜ y k → y in L ploc ( U ; R d ) . Proof of Theorem 4.1.
Fix U ∈ A L (Ω) . We start with the lim inf -inequality. Let y k , y ∈ L p (Ω; R d ) such that y k → y . We can assume that lim inf k →∞ F gε k ( y k , U ) < ∞ , because otherwise there is nothing to show. For some subsequence we then get lim inf k →∞ F gε k ( y k , U ) = lim l →∞ F gε kl ( y k l , U ) . But since F ε kl ≤ F gε kl , we can argue as in Lemma 3.4 to see that y ∈ W ,p ( U ; R d ) and,for any V ⊂⊂ U , that ˜ y k l ⇀ y in W ,p ( V ; R d ) . Using Proposition 4.2, we see that ˜ y k l converges strongly in L p ( U ; R d ) and, since ∇ ˜ y k l is now bounded in L p ( U ; R d ) , weakly in W ,p ( U ; R d ) to y . Regarding the boundary condition, there are open neighborhoods V l of ∂U , where ˜ y k l is an affine interpolation of g . Namely, V l is the interior of the union of allcells Q ε kl , with Q ε kl ∩ ∂U = ∅ . Then sup x ∈ ∂U | ˜ y k l ( x ) − g ( x ) | ≤ sup x ∈ V l | ˜ y k l ( x ) − g ( x ) | ≤ Cε k l since g is Lipschitz. Denoting the trace operator by T , we thus have T ˜ y k l ⇀ T y = T g in L p ( ∂U ; R d ) and hence y ∈ g + W ,p ( U ; R d ) . But then, we can calculate F g ( y, U ) = F ( y, U ) ≤ lim inf k →∞ F ε k ( y k , U ) ≤ lim inf k →∞ F gε k ( y k , U ) , and have indeed proven the lim inf -inequality.To get the Γ -convergence result, we now proof the lim sup -inequality. Let us first assume y ( x ) = g ( x ) + ψ ( x ) , for every x ∈ U and some ψ ∈ C ∞ c ( U ; R d ) . Then F g ( y, U ) = F ( y, U ) < ∞ . So, there exists a sequence u k ∈ B ε k ( U ) such that u k → y in L p (Ω; R d ) and lim k →∞ F ε k ( u k , U ) = F ( y, U ) . δ > , and then choose U ′ such that supp ψ ⊂ U ′ ⊂⊂ U and | U \ U ′ | ≤ δ . We now usea cut-off argument similarly as in the proof of Lemma 3.6. Fix N ∈ N and define U j = (cid:26) x ∈ U : dist( x, U ′ ) < j dist( U ′ , U c ) N (cid:27) . Then choose the cut-off functions ϕ j ∈ C ∞ c ( U j +1 ; [0 , with ϕ j ≡ on U j and k∇ ϕ j k ∞ ≤ CN and set ˆ g k ( x ) = g ( a ) , if a ∈ Q ε k ( x ) ∩ L ε k and w n,j ( x ) = ( ϕ j (¯ x ) u k ( x ) + (1 − ϕ j (¯ x ))ˆ g k ( x ) , if x ∈ U ε k ,u k ( x ) otherwise.As in the proof of Lemma 3.6 we calculate F ε k ( w k,j , U ) ≤ F ε k ( u k , U ) + C ( k∇ g k p ∞ + 1) (cid:12)(cid:12) U \ U ′ (cid:12)(cid:12) + ε dk X ¯ x ∈ ( L ′ εk ( W j )) ◦ W cell ( ¯ ∇ w k,j (¯ x )) | {z } := S j,k , with W j = U j +2 \ U j − , estimate S j,k by averaging, choose j ( k ) suitably and thus get lim sup k →∞ F ε k ( w k,j ( k ) , U ) ≤ F ( y, U ) + Cδ + CN − . Since we choose j ( k ) ≤ N − , we have w k,j ( k ) ∈ B ε k ( U, g ) for any k large enough. Fur-thermore, w k,j ( k ) → y in L p (Ω; R d ) since ψ has support in U ′ . Hence, Γ - lim sup k →∞ F gε k ( y, U ) ≤ F g ( y, U ) + δC + CN − . Let δ → and N → ∞ .In the general case y | U ∈ g + W ,p ( U ; R d ) , take y l such that y l | U ∈ g + C ∞ c ( U ; R d ) and y l → y in W ,p ( U ; R d ) and in L p (Ω; R d ) . We get Γ - lim sup k →∞ F gε k ( y, U ) ≤ lim inf l →∞ (Γ - lim sup k →∞ F gε k ( y l , U )) ≤ lim inf l →∞ F g ( y l , U )= F g ( y, U ) by the lower semicontinuity of the Γ - lim sup with respect to L p (Ω; R d ) -convergence andcontinuity of F g ( · , U ) with respect to W ,p ( U ; R d ) -convergence. The following theorem is important in two ways. On the one hand we gain insight into the Γ -convergence result, on the other hand we will directly need it to get the homogenizationresult in Section 5. Theorem 4.4.
Under the assumptions of Theorem 4.1, we have min y F g ( y, U ) = lim k →∞ (inf y F gε k ( y, U )) . urthermore, any sequence y k with equibounded energy is pre-compact in L p ( U ; R d ) and ifwe have a sequence satisfying lim k →∞ (inf y F gε k ( y, U )) = lim k →∞ F gε k ( y k , U ) , then every limit of a converging subsequence is a minimizer of F g ( · , U ) .Proof. Fix g, U and write G k ( y ) = F gε k ( y, U ) , G ( y ) = F g ( y, U ) . Let y k be a sequence withequibounded energy G k ( y k ) . By Assumption 1 and Proposition 3.2 we obtain that ˆ U εk |∇ ˜ y k | p dx ≤ C. Furthermore, using the boundary condition, we have ˆ U |∇ ˜ y k | p dx ≤ C. A Poincaré-type inequality involving the trace yields k ˜ y k k W ,p ( U ; R d ) ≤ C ( k∇ ˜ y k k L p ( U ; R d ) + k T ˜ y k k L p ( ∂U ; R d ) ) ≤ C + C k g k ∞ H d − ( ∂U ) p ≤ C and so ˜ y k l → y in L p ( U ; R d ) for some subsequence k l and some y ∈ W ,p ( U ; R d ) . Then,by Proposition 4.2, y k l → y in L p ( U ; R d ) .Now from Theorem 4.1 we infer that G k Γ( L p (Ω; R d )) -converges to G . But then G k also Γ( L p ( U ; R d )) -converges to G . Here the existence of recovery sequences is immediateas L p (Ω; R d ) - implies L p ( U ; R d ) -convergence. As for the lim inf -inequality, if y k → y in L p ( U ; R d ) , where the energies G k ( y k ) are without loss of generality assumed to be equi-bounded and, in particular, in y k ∈ B ε k ( U, g ) , we can extend the functions by defining y k ( x ) := g (¯ x ) and y ( x ) := g ( x ) for x ∈ Ω \ U without changing their respective energies.Since then y k → y in L p (Ω; R d ) , we have indeed that lim inf k →∞ G k ( y k ) ≥ G ( y ) . Theremaining part of the proof now directly follows from Theorem 2.3. Γ -convergence results To simplify notations, we define P h ( x ) = x + A (0 , h ) d and P h = P h (0) . First, we will provethe following lemma. Lemma 5.1.
The limit lim N →∞ N d inf X x ∈ ( L ′ ( P N )) ◦ W cell ( ¯ ∇ y ( x )) : y ∈ B ( P N , y M ) exists for every M ∈ R d × d . roof. Let us define G ( y, U ) = X x ∈ ( L ′ ( U )) ◦ W cell ( ¯ ∇ y ( x )) and f k ( M ) = 1 k d inf { G ( y, P k ) : y ∈ B ( P k , y M ) } . Fix M ∈ R d × d and let k, n ∈ N with k > n . Choose v n ∈ B ( P n , y M ) such that n d G ( v n , P n ) ≤ f n ( M ) + 1 n . Now, we can define u k ( x ) = v n ( x − nα ) + nM α if x ∈ P n ( nα ) for some α ∈ A (cid:26) , , . . . , (cid:20) kn (cid:21) − (cid:27) d ,M ¯ x otherwise.Since v n satisfies the boundary condition, u k is constant on every cell. Moreover, u k ∈B ( P k , y M ) and we can estimate f k ( M ) ≤ k d G ( u k , P k ) ≤ k d (cid:20) kn (cid:21) d G ( v n , P n ) + c ( | M | p + 1) L ′ ( P k )) ◦ − (cid:20) kn (cid:21) d L ′ ( P n ( αn ))) ◦ !! ≤ n d G ( v n , P n ) + c ( | M | p + 1) k d | P k | − (cid:12)(cid:12)(cid:12) P n [ kn ] (cid:12)(cid:12)(cid:12) | P | + (cid:20) kn (cid:21) d (cid:16) n d − ( n − d (cid:17) ≤ f n ( M ) + 1 n + c ( | M | p + 1) k d k d − (cid:18) n (cid:20) kn (cid:21)(cid:19) d + k d − (cid:18) − n (cid:19) d !! ≤ f n ( M ) + 1 n + c ( | M | p + 1) − (cid:16) − nk (cid:17) d + 1 − (cid:18) − n (cid:19) d ! . Thus, for every n ∈ N , lim sup k →∞ f k ( M ) ≤ f n ( M ) + 1 n + c ( | M | p + 1) − (cid:18) − n (cid:19) d ! , hence, lim sup k →∞ f k ( M ) ≤ lim inf n →∞ f n ( M ) . Now, we can prove our first main theorem.
Proof of Theorem 1.1.
We will first show that F ε ( · , Ω) Γ( L p (Ω; R d )) -converges to F . Ac-cording to Lemma 5.1, W cont is well-defined. By the Urysohn property of Γ -convergencein Proposition 2.2, it is enough to show that, for any sequence ε n → , the function f of26heorem 3.1 equals W cont . Fix such a sequence, the subsequence ε k and the associated f .Since f is quasiconvex, we have for every M ∈ R d × d and U ∈ A L (Ω) f ( M ) = 1 | U | min ˆ U f ( ∇ y ( x )) dx : y − y M ∈ W ,p ( U ; R d ) = 1 | U | min n F ( y, U ) : y − y M ∈ W ,p ( U ; R d ) o . If we restrict y M to a ball that contains some neighborhood of Ω , we can extend it toa function in W , ∞ ( R d ; R d ) ∩ C ( R d ; R d ) , so y M is admissible as a boundary condition inTheorem 4.1 and we get the Γ -convergence result with boundary condition. Hence byTheorem 4.4, for h > and x ∈ R d such that P h ( x ) ⊂⊂ Ω f ( M ) = 1 | P h ( x ) | lim k →∞ (inf { F ε k ( y, P h ( x )) : y ∈ B ε k ( P h ( x ) , y M ) } )= 1 | P h ( x ) | lim k →∞ inf ε dk X x ∈ ( L ′ εk ( P h ( x ))) ◦ W cell ( ¯ ∇ y ( x )) : y ∈ B ε k ( P h ( x ) , y M ) It is easy to see, that we can always find h k > and x k ∈ L ε k such that P h k ( x k ) = [ x ∈ P h ( x ) Q ε k ( x ) ◦ . We then know P h k ( x k ) ⊂ Ω for all k large enough, | x − x k | ≤ diam Q ε k = ε k diam Q , h ≤ h k ≤ h + 2 ε k and, that there are N k ∈ N satisfying h k = N k ε k . Furthermore, L ′ ε k ( P h ( x )) = L ′ ε k ( P h k ( x k )) and ( L ′ ε k ( P h ( x ))) ◦ = ( L ′ ε k ( P h k ( x k ))) ◦ . Hence, B ε k ( P h ( x ) , y M ) and B ε k ( P h k ( x k ) , y M ) are equal up to extending the functions in B ε k ( P h ( x ) , y M ) constant on cells that intersect P h k ( x k ) \ P h ( x ) . It follows that f ( M ) = 1 | P | lim k →∞ N dk h dk h d inf X x ∈ ( L ′ εk ( P hk ( x k ))) ◦ W cell ( ¯ ∇ y ( x )) : y ∈ B ε k ( P h k ( x k ) , y M ) = 1 | P | lim k →∞ N dk inf X x ∈ ( L ′ εk ( P hk )) ◦ W cell ( ¯ ∇ y ( x )) : y ∈ B ε k ( P h k , y M ) , where we used, that y ∈ B ε k ( P h k ( x k ) , y M ) , if and only if y ( · + x k ) − M x k ∈ B ε k ( P h k , y M ) andthat the discrete gradient of y at a point x equals the discrete gradient of y ( · + x k ) − M x k at x − x k . In a similar way y ∈ B ε k ( P h k , y M ) if and only if y ′ ∈ B ( P N k , y M ) and ¯ ∇ y ′ ( x ) =¯ ∇ y ( ε k x ) , where y ′ ( x ) = ε k y ( ε k x ) . Hence, f ( M ) = 1 | P | lim k →∞ N dk inf X x ∈ ( L ′ ( P Nk )) ◦ W cell ( ¯ ∇ y ( x )) : y ∈ B ( P N k , y M ) = W cont ( M ) .
27n order to prove that also F ε ( · , Ω) Γ( L ploc (Ω; R d ) / R ) -converges to F , we only needto verify the lim inf -inequality as the existence of recovery sequences immediately followsfrom the first part of the proof since convergence in L p (Ω; R d ) implies convergence in L ploc (Ω; R d ) / R . But if ε n → and y ε n → y in L ploc (Ω; R d ) / R , then there exist c n ∈ R such that, for every U ∈ A L (Ω) with U ⊂⊂ Ω , y ε n − c n → y in L p ( U ; R d ) , so that by theprevious result lim inf n →∞ F ε n ( y ε n , Ω) = lim inf n →∞ F ε n ( y ε n − c n , Ω) ≥ lim inf n →∞ F ε n ( y ε n − c n , U ) ≥ F ( y, U ) . Without loss of generality we may assume that lim inf k →∞ F ε k ( y ε k , Ω) < ∞ . Since for any V ∈ A (Ω) with V ⊂⊂ Ω there exists U ∈ A L (Ω) with V ⊂ U ⊂⊂ Ω , we then deduce fromLemma 3.4 that y ∈ W ,p ( V ; R d ) with k y k W ,p ( V ; R d ) bounded uniformly in V ∈ A with V ⊂⊂ Ω , hence y ∈ W ,p (Ω; R d ) . Then invoking Lemma 3.7 and passing to the supremumover U ∈ A L (Ω) in the above inequality yields lim inf k →∞ F ε k ( y ε k , Ω) ≥ F ( y, Ω) . Proof of Theorem 1.3.
Theorem 1.3 is a direct consequence of Theorem 4.1 and Theorem1.1, where the limiting energy density f has been identified as W cont . Proof of Theorem 1.2.
Suppose y k is a sequence with equibounded energies F ε k ( y k ) . ByProposition 3.2 and the growth assumptions on W cell , for every U ∈ A (Ω) with U ⊂⊂ Ω we have ˆ U |∇ ˜ y k | p ≤ CF ε k ( y k ) + C | Ω | uniformly bounded for sufficiently large k . Choose U ∈ A L (Ω) connected and with ∅ 6 = U ⊂⊂ Ω . As U is connected, by Poincaré’s inequality we find c k ∈ R such that ˜ y k − c k ispre-compact in L p ( U ; R d ) . But then indeed for any connected U ∈ A L with U ⊂ U ⊂⊂ Ω the Poincaré inequality k ˜ y k − c k k W ,p ( U ; R d ) ≤ C k∇ ˜ y k k L p ( U ; R d ) + k ˜ y k − c k k L p ( U ; R d ) yields that ˜ y k − c k is pre-compact in L p ( U ; R d ) . Exhausting Ω with a countable numberof such domains and passing to a diagonal sequence, we find a subsequence y k n such that ˜ y k n − c k n converges in L ploc (Ω; R d ) . By Remark 4.3 we finally obtain that y k n − c k n convergesin L ploc (Ω; R d ) . Proof of Theorem 1.4.
This is immediate from Theorem 4.4.
Proof of Corollary 1.5.
This is a direct consequence of Theorems 1.1, 1.2, 1.3, 1.4 and2.3.
Proof of Theorem 1.8.
If in addition to Assumptions 1 and 2 Assumption 3 holds true,we can apply [CDKM06, Theorem 4.2] with
Λ = ( L ′ ( P N k )) ◦ . It is easy to see that theboundary of Λ as defined in [CDKM06] equals ∂ L ( P N k ) ∪ L \L ′ ( P N k ) , but of course the28econd part does not change anything. This shows that there is a neighborhood U of SO ( d ) , such that for every M ∈ U W cont ( M ) = 1 | P | lim k →∞ N dk inf X x ∈ ( L ′ ( P Nk )) ◦ W cell ( ¯ ∇ y ( x )) : y ∈ B ( P N k , y M ) = 1 | det A | lim k →∞ N dk X x ∈ ( L ′ ( P Nk )) ◦ W cell ( M Z )= 1 | det A | W cell ( M Z )= W CB ( M ) . Next we prove Propositions 1.9 and 1.10.
Proof of Proposition 1.9.
At variance with our previous decomposition procedure, we nowchoose any simplicial decomposition S of the cell A [ − , ) d into d -simplices all of whosecorners lie in A {− , } d . For F = ( f , . . . , f d ) ∈ R d × d we then interpolate the mapping A (cid:26) − , (cid:27) d → R d , x i f i affine on each simplex in order to obtain u F : A (cid:20) − , (cid:19) d → R d . Then W cell is defined by W cell ( F ) := ˆ A [ − , ) d V ( ∇ u F ) dx. As every corner z i , . . . , z i d of S ∈ S lies in A {− , } d , we have c d X j =1 (cid:12)(cid:12) f i j − f i (cid:12)(cid:12) ≤ |∇ u F | ≤ C | F | on S . Thus, k F k = max x ∈ A [ − , ) d |∇ u F ( x ) | is a norm on V and we calculate W cell ( F ) ≥ ˆ A [ − , ) d c |∇ u F | p − c ′ dx ≥ c k F k p − c ′ ≥ c | F | p − c ′ , and on the other hand W cell ( F ) ≤ C ( | F | p + 1) . W cell satisfies Assumptions 1 and 2. From Theorem 1.1 we then deduce that W cont ( M ) = 1 | det A | lim N →∞ N d inf X x ∈ ( L ′ ( P N )) ◦ W cell ( ¯ ∇ y ( x )) : y ∈ B ( P N , y M ) = 1 | det A | lim N →∞ N d inf ( ˆ ( P N ) V ( ∇ u ¯ ∇ y ( x ) ) : y ∈ B ( P N , y M ) ) = 1 | det A | lim N →∞ N d | det( A ) | ( N − d V ( M )= V ( M ) due to the quasiconvexity of V . Proof of Proposition 1.10.
Any F ∈ V can be decomposed orthogonally as F = F ′ Z + F ′′ with unique F ′ ∈ R d × d and F ′′ ∈ ( R d × d Z ) ⊥ . Set W cell ( F ) = | det A | Q (cid:18)q ( F ′ ) T F ′ − Id (cid:19) + | F ′′ | + χ ( F ) , where χ is any frame indifferent function satisfying Assumptions 1 and 2 with p ≥ d whichis non-negative, vanishes near ¯ SO ( d ) and is bounded from below by a positive constant on ¯ O ( d ) \ ¯ SO ( d ) , ¯ O ( d ) = O ( d ) Z . Then also W cell satisfies Assumptions 1 and 2 with the same p . Noting that, for M ∈ R d × d , ( M F ) ′ = M F ′ and ( M F ) ′′ = M F ′′ , it is not hard to verifythat W cell also satisfies Assumption 3 with D W cell ( Z )( F, F ) = 2 | det A | Q (cid:18) ( F ′ ) T + F ′ (cid:19) + 2 | F ′′ | . But then D W CB (Id)( M, M ) = 12 | det A | D W cell ( Z )( M Z, M Z ) = Q (cid:18) M T + M (cid:19) = Q ( M ) . We briefly comment on more general long-range interactions. Suppose
Λ = { z , . . . , z d , . . . , z N } ⊂L ′ is any fixed finite set, where z , . . . , z d still denote A {− , } d . For y ∈ B ε (Ω) we define y i = y (¯ x + εz i ) . With ¯ x and ¯ y as before, i.e., only depending on y , . . . y d , let now ¯ ∇ y ( x ) = 1 ε ( y − ¯ y, . . . , y N − ¯ y ) ∈ R d × N . The lattice interior ( L ′ ε ( U )) ◦ and boundary ∂ L ′ ε ( U ) now have to be shrunk respectivelyenlarged to a whole boundary layer, according to the maximal interaction length in Λ .Assumptions 1 and 2 are then replaced by the estimate c (cid:12)(cid:12) F ′ (cid:12)(cid:12) p − c ′ ≤ W super − cell ( F ) ≤ c ′′ ( | F | p + 1) for constants c, c ′ , c ′′ > and all F ∈ R d × N which satisfy F ′ ∈ V , where F ′ ∈ R d × d denotes the left d × d submatrix of F . Note that the lower bound in particular allowsfor arbitrarily weak long range interactions. As the interpolation we used only depends on30he d × d values of the corresponding lattice cell, this implies that we get the standardestimates for the gradients in Proposition 3.2 only on this part of the discrete gradient.It is important that the interaction range is bounded by Cε , so that, e.g., Lemma3.5 and its proof still work. In the estimates of the error S j,n in, e.g., Lemma 3.6, it isimportant that ε − | u (¯ x + εz i ) − u (¯ x ) | and thus the discrete gradient can be bounded bya fixed finite sum of smaller d × d discrete gradients of some cells near x . Hence, we stillhave the estimate ε dn X x ∈ ( L ′ εn ( U )) ◦ (cid:12)(cid:12) ¯ ∇ u ( x ) (cid:12)(cid:12) p ≤ C ˆ U |∇ ˜ u ( x ) | p dx Note that according to our enlarging of the lattice boundaries, also the cell formula forthe limit density will now involve a sequence of minimizing problems with affine boundaryconditions on a boundary layer.We finally remark that the statement on the applicability of the Cauchy-Born ruletranslates naturally, as the main ingredient does, see [CDKM06, Theorem 5.1].
It is also possible to generalize these results to certain non-Bravais lattices, namely tomulti-lattices of the form
L ∪ ( s + L ) ∪ · · · ∪ ( s m + L ) , in the following way: We stillconsider L to be our main lattice. But now we have m additional atoms in each cell, whichwe describe by the ‘internal variable’ s ( x ) ∈ R d × m , such that εs · j describes the distance ofthe j -th atom to the midpoint of the cell. Of course s can be identified with a function,that is constant on every interior cell and is outside and thus lies in some L q (Ω; R d × m ) , < q < ∞ . The new cell energy depends on md additional variables and we now considerthe growth condition c ( (cid:12)(cid:12) M ′ (cid:12)(cid:12) p + | s | q ) − c ′ ≤ W super − cell ( M, s ) ≤ c ′′ ( | M | p + | s | q + 1) for M ∈ R d × N and s ∈ R d × m . It is now natural to have a Γ -convergence result with respectto strong- L p -convergence in the first and weak- L q -convergence in the second component.As we will see in a moment, it turns out that we have to consider a combined boundaryvalue and mean value problem. For this we define B ε ( U, g, s ) to consist of all pairs ( y, s ) ,such that y ∈ B ε ( U, g ) and s ∈ L q (Ω; R d × m ) is constant on every interior cell of U , is outside and has mean value s on the union of interior cells of U .In analogy to Theorem 1.1, we now have Theorem 1.6. The proof of this theorem issimilar to the proof of Theorem 1.1. But there are several things that need to be addressed:First of all, the weak topology on L q is not given by a metric. But, as discussed in[DM93] in detail, this is not a big problem, since our functionals are equicoercive and thedual of L q is separable. In particular, we can describe Γ -convergence by sequences and thecompactness and the Urysohn property are still true. Next, we need an advanced versionof our integral representation result: Theorem 5.2.
Let ≤ p, q < ∞ and let F : W ,p (Ω; R d ) × L q (Ω; R d × m ) × A (Ω) → [0 , ∞ ] satisfy the following conditions:(i) (locality) F ( y, s, U ) = F ( v, t, U ) , if y ( x ) = v ( x ) and s ( x ) = t ( x ) for a.e. x ∈ U ;(ii) (measure property) F ( y, s, · ) is the restriction of a Borel measure to A (Ω) ;(iii) (growth condition) there exists c > such that F ( y, s, U ) ≤ c ˆ U |∇ y ( x ) | p + | s | q + 1 dx ; iv) (translation invariance in y) F ( y, s, U ) = F ( y + a, s, U ) for every a ∈ R d ;(v) (lower semicontinuity) F ( · , · , U ) is sequentially lower semicontinuous with respect toweak convergence in W ,p (Ω; R d ) in the first and weak convergence in L q (Ω; R d × m ) in the second component;(vi) (translation invariance in x) With y M ( x ) = M x and s ( x ) = s we have F ( y M , s, B r ( x )) = F ( y M , s, B r ( x ′ )) for every M ∈ R d × d , s ∈ R d × m , x, x ′ ∈ Ω and r > such that B r ( x ) , B r ( x ′ ) ⊂ Ω .Then there exists a continuous f : R d × d × R d × m → [0 , ∞ ) such that ≤ f ( M, s ) ≤ C (1 + | M | p + | s | q ) for every M ∈ R d × d , s ∈ R d × m and F ( y, s, U ) = ˆ U f ( ∇ y ( x ) , s ( x )) dx for every y ∈ W ,p (Ω; R d ) , s ∈ L q (Ω; R d × m ) and U ∈ A (Ω) . The proof in [BD98] for the pure Sobolev version of this theorem readily applies to thismore general statement. (Note that continuity of f then follows from seperate convexity.)Most of the lemmata then translate naturally. We just want to comment on some detailsin Lemma 3.6. The recovery sequences now contain additionally some t n ⇀ s , r n ⇀ s corresponding to U , V respectively. We define q n,j = χ U j (¯ x ) t n ( x ) + (1 − χ U j (¯ x )) r n ( x ) , and then choose j ( n ) as before to define s n = q n,j ( n ) . The only part that is not immediatelyclear now, is the convergence s n ⇀ s . To prove this, let ϕ ∈ L q ′ (Ω; R d × m ) . We now split ϕ into several parts we can control ϕ = ψ n + ϕχ U ′ + ϕχ Ω \ U N + N − X j =0 ϕχ ( U j +1 \ U j ) εn . Here the ψ n contain all the remaining parts. We see that ψ n → strongly in L q ′ as longas | ∂U j | = 0 for every j , so this is true up to changing the sets U j a little bit. But then wealso have ϕχ ( U j +1 \ U j ) εn → ϕχ U j +1 \ U j strongly in L q ′ . The advantage is now that on each set ( U j +1 \ U j ) ε n we have either s n = t n or s n = r n , possibly changing with n . But in both cases we have weak convergence to s ,hence ˆ Ω s n ( x ) ϕ ( x ) χ ( U j +1 \ U j ) εn ( x ) dx → ˆ Ω s ( x ) ϕ ( x ) χ U j +1 \ U j ( x ) dx. And, putting it all together, we get ˆ Ω s n ( x ) ϕ ( x ) dx → ˆ Ω s ( x ) ϕ ( x ) dx. y we have a fixed mean value for s , i.e., we consider B ε ( U, g, s ) instead of B ε ( U, g ) in the discrete setting and add the constraint U s ( x ) dx = s in the continuum setting. To get the lim inf -inequality just notice that for s k ⇀ s with ( y k , s k ) ∈ B ε ( U, g, s ) we have U s ( x ) dx = lim k →∞ U s k ( x ) dx = lim k →∞ | U ε k || U | U εk s k ( x ) dx = lim k →∞ | U ε k || U | s = s . The lim sup -inequality is a little more subtle. We have a function s ∈ L q (Ω; R d × m ) with U s ( x ) dx = s and a recovery sequence without this mean value s k ⇀ s . Let us write U εk s k ( x ) dx + ξ k = s , so that ξ k → . We now adjust the s k adequately. Define t k ( x ) = s k ( x ) + ξ k | U ε k || V k | χ V k . If V k is a union of cells with some distance to the boundary of U , then, for k large enough,the t k are admissible functions and do not interact with the adjustments on y . We have tomake sure, that t k ⇀ s and lim sup k →∞ F ε k ( u k , t k , U ) ≤ lim sup k →∞ F ε k ( u k , s k , U ) . The weak convergence is true, if | V k | → and | V k | ≥ cξ k for some c > . For the secondestimate, we have to choose the V k a little more carefully to avoid concentration of theenergy. Choose sequences η k → , L k → ∞ such that η k ≥ cξ k , η k ε dk → ∞ and L k η k → .Then take L k ∈ N disjoints sets W k,l ⊂ U , that are unions of cells, such that | W k,l | isindependent of l and is roughly equal to η k , which means cη k ≤ | W k,l | ≤ Cη k , with C, c > independent of k and l . This is possible as η k ε dk → ∞ and L k η k → . Then,33e can choose l ( k ) and set V k = W k,l ( k ) , such that ˆ V k W cell ( ¯ ∇ u k ( x ) , s k ( x )) + W cell (cid:0) ¯ ∇ u k ( x ) , s k ( x ) + ξ k | U ε k || V k | (cid:1) dx ≤ L k L k X l =1 ˆ W k,l W cell (cid:0) ¯ ∇ u k ( x ) , s k ( x )) + W cell ( ¯ ∇ u k ( x ) , s k ( x ) + ξ k | U ε k || W k,l | (cid:1) dx ≤ L k C, due to the growth condition. So the error goes to zero with L k → ∞ . The rest of the prooftranslates naturally. The most important observation is the equality f ( M, s ) = 1 | U | min ( ˆ U f ( ∇ y ( x ) , s ( x )) dx : y − y M ∈ W ,p ( U ; R d ) ,s ∈ L q ( U ; R d × m ) , U s ( x ) dx = s ) = 1 | U | min ( F ( y, s, U ) : y − y M ∈ W ,p ( U ; R d ) ,s ∈ L q ( U ; R d × m ) , U s ( x ) dx = s ) , which is of course a consequence of the lower semicontinuity properties. Proof of Theorem 1.7.
Fix y ∈ W ,p (Ω; R d ) and without loss of generality fix a version of y that is finite everywhere.Due to the growth condition and the continuity, we know that the infimum in inf s ∈ R d × m W cont ( M, s ) is actually a minimum for arbitrary M and that the function M min s ∈ R d × m W cont ( M, s ) is continuous. Obviously, we always have the inequality ˆ Ω W cont ( ∇ y ( x ) , s ( x )) dx ≥ ˆ Ω min s ∈ R d × m W cont ( ∇ y ( x ) , s ) dx. We now want to show, that there always exists an L q -function s where this is an equality.The idea is of course to choose s ( x ) as a minimizer of s W cont ( ∇ y ( x ) , s ) . The key pointis to ensure measurability. We will do this by using the theory of measurable multifunctionsas developed, e.g., in [FL07]. Define Θ( M ) = { s ∈ R d × m : W cont ( M, s ) = min t ∈ R d × m W cont ( M, t ) } and set Γ( x ) = Θ( ∇ y ( x )) . Due to the continuity and the growth of W cont , the set Γ( x ) is closed and non-empty for every x ∈ Ω , hence Γ : Ω → P ( R d × m ) is a closed-valuedmultifunction. 34ext, we want to show that Γ is measurable, in the sense that Γ − ( C ) = { x ∈ Ω : Γ( x ) ∩ C = ∅} is Lebesgue-measurable for every closed set C ⊂ R d × m . To this end, we will first showthat Θ − ( C ) is closed. Let M n ∈ Θ − ( C ) , M n → M and choose s n ∈ Θ( M n ) ∩ C . Usingthe growth of W cont and since the M n are bounded, the s n are also bounded. So for somesubsequence we have s n k → s and s ∈ C . Furthermore, W cont ( M, s ) = lim k →∞ W cont ( M n k , s n k )= lim k →∞ min t ∈ R d × m W cont ( M n k , t )= min t ∈ R d × m W cont ( M, t ) . This proves s ∈ Θ( M ) ∩ C , so Θ − ( C ) is closed and Γ − ( C ) = ( ∇ y ) − (Θ − ( C )) is Lebesgue-measurable. Now, we can apply [FL07, Thm. 6.5], to get a measurable s : Ω → R d × m ,with W cont ( ∇ y ( x ) , s ( x )) = min t ∈ R d × m W cont ( ∇ y ( x ) , t ) and s ∈ L q (Ω; R d × m ) , since ˆ Ω | s ( x ) | q dx ≤ C ˆ Ω W cont ( ∇ y ( x ) , s ( x )) + 1 dx = C ˆ Ω min s ∈ R d × m W cont ( ∇ y ( x ) , s ) + 1 dx ≤ C ˆ Ω min s ∈ R d × m |∇ y ( x ) | p + | s | q + 1 dx ≤ C ˆ Ω |∇ y ( x ) | p + 1 dx. It remains to justify the Γ -convergence result for F s − min ε . Suppose y k → y ∈ W ,p (Ω; R d ) strongly in L p (Ω; R d ) . Choose s k ∈ L q (Ω; R d × m ) with F s − min ε k ( y k , Ω) ≤ F ε k ( y k , s k , Ω)+ k − .Without loss of generality assuming that F s − min ε ( y k , Ω) is bounded, by passing to a subse-quence (not relabeled) we may assume that s k ⇀ s in L q . But then Theorem 1.6 showsthat lim inf k →∞ F s − min ε k ( y k , Ω) = lim inf k →∞ F ε k ( y k , s k , Ω) ≥ F ( y, s , Ω) ≥ F s − min ( y, Ω) by the first part of the proof. On the other hand, if y ∈ W ,p (Ω; R d ) is given, choose s ∈ L q (Ω; R d × m ) according to the first part of the proof such that F s − min ( y, Ω) = F ( y, s, Ω) .Then if ( y k , s k ) is a recovery sequence for ( y, s ) from Theorem 1.6, we obtain lim sup k →∞ F s − min ε k ( y k , Ω) ≤ lim sup k →∞ F ε k ( y k , s k , Ω) = F ( y, s, Ω) = F s − min ( y, Ω) . eferences [AC04] Roberto Alicandro and Marco Cicalese. A general integral representation resultfor continuum limits of discrete energies with superlinear growth. SIAM J.Math. Anal. , 36(1):1–37, 2004.[ACG11] Roberto Alicandro, Marco Cicalese, and Antoine Gloria. Integral representa-tion results for energies defined on stochastic lattices and application to non-linear elasticity.
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