An integral-representation result for continuum limits of discrete energies with multi-body interactions
AAn integral-representation result for continuum limitsof discrete energies with multi-body interactions
Andrea BraidesDipartimento di Matematica, Universit`a di Roma Tor Vergatavia della ricerca scientifica 1, 00133 Roma, Italy e-mail: [email protected]
Leonard KreutzGran Sasso Science Institute, Viale F. Crispi 7, 67100 L’Aquila, Italy e-mail: [email protected]
Abstract
We prove a compactness and integral-representation theorem for sequences offamilies of lattice energies describing atomistic interactions defined on lattices withvanishing lattice spacing. The densities of these energies may depend on interactionsbetween all points of the corresponding lattice contained in a reference set. We giveconditions that ensure that the limit is an integral defined on a Sobolev space. A ho-mogenization theorem is also proved. The result is applied to multibody interactionscorresponding to discrete Jacobian determinants and to linearizations of Lennard-Jones energies with mixtures of convex and concave quadratic pair-potentials.
Keywords: lattice energies, discrete-to-continuum, multibody interactions, homoge-nization, Lennard-Jones energies
This paper focuses on the passage from lattice theories to continuum ones in the frameworkof variational problems, such as for atomistic systems in Computational Materials Science(see e.g. [8]). For notational convenience we will state our results for energies defined onfunctions u parameterized on a portion of Z N (with values in R n ), but our assumptionsmay be immediately extended to more general lattices. For central interactions suchenergies may be written as E ( u ) = (cid:88) i,j ψ ij ( u i − u j ) , (1)where i, j are points in the domain under consideration. We are interested in the behaviourof such an energy when the dimensions of the domain are much larger than the lattice1 a r X i v : . [ m a t h . A P ] M a r pacing. In the discrete-to-continuum approach this can be done by approximation with acontinuum energy obtained as a limit after a scaling argument. To that end, we introducea small parameter ε (which, for the unscaled energy E is the inverse of the linear dimensionof the domain) and scale the energies as E ε ( u ) = (cid:88) i,j ε N ψ εij (cid:16) u i − u j ε (cid:17) , (2)where now i, j belong to a domain Ω independent of ε , and the domain of u is Ω ∩ ε Z N ;accordingly, we set ψ εij = ψ i/ε j/ε . Both scalings, ε N of the energy, and u i /ε of the function,are important in this process and highlight that in this case we are regarding the energy asa volume integral ( ε N being the volume element of a lattice cell) depending on a gradient(( u i − u j ) /ε being interpreted as a scaled difference quotient or discrete gradient). Otherscalings are possible and give rise to different types of energies, depending on the formof ψ εij , highlighting the multiscale nature of the problem. In the present context we focuson this particular “bulk” scaling (for an account of other scaling limits see [11, 12]).The continuum approximation of E ε is obtained by taking a limit as ε →
0. Thishas been done in different ways, using a pointwise limit in [7] (where lattice functionsare considered as restrictions of a smooth function to Z N ) or a Γ-limit in [2] (in thiscase lattice functions are extended as piecewise-constant functions and embedded in somecommon Lebesgue space) to obtain an energy of the form F ( u ) = (cid:90) Ω f ( x, ∇ u )d x (3)with domain a Sobolev space. We focus on the result of [2], which relies on the localizationmethods of Γ-convergence (see [10] Chapter 12) envisaged by De Giorgi to deduce theintegral form of the Γ-limit from its behaviour both as a function of u and Ω. Conditionsthat allow to apply those methods are(i) ( coerciveness ) growth conditions from below that allow to deduce that the limit isdefined on some Sobolev space; e.g. that ψ εij ( w ) ≥ c ( | w | p −
1) for nearest-neighbours and ψ εij ≥ i, j ;(ii) ( finiteness ) growth conditions from above that allow to deduce that the limit isfinite on the same Sobolev space; e.g. that ψ εij ( w ) ≤ c εij ( | w | p + 1) for all ij , with somesummability conditions on c εij uniformly in ε ;(iii) ( vanishing non-locality ) conditions that allow to deduce that the Γ-limit is a mea-sure in its dependence on Ω. This is again obtained from some uniform decay conditionson the coefficients c εij .Hypotheses (i)–(iii) are sharp, in the sense that failure of any of these conditions mayresult in a Γ-limit that cannot be represented as in (3). The result in [2] has been successfulin many applications, among which the computation of optimal bounds for conductingnetworks [16], the derivation of nonlinear elastic energies from atomistic systems [2, 24], oftheir linear counterpart [19], and of Q -tensor theories from spin interactions [14], numericalhomogenization [23], the analysis of the pile-up of dislocations [22], and others. Moreover,it has been extended to cover stochastic lattices [4] and dimension-reduction problems [1].However, its range of applicability is restricted to pairwise interactions, which implies2onstraints on the possible energy densities. The main motivation of the present work isto overcome some of those limitations. More precisely, we focus on two issues: • the extension to the result to many-body interactions . In principle, a point in thelattice may interact with all other points in the domain Ω. As a particular case, we maythink of k -body interactions corresponding to the minors of the lattice transformation(which is affine at the lattice level), such as the discrete determinant in two dimensions,which can be viewed as a three-point interaction. Some works in this direction are alreadypresent in the literature for particular cases [20, 25, 26]; • the use of averaged growth conditions on the energy densities . Some lattice energiesare obtained as an approximation of non-convex long-range interactions. As such, evenwhen considering pair interactions, they may fail to satisfy coerciveness conditions forsome ψ ij . As an example we can think of the linearization of Lennard-Jones interactions,which gives concave quadratic energies for distant i and j . The coerciveness of the energycan nevertheless be recovered using the fast decay of the potential so that short-rangeconvex interactions dominate long-range concave ones. In general, coerciveness can beobtained by substituting a growth conditions on each of the interactions with an averagedgrowth condition.In order to achieve the greatest generality, we assume that energy densities may indeeddepend on all points in Ω ∩ ε Z N . An energy density φ εi will describe the interaction of apoint i ∈ Ω ∩ ε Z N with all other points in the domain. This standpoint, already used in[13] for surface energies in a simpler setting (see also [18] in a one-dimensional setting),brings some notational complications (except for the case Ω = R N ) since it is convenientto regard each such function as defined on a different set (Ω − i ) ∩ ε Z N . This complicationis anyhow present each time that we consider more-than-two-body interactions. Theenergies are then defined as F ε ( u ) = (cid:88) i ∈ Ω ∩ ε Z N ε N φ εi ( { u j + i } j ∈ (Ω − i ) ∩ ε Z N ) . (4)An important remark to make is that there are many ways to define energy densitiesgiving the same F ε . Note for example that for central interactions as above φ εi may besimply given by φ εi ( { z j } ) = (cid:88) j ∈ (Ω − i ) ∩ ε Z N ψ εij (cid:16) z j − z ε (cid:17) = (cid:88) j ∈ (Ω − i ) ∩ ε Z N ψ i/ε j/ε (cid:16) z j − z ε (cid:17) , (5)but the interactions may also be regrouped differently and in principle φ εi may includesome ψ εkj with k (cid:54) = i . This is important in order to allow that some ψ εij be unboundedfrom below, up to satisfying a lower bound when considered together with the otherinteractions.The set of hypotheses we are going to list for φ εij will allow to treat a larger class ofenergies than those of the form (2), but they must be stated with some care. The precisestatements are given in Section 3. Here we give a simplified description as follows:(o) ( translational invariance in the codomain ) φ εi ( { z j + w } ) = φ εi ( { z j } ) for all i , { z j } and vector w . This condition is automatically satisfied for interactions depending ondifferences z i − z j ; 3i) ( coerciveness ) the energy must be estimated from below by a nearest-neighbourpair energy and φ εi ≥ i . This condition is less restrictive than the correspondingone for pair interactions since it refers to an already averaged energy density;(ii) ( Cauchy-Born hypothesis ) we assume a polynomial upper bound for F ε ( u ) onlywhen u is linear. For energy densities as in (5) this in general rewritten in terms of ψ ij asΨ( M ) := (cid:88) j ψ i i + j ( M j ) ≤ C (1 + | M | p ) , (6)for all i ∈ Z N , and all n × N matrices M . This condition is in principle weaker than thefiniteness property (ii) for pair interactions. Examining this condition separately goes inthe direction of analyzing first pointwise convergence (as in [7]) and then Γ-convergence;(iii) ( vanishing non-locality ) we assume that if u = v on a square of centre i andside-length δ then φ εi ( { u j + i } j ∈ (Ω − i ) ∩ ε Z N ) ≤ φ εi ( { v j + i } j ∈ (Ω − i ) ∩ ε Z N ) + r ( ε, δ, (cid:107)∇ u (cid:107) p )( u is identified with a piecewise-affine interpolation), where the rest r is negligible as ε → (cid:107)∇ u (cid:107) p bounded. Note that this condition is automatically satisfied with r = 0 if therange of the interactions is finite, and can be deduced from the corresponding condition(iii) for central interactions;(iv) ( controlled non-convexity ) a final condition must be added to ensure that thelimit be a measure as a function of Ω. For central interactions, this condition is hiddenin the previous (i) and (ii), which imply a convex growth condition on Ψ; more preciselya polynomial growth of the form c ( | M | p − ≤ Ψ( M ) ≤ C (1 + | M | p ) . This double inequality allows to use classical convex-combination arguments with cut-offfunctions even though Ψ may not be convex. In our case this compatibility with convexarguments must be required separately, and is formalized in condition (H5) in Section 3.1.Under the conditions above we again deduce that Γ-limits of energies F ε are integralfunctionals F as in (3) defined on a Sobolev space. The integrand f can be described bya derivation formula, which is allowed by the study of suitably defined boundary-valueproblems. This derivation formula can also be used to prove a periodic-homogenizationresult. In the generality of energies possibly depending on the interaction of all points inΩ some care must be used to define periodicity for the energy densities. In the case offinite-range interactions we require that in the interior of Ω we have φ εi = φ ε/i , where φ k is periodic in k . For infinite-range interactions the definition is given by approximationwith periodic energy densities with finite-range interactions.The paper is organized as follows. After some notation, in Section 3 we rigor-ously state the hypotheses outlined above and prove the main compactness and integral-representation theorem. Section 4 is devoted to formalizing and proving the convergenceof Dirichlet boundary-value problems, which is used in the following Section 5 to stateand derive a homogenization formula. Finally, Section 6 is devoted to examples. Moreprecisely, we show how our hypotheses are satisfied by functions depending on discretedeterminants and by a linearization of Lennard-Jones energies mixing convex and concavequadratic pair energy densities. Finally, in the same section we recover the result in [2]as a particular case of our main theorem. 4 Notation and preliminaries
We denote by Ω an open and bounded subset of R N with Lipschitz boundary. We set Q tobe the unit cube with sides orthogonal to the canonical orthonormal basis { e , . . . , e N } , Q = { x ∈ R N : |(cid:104) x, e i (cid:105)| ≤ , for all i = 1 , . . . , N } and for δ > Q δ = δQ .Moreover, for x ∈ R N we set Q ( x ) = Q + x and Q δ ( x ) = Q δ + x . We set A (Ω) = { A ⊂ Ω : A open } , A reg (Ω) = { A ∈ A (Ω) : ∂A Lipschitz } , and for δ > A δ = { x ∈ Ω :dist ∞ ( x, A ) < δ } and A δ = { x ∈ A : dist ∞ ( x, A c ) > δ } . For B ⊂ R N we write | B | for the N -dimensional Lebesgue measure of B . For a vector x ∈ R N we set (cid:98) x (cid:99) = ( (cid:98) x (cid:99) , . . . , (cid:98) x N (cid:99) ) . We define for u : R N → R n , ξ ∈ Z N , x ∈ R N and ε > D ξε u ( x ) := u ( x + εξ ) − u ( x ) ε | ξ | the discrete difference quotient of u at x in direction ξ .For a function u we set C ( u ) to be a constant depending on u , the dimension and itsdomain of definition and which may vary from line to line. Slicing.
We recall the standard notation for slicing arguments (see [6]). Let ξ ∈ S N − ,and let Π ξ = { y ∈ R N : (cid:104) y, ξ (cid:105) = 0 } be the linear hyperplane orthogonal to ξ . If y ∈ Π ξ and E ⊂ R N we define E ξ = { y ∈ Π ξ such that ∃ t ∈ R : y + tξ ∈ E } and E ξy = { t ∈ R : y + tξ ∈ E } . Moreover, if u : E → R n we set u ξ,y : E ξy → R n to u ξ,y ( t ) = u ( y + tξ ).Γ -convergence. A sequence of functionals F n : L p (Ω; R n ) → [0 , + ∞ ] is said to Γ-converge to a functional F : L p (Ω; R n ) → [0 , + ∞ ] at u ∈ L p (Ω; R n ) as n → ∞ and wewrite F ( u ) = Γ- lim n →∞ F n ( u ) if the following two conditions are satisfied:(i) For every u n converging to u in L p (Ω; R n ) we have lim inf n →∞ F n ( u n ) ≥ F ( u ) . (ii) There exists a sequence { u n } n ⊂ L p (Ω; R n ) converging to u in L p (Ω; R n ) such thatlim sup n →∞ F n ( u n ) ≤ F ( u ) . We say that F n Γ-converges to F if F ( u ) = Γ- lim n →∞ F n ( u ) for all u ∈ L p (Ω; R n ).If { F ε } ε> is a family of functionals indexed by a continuous parameter ε > F ε Γ-converges to F as ε → + if for all ε n → F ε n Γ-converges to F .We define the Γ-lim inf F (cid:48) : L p (Ω; R n ) → [0 , ∞ ] and the Γ-lim sup F (cid:48)(cid:48) : L p (Ω; R n ) → [0 , ∞ ]respectively by F (cid:48) ( u ) = Γ- lim inf ε → F ε ( u ) = inf (cid:110) lim inf ε → F ε ( u ε ) : u ε → u (cid:111) ,F (cid:48)(cid:48) ( u ) = Γ- lim sup ε → F ε ( u ) = inf (cid:110) lim sup ε → F ε ( u ε ) : u ε → u (cid:111) . Note that the functionals F (cid:48) , F (cid:48)(cid:48) are lower semicontinuous and F ε Γ-converges to F as ε → + if and only if F = F (cid:48) = F (cid:48)(cid:48) . Lattice functions.
For A ∈ A (Ω) , we set Z ε ( A ) = ε Z N ∩ A We set A ε (Ω , R n ) := { u : Z ε (Ω) → R n } . 5 efinition 2.1. (Convergence of discrete functions) Functions u ∈ A ε (Ω; R n ) can beinterpreted by functions belonging to the space L p (Ω; R n ) by setting (with slight abuse ofnotation) u ( z ) = 0 for all z ∈ Z ε (Ω c ) and u ( x ) = u ( z εx )where z εx is the closest point of Z ε ( R N ) to x (which is uniquely defined up to a set ofmeasure 0). We then say that u ε → u in L p (Ω; R n ) if the interpolations of u ε converge to u in L p (Ω; R n ). Integral representation.
We will use the following integral representation result (see[15]).
Theorem 2.2.
Let F : W ,p (Ω; R n ) × A (Ω) → [0 , + ∞ ] satisfy the following propertiesi) (measure property) For every u ∈ W ,p (Ω; R n ) we have that F ( u, · ) is the restrictionof a Radon measure to the open sets.ii) (lower semicontinuity) For every A ∈ A (Ω) we have that F ( · , A ) is weakly- W , (Ω; R n ) lower semicontinuous.iii) (bounds) For every ( u, A ) ∈ W ,p (Ω; R n ) × A (Ω) it holds that ≤ F ( u, A ) ≤ C (cid:18)(cid:90) A |∇ u | p d x + | A | (cid:19) iv) (translational invariance) For every ( u, A ) ∈ W ,p (Ω; R n ) × A (Ω) and for every c ∈ R n it holds F ( u, A ) = F ( u + c, A ) .v) (locality) For every A ∈ A (Ω) and every u, v ∈ W ,p (Ω; R n ) such that u = v a.e. in A , we have that F ( u, A ) = F ( v, A ) .Then there exists a Carath`eodory function f : Ω × R n × N → [0 , + ∞ ] such that F ( u, A ) = (cid:90) A f ( x, ∇ u )d x for every ( u, A ) ∈ W ,p (Ω; R n ) × A (Ω) .vi) (translational invariance in x ) if for every M ∈ R n × N , z, y ∈ Ω and for every ρ > such that Q ρ ( z ) ∪ Q ρ ( y ) ⊂ Ω we have that F ( M x, Q ρ ( y )) = F ( M x, Q ρ ( z )) , then f does not depend on x . The main result
For all i ∈ Ω, we denote by Ω i = Ω − i the translation of the set Ω with i at the origin, andwe consider a function φ εi : ( R n ) Z ε (Ω i ) → [0 , + ∞ ). Let F ε : A ε (Ω , R n ) × A (Ω) → [0 , + ∞ )be defined by F ε ( u, A ) = (cid:88) i ∈ Z ε ( A ) ε N φ εi ( { u j + i } j ∈ Z ε (Ω i ) ) . (7)In this section we give hypothesis on the energy densities φ εi in order to ensure that theΓ-limit of the energies defined in (7) be finite only on W ,p ( A, R n ) ∩ L p (Ω; R n ) and thereexists a Carath´eodory function f : Ω × R n × N → [0 , ∞ ) such that F ( u, A ) = (cid:90) A f ( x, ∇ u ( x ))d x (8)for all ( u, A ) ∈ W ,p ( A, R n ) ∩ L p (Ω; R n ) × A (Ω). A corresponding problem on the con-tinuum is one of the first formalized in the theory of Γ-convergence, when F ε themselvesare integral energies. In that approach integral functionals are interpreted as dependingon a pair ( u, A ) with u a Sobolev function and A a subset of Ω, when the integration isperformed on A only. The compactness property of Γ-convergence then ensures that aΓ-converging subsequence exits on a dense family of open sets by a simple diagonal argu-ment. Showing that the dependence of the limit on the set variable is that of a regularmeasure, the convergence is extended to a larger family of sets, and an integral repre-sentation result can be applied. The type of conditions singled out in that case can beadapted to the discrete setting, taking into account that discrete energies are “nonlocal”in nature since they depend on the interactions of points at a finite distance. The localityof the limit energy F must then be assured by a requirement of “vanishing nonlocality”as ε → A first requirement is that F ε be invariant under addition of constants to u ; namely(H1) ( translational invariance ) for all w ∈ R n we have φ εi ( { z j + w } j ∈ Z ε (Ω i ) ) = φ εi ( { z j } j ∈ Z ε (Ω i ) ) (9)for all ε > i ∈ Z ε (Ω) and z : Z ε (Ω) → R n .A second requirement is that F ε ( u ε ) be finite if (cid:98) u ε are a discretization of a W ,p function. In particular this should hold for affine functions.(H2) ( upper bound for the Cauchy-Born approximation ) there exists C >
0, such thatfor every M ∈ R n × N and M x ( i ) = M i we have φ εi ( { ( M x ) j } j ∈ Z ε (Ω i ) ) ≤ C ( | M | p + 1) (10)for all ε > i ∈ Z ε (Ω). 7e then also require that the limit domain be exactly W ,p functions, with p >
1. Tothat end a coerciveness condition should be imposed.(H3) ( equi-coerciveness ) there exists c > c (cid:16) N (cid:88) n =1 | D e n ε z (0) | p − (cid:17) ≤ φ εi ( { z j } j ∈ Z ε (Ω i ) ) (11)for all ε and i such that i + e n ∈ Z ε (Ω) for all n ∈ { , · · · , N } .Next, we have to impose that the approximating continuum energy be local. Indeed,in principle discrete interactions are non-local, in that they take into account nodes ofthe lattice at a finite distance. This condition ensures that we can always find recoverysequences for a set A ∈ A (Ω) that will not oscillate too much a finite distance awayfrom A . We expect the limit to depend on ∇ u if only the interactions for small distancesare relevant, or, in other words, if the decay of interactions is fast enough. This can beformulated otherwise: we may require that the overall effect of long-range interactions ata point decay sufficiently fast as follows.(H4) ( decaying non-locality ) There exist { C j,ξε,δ } ε> ,δ> ,j ∈ ε Z N ,ξ ∈ Z N , C j,ξε,δ ≥ ε → (cid:88) j ∈ Z ε ( R N ) ,ξ ∈ Z N C j,ξε,δ = 0 ∀ δ > δ > z, w ∈ A ε (Ω , R n ) satisfying z ( j ) = w ( j ) for all j ∈ Z ε ( Q δ ( i )) wehave φ εi ( { z j } j ∈ Z ε (Ω i ) ) ≤ φ εi ( { w j } j ∈ Z ε (Ω) ) + (cid:88) j ∈ Z ε (Ω i ) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω i ) C j,ξε,δ (cid:0) | D ξε z ( j ) | p + 1 (cid:1) . The final condition is the most technical and derives from our requirement that thelimit can be expressed in terms of an integral. This is the most restrictive in the vectorialcase d > ψ : Z ε (Ω) → R iscalled a cut-off function if 0 ≤ ψ ≤ controlled non-convexity ) There exist C > { C j,ξε } ε> ,j ∈ ε Z N ,ξ ∈ Z N , C j,ξε ≥ ε → (cid:88) j ∈ Z ε ( R N ) ,ξ ∈ Z N C j,ξε < + ∞ , ∀ δ > ε → (cid:88) max { ε | ξ | , | j |} >δ C j,ξε = 0 (13)such that for all z, w ∈ A ε (Ω , R n ) and ψ cut-off functions we have φ εi ( { ψ j z j + (1 − ψ j ) w j } j ∈ Z ε (Ω i ) ) ≤ C (cid:0) φ εi ( { z j } j ∈ Z ε (Ω i ) ) + φ εi ( { w j } j ∈ Z ε (Ω i ) ) (cid:1) + R εi ( z, w, ψ )where R εi ( z, w, ψ ) = (cid:88) j ∈ Z ε (Ω i ) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω i ) C j,ξε (cid:16) ( sup k ∈ Z ε (Ω i ) n ∈{ ,...,N } | D e n ε ψ ( k ) | p + 1) | z ( j + εξ ) − w ( j + εξ ) | p (cid:17) + (cid:88) j ∈ Z ε (Ω i ) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω i ) C j,ξε (cid:16) | D ξε z ( j ) | p + | D ξε w ( j ) | p + 1 (cid:17) . emark 3.1. (observations on the assumptions) If condition (H1) fails we expect thelimit not to be translational invariant anymore and if a integral representation exists it isexpected to be of the form F ( u, A ) = (cid:90) A f ( x, u, ∇ u )d x. However, integral-representation theorems for non-translation-invariant functionals in gen-eral require restrictive hypotheses that should be added to (H2)–(H5).If condition (H2) fails the Γ-limit may not be finite on W ,p (Ω; R n ). Condition (H3)allows to estimate nearest-neighbour interactions centered in i in terms of φ εi . Note thatthis estimate may still be true even if there are no interactions of the type | D e n ε u | p takeninto account by φ εi . Indeed if d = 1 we may take c , c > φ εi ( { z j } j ∈ Z ε (Ω i ) ) = c (cid:12)(cid:12)(cid:12)(cid:12) z − z ε (cid:12)(cid:12)(cid:12)(cid:12) + c (cid:12)(cid:12)(cid:12)(cid:12) z − z ε (cid:12)(cid:12)(cid:12)(cid:12) . If we assume a finite range R of interactions and assume that the potential φ εi is wellbehaved in some sense condition (H4) is always satisfied and in the definition of R εi thesummation is only taken over Q R ( i ). If condition (H4) fails the Γ-limit may be non-local.Indeed there are examples where functionals of the form F ( u ) = (cid:90) Ω |∇ u | d x + (cid:90) Ω × Ω k ( x, y ) | u ( x ) − u ( y ) | d x can be obtained as the Γ-limit of energies of the form F ε ( u ) = (cid:88) i ∈ Z ε (Ω) (cid:88) ξ ∈ Z N i + εξ ∈ Z ε (Ω) ε N c εi,ξ | D ξε u ( i ) | . Note that (H1) is still satisfied. Condition (H5) mimics the so-called fundamental estimate in the continuum and ensures that the limit F ( u, · ) be subadditive as a set function.Note that this condition is satisfied for subadditive potentials with appropriate growthconditions. In particular, in Section 6.3 we show how the hypotheses above can be deducedfrom those in [2] in the case of pair potentials. The goal of this section is to establish the proof of Theorem 3.2.
Theorem 3.2. ( Integral Representation)
Let F ε : L p (Ω; R n ) → [0 , + ∞ ] be definedby (7) , where φ εi : ( R n ) Z ε (Ω) → [0 , + ∞ ) satisfy (H1)–(H5) . Then for every sequence ( ε j ) of positive numbers converging to , there exists a subsequence ε j k and a Carath´eodoryfunction f : Ω × R n × N → [0 , + ∞ ) , quasiconvex in the second variable satisfying c ( | ξ | p − ≤ f ( x, ξ ) ≤ C ( | ξ | p + 1)9 ith < c < C , such that F ε jk ( · ) Γ -converges with respect to the L p (Ω; R n ) -topology tothe functional F : L p (Ω; R n ) → [0 , + ∞ ] defined by F ( u ) = (cid:90) Ω f ( x, ∇ u )d x if u ∈ W ,p (Ω; R n )+ ∞ otherwise.Moreover, for any u ∈ W ,p (Ω; R n ) and any A ∈ A (Ω) we have Γ - lim k → + ∞ F ε jk ( u, A ) = (cid:90) A f ( x, ∇ u )d x. We will derive the proof of Theorem 3.2 as a consequence of some propositions andlemmas, which are fundamental in order to show that our limit functionals satisfies all theassumption of Theorem 2.2. In the next two proposition we show with the use of (H1)–(H5) that assumptions (ii) and (iii) of Theorem 2.2 are satisfied. Note that property(14) below allows to deduce weak lower-semicontinuity in W ,p even though we prove theΓ-convergence of the discrete energies with respect to the strong L p (Ω; R n )-topology, sothat assumption (ii) is satisfied.Note that the proof of Proposition 3.3 is the same as the proof of Proposition 3.4 in[2]. We repeat it here only for completeness and the reader’s convenience. Proposition 3.3.
Let φ εi : ( R n ) Z ε (Ω i ) → [0 , + ∞ ) satisfy (H3) . If u ∈ L p (Ω , R n ) is suchthat F (cid:48) ( u, A ) < + ∞ , then u ∈ W ,p ( A, R n ) and F (cid:48) ( u, A ) ≥ c (cid:16) ||∇ u || pL p ( A ; R n × N ) − | A | (cid:17) (14) for some positive constant c independent on u and A .Proof. Let ε n → + and let u n → u in L p (Ω; R n ) be such that lim inf n F ε n ( u n , A ) < + ∞ .By (H3) we get F ε n ( u n , A ) ≥ c (cid:88) i ∈ Z ε ( A ) N (cid:88) k =1 ε N | D e k ε n u n ( i ) | p − cN | A | . (15)For any k ∈ { , · · · , N } , consider the sequence of piecewise-affine functions ( v kn ) definedas follows v kn ( x ) = u n ( i ) + D e k ε n u n ( i )( x k − i k ) x ∈ ( i + [0 , ε n ) N ) ∩ Ω , i ∈ Z ε ( A ) . Note that v kn is a function of bounded variation and we will denote by ∂v kn ∂x k the density of theabsolutely continuos part of D x k v kn with respect to the Lebesgue measure. Moreover, for H N − -a.e. y ∈ ( A ) e k the slices ( v kn ) e k ,y belong to W ,p (( A ) e k y ; R n ). Note that, for any fixed η > v kn → u in L p ( A η ; R n ) for every k ∈ { , · · · , N } . Moreover, since ∂v kn ∂x k ( x ) = D e k ε n u n ( i )for x ∈ i + [0 , ε n ) N , we get F ε n ( u n , A ) ≥ c N (cid:88) k =1 (cid:90) A η (cid:12)(cid:12)(cid:12)(cid:12) ∂v kn ∂x k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p d x − cN | A | .
10e now apply a standard slicing argument. By Fubini’s Theorem and Fatou’s Lemma forany k we getlim inf n (cid:90) A η (cid:12)(cid:12)(cid:12)(cid:12) ∂v kn ∂x k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p d x ≥ (cid:90) ( A η ) ek lim inf n (cid:90) ( A ) eky | ( v kn ) (cid:48) e k ,y ( t ) | p dtd H N − ( y ) . Since, up to passing to a subsequence, we may assume that, for H N − -a.e. y ∈ ( A η ) e k ( v kn ) e k ,y → u e k ,y in L p (( A η ) e k y ; R n ), we deduce that u e k ,y ∈ W ,p (( A η ) e k y ; R n ) for H N − -a.e. y ∈ ( A η ) e k andlim inf n (cid:90) A η (cid:12)(cid:12)(cid:12)(cid:12) ∂v kn ∂x k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p d x ≥ (cid:90) ( A η ) ek (cid:90) ( A ) eky | u (cid:48) e k ,y ( t ) | p dtd H N − ( y ) . Then by (15), we havelim inf n F ε n ( u n , A ) ≥ c N (cid:88) k =1 (cid:90) ( A η ) ek (cid:90) ( A ) eky | u (cid:48) e k ,y ( t ) | p dtd H N − ( y ) − cN | A | . Since, in particular, the previous inequality implies that N (cid:88) k =1 (cid:90) ( A η ) ek (cid:90) ( A ) eky | u (cid:48) e k ,y ( t ) | p dtd H N − ( y ) < + ∞ , thanks to the characterization of W ,p by slicing we obtain that u ∈ W ,p ( A η , R n ) andlim inf n F ε n ( u n , A ) ≥ c N (cid:88) k =1 (cid:90) A η (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂x k ( x ) (cid:12)(cid:12)(cid:12)(cid:12) p d x − cN | A |≥ c (cid:16) ||∇ u || pL p ( A η ; R n × N ) − | A | (cid:17) Letting η → + , we get the conclusion. Proposition 3.4.
Let φ εi : ( R n ) Z ε (Ω i ) → [0 , + ∞ ) satisfy (H2),(H4) and (H5) . We thenhave F (cid:48)(cid:48) ( u, A ) ≤ C (cid:16) ||∇ u || pL p ( A ; R n × N ) + | A | (cid:17) (16) for some positive constant C independent on u and A .Proof. We first show that the inequality holds for u ∈ W ,p (Ω; R n ) piecewise affine andthen we recover the inequality for any u ∈ W ,p (Ω; R n ) by a density argument. Let u ∈ W ,p (Ω; R n ) be piecewise affine, that means u ( x ) = K (cid:88) k =1 χ Ω k ( x )( M k x + b k ) = K (cid:88) k =1 χ Ω k ( x ) u k ( x ) , where Ω k = U k ∩ Ω, with U k disjoint open simplices such that | Ω \ (cid:83) k Ω k | = 0, M k ∈ R n × N , b k ∈ R n , k = 1 , . . . , K . In the following, for such u ∈ W ,p (Ω; R n ) we construct11 δ ∈ W , ∞ (Ω; R n ), which agrees with u on Ω δk for all k ∈ { , . . . , K } , u δ = u on (Ω k ) δ for all k ∈ { , . . . , K } and close to ∂ Ω k we have that u δ = ψ δk u j + (1 − ψ δk ) u δj − for some j ∈ { , . . . , K } and ||∇ u j || ∞ + ||∇ u δj − || ∞ ≤ C ||∇ u || ∞ independent on δ . The way weconstruct u δj , it satisfies the same property close to the boundary so that (H5) or (H4)can be applied repeatedly. In (cid:83) k Ω δk we estimate the interactions separately with (H4).Let d = min ∂ Ω k ∩ Ω j = ∅ dist ∞ ( ∂ Ω k , ∂ Ω j )and let δ < d4 . For k = 1 , . . . , K choose inductively ϕ kj,δ ⊂ C ∞ (Ω) , j = 1 , . . . , k such that0 ≤ ϕ kj,δ ≤ , k (cid:88) j =1 ϕ kj,δ = 1 , supp( ϕ kj,δ ) ⊂ (Ω j ) δ , ϕ kj,δ = (1 − ϕ kk,δ ) ϕ k − j,δ for all j < kϕ kj,δ ( x ) = 1 if x ∈ (Ω j ) δ , ||∇ ϕ kj,δ || ∞ ≤ Cδ and define u kδ ( x ) = ϕ kk,δ ( x ) u k ( x ) + (1 − ϕ kk,δ ( x )) u k − δ ( x ) , u δ ( x ) = u ( x ) . Set u δ ( x ) = u Kδ ( x ). We then have ||∇ u kδ || ∞ ≤ C ||∇ u || ∞ for all k ∈ { , . . . , K } and u δ → u strongly in W ,p (Ω; R n ) and we claim thatlim inf δ → F (cid:48)(cid:48) ( u δ , A ) ≤ C ( ||∇ u || pL p ( A ; R d × N ) + | A | ) . (17)To this end define u εδ ( i ) = u δ ( i ) , i ∈ Z ε (Ω) . We have that u εδ → u strongly in L p (Ω; R n ) and therefore F (cid:48)(cid:48) ( u δ , A ) ≤ lim inf ε → F ε ( u εδ , A ) . (18)We divide the energy into the energy of points which are far away from the boundary ofall the Ω k and to the points which are close to some of the boundaries of Ω k : F ε ( u εδ , A ) = (cid:88) i ∈ Z ε ( A ) ε N φ εi ( { ( u εδ ) j + i } j ∈ Z ε (Ω i ) ) = K (cid:88) k =1 (cid:88) i ∈ Z ε ( A ∩ (Ω k ) δ ) ε N φ εi ( { ( u εδ ) j + i } j ∈ Z ε (Ω i ) )+ (cid:88) i ∈ Z ε ( A \ ( (cid:83) Kk =1 (Ω k ) δ )) ε N φ εi ( { ( u εδ ) j + i } j ∈ Z ε (Ω i ) )= I ε,δ + I ε,δ . Now, note that for M ∈ R n × N , z ∈ R n and ( M x + z )( i ) = M i + z we have | D e n ε ( M x + z )( i ) | = (cid:12)(cid:12)(cid:12)(cid:12) M ( i + εe n ) + z − ( M i + z ) ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ | M | , ∀ n ∈ { , . . . , N } . v ∈ W , ∞ (Ω; R n ) we have that | D ξε v | p ≤ ||∇ v || p ∞ . Using(H4) and (H2), noting that ∇ u ( x ) = ∇ u k ( x ) = M k for x ∈ Ω k and using the fact that u εδ ( j ) = u k ( j ) for all j ∈ Z ε ( Q δ ( i )) , i ∈ Z ε ((Ω k ) δ ) (with slight abuse of notation we write u k for the discrete function as well as for the function defined in the continuum) we canestimate the first term by I ε,δ ≤ K (cid:88) k =1 (cid:88) i ∈ Z ε ( A ∩ (Ω k ) δ ) ε N φ εi ( { ( u k ) j + i } j ∈ Z ε (Ω i ) )+ K (cid:88) k =1 (cid:88) i ∈ Z ε ( A ∩ (Ω k ) δ ) ε N (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε,δ ( | D ξε u εδ ( j ) | p + 1) ≤ K (cid:88) k =1 (cid:88) i ∈ Z ε ( A ∩ Ω k ) ε N C ( | M k | p + 1) + C ( u ) (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N C j − i,ξε,δ ≤ C ( ||∇ u || L p ( A ε ; R n × N ) + | A ε | ) + C ( u ) (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N C j − i,ξε,δ . Taking the lim sup as ε → ε → I ε,δ ≤ C ( ||∇ u || L p ( A ; R n × N ) + | A | ) . (19)Now let i ∈ Z ε ( A \ ( (cid:83) Kk =1 (Ω k ) δ )), that means dist ∞ ( i, ∂ Ω k ) ≤ δ for some k ∈ { , . . . , K } .We prove φ εi ( { ( u εδ ) j + i } j ∈ Z ε (Ω i ) ) ≤ C ( u ) (20)for some constant depending on u . Recall that u δ = u Kδ , take k ∈ { , . . . , K } and assumethat we have proved already that φ εi ( { ( u εδ ) j + i } j ∈ Z ε (Ω i ) ) ≤ C ( u ) (cid:0) φ εi ( { (( u kδ ) ε ) j + i } j ∈ Z ε (Ω i ) ) + 1 (cid:1) ;we then prove that φ εi ( { ( u εδ ) j + i } j ∈ Z ε (Ω i ) ) ≤ C ( u ) (cid:0) φ εi ( { (( u k − δ ) ε ) j + i } j ∈ Z ε (Ω i ) ) + 1 (cid:1) . (21)We either have ϕ kk,δ = 0 in Q δ ( i ). Then by using ||∇ u k − δ || ∞ ≤ C ||∇ u || ∞ , | D ξε u | ≤ C ||∇ u || ∞ and (12) we obtain φ εi ( { (( u kδ ) ε ) j + i } j ∈ Z ε (Ω i ) ) ≤ φ εi ( { (( u k − δ ) ε ) j + i } j ∈ Z ε (Ω i ) ) + (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε,δ (cid:16) | D ξε ( u kδ ) ε ( j ) | p + 1 (cid:17) ≤ C ( u )( φ εi ( { (( u k − δ ) ε ) j + i } j ∈ Z ε (Ω i ) ) + 1)13nd we obtain (21). Now in the case that ϕ kk,δ ( x ) > x ∈ Q δ ( j ) we use (H5)with ϕ kk,δ as a cutoff function, u k , u k − δ as z, w and the assumptions on ϕ kk,δ we obtain φ εi ( { (( u kδ ) ε ) j + i } j ∈ Z ε (Ω i ) ) ≤ C (cid:0) φ εi ( { ( u εk ) j + i } j ∈ Z ε (Ω i ) ) + φ εi ( { (( u k − δ ) ε ) j + i } j ∈ Z ε (Ω i ) ) + 1 (cid:1) + R εi ( u k , u k − δ , ϕ kk,δ )with R εi ( u k , u k − δ , ϕ kk,δ ) = (cid:18) δ p + 1 (cid:19) (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε | ( u k )( j + εξ ) − ( u k − δ )( j + εξ ) | p + (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε (cid:0) | D ξε ( u k − δ ) ε ( j ) | p + | D ξε u εk ( j ) | p (cid:1) . First, note that by (H2) and by ||∇ u k || ∞ ≤ C ||∇ u || ∞ we have φ εi ( { ( u εk ) j + i } j ∈ Z ε (Ω i ) ) ≤ C ( | M k | p + 1) ≤ C ( u )and (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε (cid:0) | D ξε ( u k − δ ) ε ( j ) | p + | D ξε u εk ( j ) | p (cid:1) ≤ C ( u ) (22)since ||∇ u k || ∞ , ||∇ u k − δ || ∞ ≤ C ||∇ u || ∞ . Now since ϕ kk,δ ( x ) > x ∈ Q δ ( i ) we havethat dist ∞ ( i, ∂ Ω k ) < δ , u k ( x ) = u k − δ ( x ) on ∂ Ω k and ||∇ u k || ∞ , ||∇ u k − δ || ∞ ≤ C ||∇ u || ∞ .We therefore have | ( u k )( j ) − ( u k − δ )( j ) | ≤ | ( u k )( j ) − ( u k )( x ) | + | ( u k )( x ) − ( u k − δ )( x ) | + | ( u k − δ )( x ) − ( u k − δ )( j ) |≤ C ( u ) δ for all j ∈ Z ε ( Q δ ( i )) and hence we have, splitting the sum into the summation over j, ξ such that max { ε | ξ | , | j − i |} > δ } and the complement and using (22) we obtain R εi ( u k − δ , u k , ϕ kk,δ ) ≤ C ( u ) (cid:16) (1 + δ p ) (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω)max { ε | ξ | , | j − i |}≤ δ C j − i,ξε + (cid:16) δ p + 1 (cid:17) (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω)max { ε | ξ | , | j − i |} >δ C j − i,ξε + 1 (cid:17) ≤ C ( u ) . for ε small enough, using (12). By summing over j and, taking the maximum over j ∈ Z ε (Ω) in the inner sum and using (12) we obtain (21). If k = 1 by (H2) and thedefinition of u δ we have that φ εi ( { (( u δ ) ε ) j + i } j ∈ Z ε (Ω i ) ) = φ εi ( { ( u ) j + i } j ∈ Z ε (Ω i ) ) ≤ C ( u )14nd (20) follows. Now for A ∈ A (Ω) we have that I ε,δ ≤ C ( u ) ε N Z ε (cid:32) A \ ( K (cid:91) k =1 (Ω k ) δ ) (cid:33) ≤ C ( u ) ε N Z ε (cid:32) Ω \ ( K (cid:91) k =1 (Ω k ) δ ) (cid:33) ≤ C ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ω \ ( K (cid:91) k =1 (Ω k ) δ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Therefore, using that | Ω \ (cid:83) Kk =1 Ω k | = 0 and the dominated-convergence theorem, we havethat lim sup δ → lim sup ε → I ε,δ = 0 (23)By (18), (19) and (23) we obtain (17) and the claim follows. Now by the lower semicon-tinuity of F (cid:48)(cid:48) ( · , A ) we have F (cid:48)(cid:48) ( u, A ) ≤ lim inf δ → F (cid:48)(cid:48) ( u δ , A ) ≤ C ( ||∇ u || pL p ( A ; R n × N ) + | A | ) . (24)Now for general u ∈ W ,p (Ω; R n ) we take { u n } ⊂ W ,p (Ω; R n ) piecewise affine such that u n → u strongly in W ,p (Ω; R n ) and again by the lower semicontinuity of F (cid:48)(cid:48) ( · , A ) we have F (cid:48)(cid:48) ( u, A ) ≤ lim inf n →∞ F (cid:48)(cid:48) ( u n , A ) ≤ lim n →∞ C ( ||∇ u n || pL p ( A ; R d × N ) + | A | ) = C ( ||∇ u || pL p ( A ; R n × N ) + | A | )and the statement is proven. Proposition 3.5.
Let φ εi : ( R n ) Z ε (Ω) → [0 , + ∞ ) satisfy (H2)–(H5) . Let A, B ∈ A (Ω) andlet A (cid:48) , B (cid:48) ∈ A (Ω) be such that A (cid:48) ⊂⊂ A and B (cid:48) ⊂⊂ B . Then for any u ∈ W ,p (Ω; R n ) wehave F (cid:48)(cid:48) ( u, A (cid:48) ∪ B (cid:48) ) ≤ F (cid:48)(cid:48) ( u, A ) + F (cid:48)(cid:48) ( u, B ) Proof.
Without loss of generality, we may suppose F (cid:48)(cid:48) ( u, A ) and F (cid:48)(cid:48) ( u, B ) finite. Let ( u ε ) ε and ( v ε ) ε converge to u in L p (Ω; R n ) and be such thatlim sup ε → + F ε ( u ε , A ) = F (cid:48)(cid:48) ( u, A ) , lim sup ε → + F ε ( v ε , B ) = F (cid:48)(cid:48) ( u, B ) , and therefore sup ε> (cid:88) i ∈ Z ε ( A ) ε N φ εi ( { ( u ε ) j + i } j ∈ Z ε (Ω i ) ) < ∞ , (25)sup ε> (cid:88) i ∈ Z ε ( B ) ε N φ εi ( { ( v ε ) j + i } j ∈ Z ε (Ω i ) ) < ∞ . (26)By (H3) we have that sup n ∈{ ,...,N } sup ε> (cid:88) i ∈ Z ε ( A (cid:48)(cid:48) ) ε N | D e n ε u ε ( i ) | p < + ∞ (27)sup n ∈{ ,...,N } sup ε> (cid:88) i ∈ Z ε ( B (cid:48)(cid:48) ) ε N | D e n ε v ε ( i ) | p < + ∞ (28)15or all A (cid:48)(cid:48) ⊂⊂ A, B (cid:48)(cid:48) ⊂⊂ B . Since u ε and v ε converge to u in L p (Ω; R n ), we have that (cid:88) i ∈ Z ε (Ω) ε N ( | u ε ( i ) | p + | v ε ( i ) | p ) ≤ || u ε || pL p (Ω; R n ) + || v ε || pL p (Ω; R n ) ≤ C < ∞ (29) (cid:88) i ∈ Z ε (Ω) ε N ( | u ε ( i ) − v ε ( i ) | p ) ≤ || u ε − v ε || L p (Ω; R n ) → . (30)Since u ∈ W ,p (Ω; R n ) there exists ˜ u ε , ˜ v ε such that ˜ u ε and ˜ v ε converge to u in L p (Ω; R n )and sup n ∈{ ,...,N } sup ε> (cid:88) i ∈ Z ε (Ω) ε N ( | D e n ε ˜ u ε ( i ) | p + | D e n ε ˜ v ε ( i ) | p ) < ∞ . (31)Take A (cid:48)(cid:48) , A (cid:48)(cid:48)(cid:48) , B (cid:48)(cid:48) , B (cid:48)(cid:48)(cid:48) ∈ A (Ω), ϕ A , ϕ B ∈ C ∞ (Ω) such that A (cid:48) ⊂⊂ A (cid:48)(cid:48) ⊂⊂ A (cid:48)(cid:48)(cid:48) ⊂⊂ A , B (cid:48) ⊂⊂ B (cid:48)(cid:48) ⊂⊂ B (cid:48)(cid:48)(cid:48) ⊂⊂ B , 0 ≤ ϕ A , ϕ B ≤ A (cid:48)(cid:48)(cid:48) ⊂ { ϕ A = 0 } , B (cid:48)(cid:48)(cid:48) ⊂ { ϕ B = 0 } , A (cid:48)(cid:48) ⊂ { ϕ A = 1 } , B (cid:48)(cid:48) ⊂ { ϕ B = 1 } and ||∇ ϕ A || ∞ , ||∇ ϕ B || ∞ ≤ C , and define u (cid:48) ε = ϕ A u ε +(1 − ϕ A )˜ u ε , v (cid:48) ε = ϕ B v ε +(1 − ϕ B )˜ v ε . Now for j ∈ Z ε (Ω), ψ cut-off function z, w ∈ A ε (Ω; R n ) v = ψz + (1 − ψ ) w we have D e n ε v ( j ) = ψ ( j ) D e n ε z ( j ) + (1 − ψ ( j )) D e n ε w ( j ) + D e n ε ψ ( j )( z ( j ) − w ( j )) (32)Since { ϕ A > } ⊂⊂ A , by (27), (31) and (32) we have thatsup n ∈{ ,...,N } sup ε> (cid:88) j ∈ Z ε (Ω) ε N | D e n ε u (cid:48) ε ( j ) | p < ∞ . (33)We can perform a similar construction for v (cid:48) ε and therefore assume that an analogousbound to (33) holds also for v (cid:48) ε . Moreover, since u (cid:48) ε and v (cid:48) ε converge to u in L p (Ω; R n ) wehave that (29) and (30) hold with u (cid:48) ε and v (cid:48) ε . Now for δ >
0, by (H4), it holds φ εi ( { ( u (cid:48) ε ) j + i } j ∈ Z ε (Ω i ) ) ≤ φ εi ( { ( u ε ) j + i } j ∈ Z ε (Ω i ) ) + (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε,δ ( | D ξε u (cid:48) ε ( j ) | p + 1) (34)as well as a similar estimate for v (cid:48) ε in B (cid:48) . Setd := dist ∞ ( A (cid:48) , A c ) and A k := ( A (cid:48) ) k K d for any k ∈ { K, . . . , K } . Let ϕ k be a cut-off function between A k and A k +1 , with ||∇ ϕ k || ∞ ≤ CK . Then for any k ∈ { K, . . . , K } consider the family of functions w kε ∈A ε (Ω; R n ) converging to u in L p (Ω; R n ), defined as w kε ( i ) = ϕ k ( i ) u (cid:48) ε ( i ) + (1 − ϕ k ( i )) v (cid:48) ε ( i ) . Given i ∈ Z ε ( A (cid:48) ∪ B (cid:48) ), then either dist ∞ ( i, A k +1 \ A k ) ≥ d3 K , in which case either w kε ( j ) = u (cid:48) ε ( j ) for j ∈ Z ε ( Q d2 K ( i )) and i ∈ Z ε ( A k ) or w kε ( j ) = v (cid:48) ε ( j ) j ∈ Z ε ( Q d2 K ( i )) and i ∈ ε (( A (cid:48) ∪ B (cid:48) ) \ A k +1 ) ⊂ Z ε ( B (cid:48) ), or dist ∞ ( i, A k +1 \ A k ) < d6 K . In the first case, using (H4),we estimate φ εi ( { ( w kε ) j + i } j ∈ Z ε (Ω) ) ≤ φ εi ( { ( u (cid:48) ε ) j + i } j ∈ Z ε (Ω) ) + (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε, d2 K ( | D ξε w kε ( j ) | p + 1) . (35)In the second case, using (H4), we estimate φ εi ( { ( w kε ) j + i } j ∈ Z ε (Ω) ) ≤ φ εi ( { ( v (cid:48) ε ) j + i } j ∈ Z ε (Ω) ) + (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε, d2 K ( | D ξε w kε ( j ) | p + 1) . (36)Using (32) and the convexity of | · | p we have for j ∈ Z ε (Ω) and ξ ∈ Z N | D ξε w kε ( j ) | p ≤| D ξε u (cid:48) ε ( j ) | p + | D ξε v (cid:48) ε ( j ) | p + CK p | u (cid:48) ε ( j + εξ ) − v (cid:48) ε ( j + εξ ) | p . (37)Now if dist ∞ ( i, A k +1 \ A k ) < d3 K we have that i ∈ Z ε ( A k +2 \ A k − ) =: Z ε ( S k ) where S k ⊂⊂ A ∩ B . By (H5) we have that for such an i it holds φ εi ( { ( w kε ) j + i } j ∈ Z ε (Ω) ) ≤ C ( φ εi ( { ( v (cid:48) ε ) j + i } j ∈ Z ε (Ω) ) + φ εi ( { ( u (cid:48) ε ) j + i } j ∈ Z ε (Ω) )) + R εi ( u (cid:48) ε , v (cid:48) ε , ϕ k )(38)where R εi ( u (cid:48) ε , v (cid:48) ε , ϕ k ) = ( CK p + 1) (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε | u ε ( j + εξ ) − v ε ( j + εξ ) | p (39)+ (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε (cid:0) | D ξε u (cid:48) ε ( j ) | p + | D ξε v (cid:48) ε ( j ) | p + 1 (cid:1) . Summing over i ∈ Z ε ( A (cid:48) ∪ B (cid:48) ) and splitting into the two cases as described above, using1735)–(39), we have F ε ( w kε , A (cid:48) ∪ B (cid:48) ) ≤ (cid:88) i ∈ Z ε ( A (cid:48) ∪ B (cid:48) )dist ∞ ( i,A k +1 \ A k ) ≥ d3 K ε N φ εi ( { ( w kε ) j + i } j ∈ Z ε (Ω i ) ) + (cid:88) i ∈ Z ε ( S k ) ε N φ εi ( { ( w kε ) j + i } j ∈ Z ε (Ω i ) ) ≤ F ε ( u ε , A ) + F ε ( v ε , B )+ CK p (cid:88) i ∈ Z ε ( S k ) ε N (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε | u (cid:48) ε ( j + εξ ) − v (cid:48) ε ( j + εξ ) | p + CK p (cid:88) i ∈ Z ε ( A (cid:48) ∪ B (cid:48) ) ε N (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε, d2 K | u (cid:48) ε ( j + εξ ) − v (cid:48) ε ( j + εξ ) | p + (cid:88) i ∈ Z ε ( A (cid:48) ∪ B (cid:48) ) ε N (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε, d2 K (cid:0) | D ξε u (cid:48) ε ( j ) | p + | D ξε v (cid:48) ( j ) | p + 1 (cid:1) + (cid:88) i ∈ Z ε ( S k ) ε N (cid:88) j ∈ Z ε (Ω) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω) C j − i,ξε (cid:0) | D ξε u (cid:48) ε ( j ) | p + | D ξε v (cid:48) ( j ) | p + 1 (cid:1) + C (cid:88) i ∈ Z ε ( S k ) ε N (cid:0) φ εi ( { ( v (cid:48) ε ) j + i } j ∈ Z ε (Ω i ) ) + φ εi ( { ( u (cid:48) ε ) j + i } j ∈ Z ε (Ω i ) ) (cid:1) . Note that { j (cid:54) = k : S k ∩ S j (cid:54) = ∅} ≤
5. Therefore summing over k ∈ { K, . . . , K − } ,averaging and taking into account (25)–(29), (33) and Lemma 3.6 in [2], we get1 K K − (cid:88) k = K F ε ( w kε , A (cid:48) ∪ B (cid:48) ) ≤ F ε ( u ε , A ) + F ε ( v ε , B ) + CK + ( K p + 1) O ( ε ) . (40)For any ε > k ( ε ) ∈ { K, . . . , K − } such that F ε ( w k ( ε ) ε , A (cid:48) ∪ B (cid:48) ) ≤ K K − (cid:88) k = K F ε ( w kε , A (cid:48) ∪ B (cid:48) ) . (41)Then, since w k ( ε ) ε still converges to u in L p (Ω; R n ), by (40) and (41), letting ε → F (cid:48)(cid:48) ( u, A (cid:48) ∪ B (cid:48) ) ≤ F (cid:48)(cid:48) ( u, A ) + F ( u, B ) + CK .
Letting K → ∞ we obtain the claim. Proposition 3.6.
Let φ εi : ( R n ) Z ε (Ω) → [0 , + ∞ ) satisfy (H2)–(H5) . Then for any u ∈ W ,p (Ω; R n ) and any A ∈ A (Ω) we have sup A (cid:48) ⊂⊂ A F (cid:48)(cid:48) ( u, A (cid:48) ) = F (cid:48)(cid:48) ( u, A ) . roof. Since F (cid:48)(cid:48) ( u, · ) is an increasing set function, it suffices to provesup A (cid:48) ⊂⊂ A F (cid:48)(cid:48) ( u, A (cid:48) ) ≥ F (cid:48)(cid:48) ( u, A ) . In order to prove this, we define an extension of the functional F ε to a functional ˜ F ε defined on a bounded, smooth, open set ˜Ω ⊃⊃ Ω such that˜ F ε (˜ u, A ) = F ε ( u, A )for all A ∈ A (Ω) and all ˜ u ∈ A ε ( ˜Ω; R n ) such that ˜ u = u in Z ε (Ω) and therefore F (cid:48)(cid:48) ( u, A ) = ˜ F (cid:48)(cid:48) (˜ u, A ) (42)for all A ∈ A (Ω), u ∈ W ,p (Ω; R n ) and ˜ u ∈ W ,p (Ω; R n ) such that ˜ u = u a.e. in Ω. Tothis end we define F ε : A ε ( ˜Ω) × A ( ˜Ω) → [0 , + ∞ ) by˜ F ε ( u, A ) = (cid:88) i ∈ Z ε ( A ) ε N ˜ φ εi ( { u j + i } j ∈ Z ε (˜Ω i ) )where ˜ φ εi : ( R n ) Z ε (˜Ω) → [0 , + ∞ ) is defined by˜ φ εi ( { z j + i } j ∈ Z ε (Ω i ) ) := (cid:40) φ εi ( { ( z (cid:98) Ω ) j + i } ) j ∈ Z ε (Ω) i ∈ Z ε (Ω) c (cid:80) Nn =1 | D e n ε z ( i ) | p i ∈ ˜Ω \ Ωwith c > φ εi satisfies (H2)–(H5). Let u ∈ W ,p (Ω; R n ), extended to˜ u ∈ W ,p ( ˜Ω; R n ). Let A ∈ A (Ω); for δ > A δ , A δ , B δ such that A δ ⊃⊃ A ⊃⊃ A δ ⊃⊃ A (cid:48) δ ⊃⊃ B δ ⊃⊃ B δ and | A δ \ B δ | + ||∇ u || L P ( A δ \ B δ ; R n × N ) ≤ δ. Applying Proposition 3.5 with U = A δ \ B δ , V = A δ , U (cid:48) = A \ B δ and V (cid:48) = A (cid:48) δ we have U (cid:48) ∪ V (cid:48) = A and therefore˜ F (cid:48)(cid:48) (˜ u, A ) ≤ ˜ F (cid:48)(cid:48) (˜ u, U ) = ˜ F (cid:48)(cid:48) (˜ u, U (cid:48) ∪ V (cid:48) ) ≤ ˜ F (cid:48)(cid:48) ( u, U ) + ˜ F (cid:48)(cid:48) ( u, V ) ≤ ˜ F (cid:48)(cid:48) (˜ u, A δ ) + ˜ F (cid:48)(cid:48) (˜ u, A δ \ B δ ) ≤ ˜ F (cid:48)(cid:48) (˜ u, A δ ) + C (cid:16) | A δ \ B δ | + ||∇ u || pL P ( A δ \ B δ ; R d × N ) (cid:17) ≤ ˜ F (cid:48)(cid:48) ( u, A δ ) + Cδ ≤ sup A (cid:48) ⊂⊂ A ˜ F (cid:48)(cid:48) (˜ u, A (cid:48) ) + Cδ Applying (42) to u, ˜ u and A, A (cid:48) we obtain F (cid:48)(cid:48) ( u, A ) ≤ sup A (cid:48) ⊂⊂ A F (cid:48)(cid:48) ( u, A (cid:48) ) + Cδ.
The claim follows as δ → + . Proposition 3.7.
Let φ εi : ( R n ) Z ε (Ω) → [0 , + ∞ ) satisfy (H2)–(H5) . Then for any A ∈A (Ω) and for any u, v ∈ W ,p (Ω; R n ) , such that u = v a.e. in A we have F (cid:48)(cid:48) ( u, A ) = F (cid:48)(cid:48) ( v, A )19 roof. Thanks to Proposition 3.6, we may assume that A ⊂⊂ Ω. We first prove F (cid:48)(cid:48) ( u, A ) ≥ F (cid:48)(cid:48) ( v, A )Given δ > A δ ⊂⊂ A such that | A \ A δ | + ||∇ u || pL p (Ω; R n × N ) ≤ δ Let v ε : Z ε (Ω) → R n , u ε : Z ε (Ω) → R n be such that v ε → v and u ε → u in L p (Ω; R n ) andlim sup ε → + F ε ( u ε , A ) = F (cid:48)(cid:48) ( u, A )lim sup ε → + F ε ( v ε , A \ A δ ) = F (cid:48)(cid:48) ( v, A \ A δ ) ≤ C (cid:16) | A \ A δ | + ||∇ u || pL p (Ω; R n × N ) (cid:17) ≤ Cδ Performing the same cut-off construction as in Proposition 3.5 we obtain a function w ε converging to v in L p (Ω; R n ) such that for ε > F ε ( w ε , A (cid:48) ) ≤ F ε ( u ε , A ) + F ε ( v ε , A \ A δ ) + C δ K + K p O ( ε )for some A (cid:48) ⊂⊂ A . Taking ε → + we obtain F (cid:48)(cid:48) ( v, A (cid:48) ) ≤ F (cid:48)(cid:48) ( u, A ) + C δ K + Cδ Letting K → + ∞ and δ → u and v we obtain the other inequality. Proof of Theorem . By the compactness property of Γ-convergence there exists a sub-sequence ε j k of ε j such that for any ( u, A ) ∈ W ,p (Ω; R n ) × A (Ω) there existsΓ( L p )- lim k F ε jk ( u, A ) =: F ( u, A )(see [15] Theorem 10.3). Moreover, by Proposition 3.4 we have thatΓ( L p )- lim k F ε jk ( u ) = + ∞ for any u ∈ L p (Ω; R n ) \ W ,p (Ω; R n ). So it suffices to check that for every ( u, A ) ∈ W ,p (Ω; R n ) × A (Ω), F ( u, A ) satisfies all the hypothesis of Theorem 2.2 in [2]. In fact thesuperaditivity property of F ε ( u, · ) is conserved in the limit. Thus, as an consequence ofPropositions (3.4)–(3.7) and thanks to De Giorgi-Letta Criterion (see [15]), hypotheses(i), (ii), (iii) hold true. Moreover, since F ε ( u, A ) is translationally invariant, hypothesis(iv) is satisfied and finally, by the lower semicontinuity property of Γ-limit, also hypothesis(v) is fulfilled. 20 Treatment of Dirichlet boundary data
In order to recover the limiting energy density we will establish the next lemma whichasserts that our energies still converge if we suitably assign affine boundary conditions.From this, one is able to recover the value of f in Theorem 3.2 by a blow-up argument.Given M ∈ R n × N , m ∈ N , ε > A ∈ A reg (Ω) set A M,mε ( A ; R n ) = (cid:8) u ∈ A ε (Ω; R n ) : u ( i ) = M i if ( i + [ − mε, mε ) N ) ∩ A c (cid:54) = ∅ (cid:9) (43)For M ∈ R d × N , m ∈ N we define F M,mε : L p (Ω; R n ) × A reg (Ω) → [0 , + ∞ ] by F M,mε ( u, A ) = (cid:40) F ( u, A ) if u ∈ A Mε ( A ; R n )+ ∞ otherwise. Proposition 4.1.
Let φ εi : ( R n ) Z ε (Ω) → [0 , + ∞ ) satisfy (H1)–(H5) . Let ε j k and f be as inTheorem . For any M ∈ R d × N and A ∈ A reg (Ω) we set F M : L p (Ω; R n ) × A reg (Ω) → [0 , + ∞ ] by F M ( u, A ) = (cid:90) A f ( x, ∇ u )d x if u − M x ∈ W ,p ( A ; R n )+ ∞ otherwise.Then for any M ∈ R d × N , m ∈ N and any A ∈ A reg we have that F M,mε jk ( · , A ) Γ -convergeswith respect to the strong L p (Ω; R n ) -topology to the functional F M ( · , A ) .Proof. We only prove the statement for m = 1, the other cases being done analogously.We first prove the Γ-lim inf inequality. Let { u k } k ⊂ A ε jk (Ω; R n ) converge to u in the L p (Ω; R n )-topology and be such thatlim inf k →∞ F M, ε jk ( u k , A ) = lim k →∞ F Mε jk ( u k , A ) < + ∞ . Since u k ∈ A M,mε jk ( A ; R n ) for all k ∈ N , and by (H3), we have that u k → M x in L p ( A \ Ω; R n )and sup ε> N (cid:88) n =1 (cid:88) i ∈ Z ε (Ω) ε N | D e n ε u k ( i ) | p < + ∞ . By the same reasoning as in Proposition 3.4 u ∈ W ,p (Ω; R n ) and u − M x ∈ W ,p ( A ; R n ).By Theorem 3.2 we therefore havelim inf k →∞ F M,mε jk ( u k , A ) ≥ lim inf k →∞ F ε jk ( u k , A ) = F M ( u, A ) . To prove the Γ-lim sup inequality we may first suppose that supp( u − M x ) ⊂⊂ A . Let { u k } k ⊂ A ε jk (Ω; R n ) converge to u in L p (Ω; R n ) and be such thatlim sup k →∞ F ε jk ( u k , A ) = F ( u, A ) . δ > A δ ⊂ A andsuitable cut-off functions ϕ k with supp( u − M x ) ⊂⊂ supp ϕ k ⊂⊂ A δ and | A \ A δ | < δ such that for w k ( i ) := ϕ k ( i ) u k ( i ) + (1 − ϕ k ( i )) M i we have that w k converges to u in L p (Ω; R n ) andlim sup k →∞ F ε jk ( w k , A ) ≤ lim sup k →∞ F ε jk ( u k , A ) + lim sup k →∞ F ε jk ( M x, A \ A δ ) + δ. Using (H2) we have that for every k ∈ N it holds F ε jk ( M x, A \ A δ ) ≤ C ( | M | p + 1) | ( A \ A δ ) ε | ≤ C ( | M | p + 1) | δ. By the definition of the Γ-lim sup we have thatΓ- lim sup k →∞ F M,mε jk ( u, A ) ≤ F M ( u, A ) + Cδ.
Letting δ → u ∈ W ,p (Ω; R n ) such that u − M x ∈ W ,p ( A ; R n )strongly in W ,p (Ω; R n ) by functions u n such that supp( u n − M x ) ⊂⊂ A and using thelower semicontinuity of the Γ-lim sup as well as the continuity of F ( · , A ) with respect tothe strong convergence in W ,p (Ω; R n ). Remark 4.2.
Let φ εi : ( R n ) Z ε (Ω) → [0 , + ∞ ) satisfy (H1)–(H5), and let ε j k be as inTheorem 3.2. For any M ∈ R d × N , m ∈ N and A ∈ A reg (Ω) we have thatlim k →∞ inf (cid:110) F ε jk ( u, A ) : u ∈ A M,mε jk ( A ; R n ) (cid:111) = inf (cid:8) F ( u, A ) : u − M x ∈ W ,p ( A ; R n ) (cid:9) , since the functionals F Mε are coercive with respect to the strong L p (Ω; R n )-topology.Note first that by extending the functional as in the proof of Proposition 3.6 we canassume that A ⊂⊂ Ω. Moreover, by the boundary conditions and by (H3) any sequence { u k } k satisfying sup k F M,mε jk ( u k , A ) < + ∞ satisfies sup k ∈ N N (cid:88) n =1 (cid:88) i ∈ Z εjk (Ω) ε N | D e n ε jk u k ( i ) | p < + ∞ . Then by the boundary conditions, Lemma 3.6 in [2] and the Riesz-Frech´et-KolmogorovTheorem there exists a function u ∈ L p (Ω; R n ) and a subsequence (not relabelled) thatconverges to u . By Proposition 3.4 we have that u ∈ W ,p (Ω; R n ). Moreover, u k → M x in L p (Ω \ A ; R n ) and therefore u − M x ∈ W ,p ( A ; R n ). This implies the coercivity.22 Homogenization
We now consider the case where i (cid:55)→ φ εi is periodic, though we have to explain whatthat means in our case, since the interaction energy at every point of the lattice maydepend on the whole configuration of the state { z j + i } j ∈ Z ε (Ω i ) . This will be done by usinga function φ i : ( R n ) Z N → [0 , + ∞ ), i ∈ Z N defined on the entire lattice. In order to definethe energy density inside Ω we assume that φ i is approximated by finite-range interaction.More precisely, we suppose that there exist φ ki : ( R n ) Z N → [0 , + ∞ ), i ∈ Z N T -periodic,satisfying (H1)–(H3) uniformly in k and(H p
4) ( locality ) For all k ∈ N and for all z, w ∈ A ( R N , R n ) satisfying z ( j ) = w ( j ) forall j ∈ Z N ∩ Q k ( i ) we have φ ki ( { z j } j ∈ Z N ) = φ ki ( { w j } j ∈ Z N ) . (H p
5) ( controlled non-convexity ) There exist
C > { C j,ξ } j ∈ Z N ,ξ ∈ Z N , C j,ξ ≥ (cid:88) j,ξ ∈ Z N C j,ξ < + ∞ and we have lim sup k →∞ (cid:88) max {| ξ | , | j |} >k C j,ξ = 0 (44)such that for all k ∈ N , z, w ∈ A ( R N , R n ) and ψ cut-off functions we have φ ki ( { ψ j z j + (1 − ψ j ) w j } j ∈ Z N ) ≤ C (cid:0) φ ki ( { z j } j ∈ Z N ) + φ ki ( { w j } j ∈ Z N ) (cid:1) + R ki ( z, w, ψ ) , where R ki ( z, w, ψ ) = (cid:88) j,ξ ∈ Z N j + ξ ∈ Z N ∩ Q k (0) C j,ξ (cid:16) ( sup k ∈ Z N ∩ Q k (0) n ∈{ ,...,N } | D e n ψ ( k ) | p + 1) | z ( j + ξ ) − w ( j + ξ ) | p (cid:17) + (cid:88) j,ξ ∈ Z N j + ξ ∈ Z N ∩ Q k (0) C j,ξ (cid:16) | D ξ z ( j ) | p + | D ξ w ( j ) | p + 1 (cid:17) . (H p
6) ( closeness ) There exist { C j,ξk } k ∈ N ,j ∈ Z N ,ξ ∈ Z N , C j,ξk ≥ C j,ξk +1 ≥ k →∞ (cid:88) j,ξ ∈ Z N C j,ξk = 0 (45)such that For all z ∈ A ( R N ; R n ) and k ≤ k we have that | φ k i ( { z j } j ∈ Z N ) − φ k i ( { z j } j ∈ Z N ) | ≤ (cid:88) j,ξ ∈ Z N ∩ Q k (0) j + ξ ∈ Z N ∩ Q k (0) C j,ξk (cid:16) | D ξ z ( j ) | p + 1 (cid:17) . p
7) ( monotonicity ) For every k ∈ N , for every i ∈ Z N and for every z ∈ A ( R N ; R n )we have φ ki ( { z j } j ∈ Z N ) ≤ φ k +1 i ( { z j } j ∈ Z N ) , φ ki ( { z j } j ∈ Z N ) → φ i ( { z j } j ∈ Z N ) as k → ∞ . (46)The monotonicity property (H p
7) may seem restrictive at a first sight, but it is notsince by the positivity of φ k and φ respectively we may reorder the interactions in a waythat we keep only adding positive interactions with increasing k .For every i ∈ Z ε (Ω) we define φ εi : ( R n ) Z ε (Ω) → [0 , + ∞ ) by φ εi ( { z j } j ∈ Z ε (Ω i ) ) = φ (cid:98) diε (cid:99) iε ( { z εj } j ∈ Z N ) , (47)where dist ∞ (Ω c , i ) = d i and z ε ( j ) = (cid:40) z ( εj ) ε j ∈ Q (cid:98) diε (cid:99) ( i ) ∩ Z N p
4) and Moreover, φ εi satisfies(H1)–(H5). Those assumptions are made to avoid the dependence of φ εi on Ω and stillinclude infinite-range interactions. Theorem 5.1.
Let φ ki : ( R n ) Z N → [0 , + ∞ ) satisfy (H1)–(H3) and (H p p and φ εi :( R n ) Z ε (Ω) → [0 , + ∞ ) be defined by (47) . Then, F ε : L p (Ω; R n ) → [0 , + ∞ ] Γ -convergeswith respect to the strong L p (Ω; R n ) -topology to the functional F : L p (Ω; R n ) → [0 , + ∞ ] defined by F ( u ) = (cid:90) Ω f hom ( ∇ u )d x if u ∈ W ,p (Ω; R n )+ ∞ otherwise,where f hom : R d × N → [0 , ∞ ) is given by f hom ( M ) = lim L →∞ L N inf (cid:110) (cid:88) i ∈ Z N ∩ Q L φ i ( { z j + i } j ∈ Z N ) : z ∈ A M, (cid:98)√ L (cid:99) ( Q L ; R n ) (cid:111) , (48) where A M,mε ( Q L ; R n ) = (cid:8) u ∈ A ε ( R N ; R n ) : u ( i ) = M i if ( i + [ − mε, mε ) N ) ∩ Q cL (cid:54) = ∅ (cid:9) . Remark 5.2.
Note that in Theorem 5.1 we have that the whole sequence F ε Γ-convergesto the limit functional F . We fix the boundary conditions of the admissible test functionson a boundary layer of width (cid:98)√ L (cid:99) in order to have the boundary effects negligible whilestill being able to use a subadditivity argument in order to prove the existence of the limitin (48). Arguing as in the proof of Proposition 5.3 to show that the error goes to 0 whensubstituting φ ki with φ i , and using the fact that the limit energy density is quasi-convex,we also have f hom ( M ) = lim L →∞ L N inf (cid:110) (cid:88) i ∈ Z N ∩ Q L φ i ( { z j + i } j ∈ Z N ) : z ∈ A M,m ( Q L ; R n ) (cid:111) for all m ∈ N and all M ∈ R d × N . 24 roof. By Theorem (3.2) for every sequence ε j there exists a subsequence ε j k such that F ε jk Γ-converges to a functional F such that for any u ∈ W ,p (Ω; R n ) and every A ∈ A (Ω)we have Γ- lim k →∞ F ε jk ( u, A ) = (cid:90) A f ( x, ∇ u )d x. By the Urysohn property of Γ-convergence the theorem is proved if we show that f doesnot depend on x and f = f hom . To prove the first claim it suffices to show that F ( M x, Q ρ ( z )) = F ( M x, Q ρ ( y ))for all M ∈ R d × N , z, y ∈ Ω and ρ > Q ρ ( z ) ∪ Q ρ ( y ) ⊂ Ω. By symmetry itsuffices to prove F ( M x, Q ρ ( z )) ≤ F ( M x, Q ρ ( y )) . By the inner-regularity property it suffices to prove for any ρ (cid:48) < ρF ( M x, Q ρ (cid:48) ( z )) ≤ F ( M x, Q ρ ( y )) . Let v k ∈ A ε jk (Ω; R n ) be such that v k → M x in L p (Ω; R n ) and such thatlim k →∞ F ε jk ( v k , Q ρ ( y )) = F ( M x, Q ρ ( y )) . Let ϕ ∈ C ∞ (Ω) be a cut-off function such that 0 ≤ ϕ ≤ ϕ ) ⊂⊂ Q ρ ( z ) , Q ρ (cid:48) ( z ) ⊂⊂ { ϕ = 1 } and ||∇ ϕ || ∞ ≤ Cρ − ρ (cid:48) . For k ∈ N define u k ∈ A ε jk (Ω; R n ) by u k ( i ) = ϕ ( i ) (cid:18) v k (cid:16) i + ε j k T (cid:98) y − zT ε j k (cid:99) (cid:17) + M ( z − y ) (cid:19) + (1 − ϕ ( i )) M i.
Thus by the periodicity assumption and the locality property we have that (cid:88) i ∈ Z εjk ( Q ρ (cid:48) ( z )) ε Nj k φ ε jk i ( { ( u k ) j + i } j ∈ Z εjk (Ω i ) ) ≤ (cid:88) i ∈ Z εjk ( Q ρ ( y )) ε Nj k φ ε jk i ( { ( v k ) j + i } j ∈ Z εjk (Ω i ) ) + O ( ε j k ) . Therefore, we obtain F ( M x, Q ρ (cid:48) ( z )) ≤ lim inf k →∞ F ε jk ( u k , Q ρ (cid:48) ( z )) ≤ lim inf k →∞ F ε jk ( u k , Q ρ ( y )) = F ( M x, Q ρ ( y )) . In order to obtain that f = f hom we note that by the lower semicontinuity with respectto the strong L p (Ω; R n )-topology and the coercivity of F we obtain that F is lower semi-continuous with respect to the weak W ,p (Ω; R n )-topology and hence f is quasiconvex.25y the growth properties of f and Remark 4.2 we obtain for Q = Q ρ ( x ) ⊂⊂ Ω f ( M ) = 1 ρ N inf (cid:110) (cid:90) Q f ( ∇ u )d x : u − M x ∈ W ,p ( Q ; R n ) (cid:111) = 1 ρ N inf (cid:110) F ( u, Q ) : u − M x ∈ W ,p ( Q ; R n ) (cid:111) = lim m →∞ lim k →∞ ρ N inf (cid:110) F ε jk ( u, Q ) : u ∈ A M,mε jk ( Q ; R n ) (cid:111) = f hom ( M ) . Where the last inequality follows from the next proposition.
Proposition 5.3.
Let φ ki : ( R n ) Z N → [0 , + ∞ ) satisfy (H1)–(H3) and (H p p , and φ εi : ( R n ) Z ε (Ω) → [0 , + ∞ ) be defined by (47) . Then f hom ( M ) = lim m →∞ lim k →∞ ρ N inf (cid:110) F ε jk ( u, Q ) : u ∈ A M,mε jk ( Q ; R n ) (cid:111) for all M ∈ R n × N .Proof. Without loss of generality, assume x = 0. We perform a change of variables i (cid:48) = iε j k , ˜ u ( i (cid:48) ) = 1 ε j k u ( ε j k i (cid:48) ) , L k = ρε j k . Set d ki (cid:48) = dist( ε jk Ω c , i (cid:48) ). We obtainlim m →∞ lim k →∞ ρ N inf (cid:110) F ε jk ( u, Q ) : u ∈ A M,mε jk ( Q ; R n ) (cid:111) = lim m →∞ lim k →∞ L Nk inf (cid:110) (cid:88) i (cid:48) ∈ Z N ∩ Q L φ (cid:98) d ki (cid:48) (cid:99) i (cid:48) ( { ˜ u j + i (cid:48) } j ∈ Z N ) : ˜ u ∈ A M,m ( Q L k ; R n ) (cid:111) . By the monotonicity property and (H2) we have that C ( | M | p + 1) ≥ lim m →∞ lim k →∞ L Nk inf (cid:110) (cid:88) i (cid:48) ∈ Z N ∩ Q L φ i (cid:48) ( { ˜ u j + i (cid:48) } j ∈ Z N ) : ˜ u ∈ A M,m ( Q L ; R n ) (cid:111) ≥ lim m →∞ lim k →∞ L Nk inf (cid:110) (cid:88) i (cid:48) ∈ Z N ∩ Q L φ (cid:98) d ki (cid:48) (cid:99) i (cid:48) ( { ˜ u j + i (cid:48) } j ∈ Z N ) : ˜ u ∈ A M,m ( Q L ; R n ) (cid:111) . On the other hand, let u k ∈ A M,m ( Q L ; R n ) be such that (cid:88) i (cid:48) ∈ Z N ∩ Q Lk φ (cid:98) d ki (cid:48) (cid:99) i (cid:48) ( { ( u k ) j + i (cid:48) } j ∈ Z N ) ≤ inf (cid:110) (cid:88) i (cid:48) ∈ Z N ∩ Q Lk φ (cid:98) d ki (cid:48) (cid:99) i (cid:48) ( { ˜ u j + i (cid:48) } j ∈ Z N ) : ˜ u ∈ A M,m ( Q L k ; R n ) (cid:111) + 1 k Now by (H p
6) and setting d k = (cid:98) dist( Q, Ω c ) ε j k (cid:99) we obtain d k → ∞ , since Q ⊂⊂ Ω, and (cid:88) i (cid:48) ∈ Z N ∩ Q Lk φ i (cid:48) ( { ( u k ) j + i (cid:48) } j ∈ Z N ) ≤ (cid:88) i (cid:48) ∈ Z N ∩ Q Lk (cid:16) φ (cid:98) d ki (cid:48) (cid:99) i (cid:48) ( { ( u k ) j + i (cid:48) } j ∈ Z N ) + (cid:88) j,ξ ∈ Z N C j − i (cid:48) ,ξd k ( | D ξ u k ( j ) | p + 1) (cid:17) .
26e have that either j, j + ξ ∈ Z N \ Q L k (0) in which case | D ξ u k | p ≤ | M | p or { j, j + ξ } ∩ Q L k (0) (cid:54) = ∅ . Now if j, j + ξ ∈ Q L k (0), by [[2],Lemma 3.6] and (H2), we have that (cid:88) j ∈ Z N j,j + ξ ∈ Q Lk (0) | D ξ u k ( j ) | p ≤ C N (cid:88) n =1 (cid:88) j ∈ Z N ∩ Q Lk (0) | D e n u k ( j ) | p ≤ C (cid:88) j ∈ Z N ∩ Q Lk (0) φ d k j ( { ( u k ) j (cid:48) + j } j (cid:48) ∈ Z N ) ≤ C ( | M | p + 1) L Nk . (49)Now either j ∈ Q L k (0), j + ξ / ∈ Q L k (0) or j / ∈ Q L k (0), j + ξ ∈ Q L k (0). We only deal withthe first case, the second one being done analogously. Now if | ξ | ∞ ≤ L k , by (H2) andusing the boundary conditions, we have that (cid:88) j ∈ Z N | D ξ u k ( j ) | p ≤ (cid:88) j ∈ Z N j,j + ξ ∈ Q Lk (0) | D ξ u k ( j ) | p ≤ C N (cid:88) n =1 (cid:88) j ∈ Z N ∩ Q Lk (0) | D e n u k ( j ) | p ≤ C (cid:88) j ∈ Z N ∩ Q Lk (0) φ d k j ( { ( u k ) j (cid:48) + j } j (cid:48) ∈ Z N ) + (cid:88) j ∈ Z N ∩ Q Lk (0) \ Q Lk (0) | D e n u k ( j ) | p ≤ C (cid:88) j ∈ Z N ∩ Q Lk (0) φ d k j ( { ( u k ) j (cid:48) + j } j (cid:48) ∈ Z N ) + CL Nk | M | p ≤ C ( | M | p + 1) L Nk . (50)If | ξ | ∞ > L k for every j we choose a path γ jξ = ( j h ) || ξ || +1 h =1 ⊂ Z N by defining j || ξ || +1 = j + ξ, j = j, j h +1 = j h + e n ( h ) , e n ( h ) = sign( ξ k ) e k if 1 + k − (cid:88) n =1 | ξ n | ≤ h ≤ k (cid:88) n =1 | ξ n | . For this path it holds | D ξ u ( j ) | p ≤ C ( p, N ) || ξ || || ξ || (cid:88) h =1 | D e n ( h )1 u ( j h ) | p . Now for every i ∈ Z N and for every n ∈ { , . . . , N } we set N ξ,ki,n = (cid:110) j ∈ Q L k (0) : ∃ h ∈ { , . . . , | ξ |} , n ∈ { , . . . , N } such that i = j h ∈ γ ξj and e n ( h ) = sign( ξ n ) e n } . We have that N ξ,ki,n ≤ L k for i ∈ Z N ∩ Q L k (0), using | D e n u k ( i ) | ≤ | M | for every i ∈ N \ Q L k (0) and using Fubini’s Theorem we obtain (cid:88) j ∈ Z N ∩ Q Lk (0) | D ξ u k ( j ) | p ≤ C || ξ || (cid:88) j ∈ Z N ∩ Q Lk (0) || ξ || (cid:88) h =1 | D e n ( h )1 u k ( j h ) | p ≤ C || ξ || N (cid:88) n =1 (cid:88) i ∈ Z N ∩ Q Lk (0) N ξ,ki,n | D e n u k ( i ) | p + | M | p L Nk ≤ C N (cid:88) n =1 (cid:88) i ∈ Z N ∩ Q Lk (0) | D e n u k ( i ) | p + | M | p L Nk ≤ C (cid:88) j ∈ Z N ∩ Q Lk (0) φ d k i ( { ( u k ) j + i } j ∈ Z N ) + | M | p L Nk ≤ C ( | M | p + 1) L Nk . (51)Now if j, j + ξ ∈ Q L k (0), using Fubini’s Theorem and (49), we obtain (cid:88) i (cid:48) ∈ Z N ∩ Q Lk (0) (cid:88) j,ξ ∈ Z N j,j + ξ ∈ Q Lk (0) C j − i (cid:48) ,ξd k | D ξ u k ( j ) | p ≤ (cid:88) i (cid:48) ,ξ ∈ Z N C j − i (cid:48) ,ξd k (cid:88) j ∈ Z N ∩ Q Lk (0) j + ξ ∈ Q Lk (0) | D ξ u k ( j ) | p ≤ CL Nk (cid:88) i (cid:48) ,ξ ∈ Z N C j − i (cid:48) ,ξd k ( | M | p + 1) . (52)Now if j ∈ Q L k (0) | ξ | ∞ ≤ L k , using Fubini’s Theorem and (50), we obtain (cid:88) i (cid:48) ∈ Z N ∩ Q Lk (0) (cid:88) j,ξ ∈ Z N j ∈ Q Lk (0) | ξ | ∞ ≤ L k C j − i (cid:48) ,ξd k | D ξ u k ( j ) | p ≤ (cid:88) i (cid:48) ,ξ ∈ Z N | ξ | ∞ ≤ L k C j − i (cid:48) ,ξd k (cid:88) j ∈ Z N ∩ Q Lk (0) | D ξ u k ( j ) | p ≤ CL Nk (cid:88) i (cid:48) ,ξ ∈ Z N C j − i (cid:48) ,ξd k ( | M | p + 1) . (53)If j ∈ Q L k (0) | ξ | ∞ > L k , using Fubini’s Theorem and (51), we obtain (cid:88) i (cid:48) ∈ Z N ∩ Q Lk (0) (cid:88) j,ξ ∈ Z N j ∈ Q Lk (0) | ξ | ∞ >L k C j − i (cid:48) ,ξd k | D ξ u k ( j ) | p ≤ (cid:88) i (cid:48) ,ξ ∈ Z N | ξ | ∞ >L k C j − i (cid:48) ,ξd k (cid:88) j ∈ Z N ∩ Q Lk (0) | D ξ u k ( j ) | p ≤ CL Nk (cid:88) i (cid:48) ,ξ ∈ Z N C j − i (cid:48) ,ξd k ( | M | p + 1) . (54)Now, dividing by L Nk , using (45),(52)–(54) and taking the limit as k → ∞ , we obtainlim k →∞ L Nk (cid:88) i (cid:48) ∈ Z N ∩ Q Lk (cid:88) j,ξ ∈ Z N C j − i (cid:48) ,ξd k ( | D ξ u k ( j ) | p + 1) = 028t remains to show that the limit (48) exists and f hom ( M ) = lim m →∞ lim L →∞ L N inf (cid:110) (cid:88) i ∈ Z N ∩ Q L (0) φ i ( { z j + i } j ∈ Z N ) : z ∈ A M,m ( Q L ; R n ) (cid:111) . (55)Since A M, (cid:98)√ L (cid:99) ( Q L ; R n ) ⊂ A M,m ( Q L ; R n ) we have that f hom ( M ) ≥ lim m →∞ lim L →∞ L N inf (cid:110) (cid:88) i ∈ Z N ∩ Q L (0) φ i ( { z j + i } j ∈ Z N ) : z ∈ A M,m ( Q L ; R n ) (cid:111) . On the other hand, for every u L ∈ A M,m ( Q L ; R n ), also u L ∈ A M, (cid:98) √ L + √ L (cid:99) ( Q L + (cid:98)√ L (cid:99) ; R n ),so that for ˜ L = L + (cid:98)√ L (cid:99) we have (cid:88) i ∈ Z N ∩ Q ˜ L (0) φ i ( { ( u L ) j + i } j ∈ Z N ) = (cid:88) i ∈ Z N ∩ Q L (0) φ i ( { ( u L ) j + i } j ∈ Z N )+ (cid:88) i ∈ Z N ∩ ( Q ˜ L (0) \ Q L (0)) φ i ( { ( u L ) j + i } j ∈ Z N ) . Note that lim L →∞ ˜ LL = 1 and therefore we are done if we can show that1 L N (cid:88) i ∈ Z N ∩ ( Q ˜ L (0) \ Q L (0)) φ i ( { ( u L ) j + i } j ∈ Z N ) → L → ∞ and then m → ∞ . By the locality property (H p
4) and the boundary conditionswe have for all i ∈ Z N ∩ ( Q ˜ L (0) \ Q L (0)) φ i ( { ( u L ) j + i } j ∈ Z N ) ≤ φ i ( { M x j + i } j ∈ Z N ) + (cid:88) j,ξ ∈ Z N C j − i,ξm ( | D ξ u L ( j ) | p + 1) ≤ C ( | M | p + 1) + (cid:88) j,ξ ∈ Z N C j − i,ξm ( | D ξ u L ( j ) | p + 1) . Using similar arguments as for (52)–(54) we obtain1 L N (cid:88) i ∈ Z N ∩ ( Q ˜ L (0) \ Q L (0)) (cid:88) j,ξ ∈ Z N C j − i,ξm ( | D ξ u L ( j ) | p + 1) → L → ∞ and then m → ∞ and hence (55). We are done if we show that the limit inthe definition of (48) exists. To this end set F L ( M ) = 1 L N inf (cid:110) (cid:88) i ∈ Z N ∩ Q L φ i ( { z j + i } j ∈ Z N ) : z ∈ A M, √ L ( Q L ; R n ) (cid:111) . L ∈ N and let k ∈ N be such that kT ≤ L ≤ ( k + 1) T . For any u ∈ A M, (cid:98)√ L (cid:99) ( Q L ; R n )we have that u ∈ A M, (cid:98) √ ( k +1) T (cid:99) ( Q ( k +1) T ; R n ) and1 L N (cid:88) i ∈ Z N ∩ Q ( k +1) T (0) φ i ( { u j + i } j ∈ Z N ) ≤ L N (cid:88) i ∈ Z N ∩ Q L (0) φ i ( { u j + i } j ∈ Z N )+ 1 L N (cid:88) i ∈ Z N ∩ ( Q ( k +1) T (0) \ Q L (0) φ i ( { u j + i } j ∈ Z N ) , where the last term tends to 0 as L → ∞ , again using similar arguments as to prove (56).Noting that for every k ∈ N the function u ∈ A M, (cid:98)√ kT (cid:99) ( Q kT ; R n ) can also be used as atest function u ∈ A M, (cid:98)√ L (cid:99) ( Q L ; R n ) in the minimum on Q L we obtain thatlim k →∞ F kT ( M ) = lim L →∞ F L ( M ) . Hence, we can assume that
L, S ∈ T N , 1 << L << S and u L ∈ A M, (cid:98)√ L (cid:99) ( Q L ; R n ) be suchthat 1 L N (cid:88) i ∈ Z N ∩ Q L (0) φ i ( { ( u L ) j + i } j ∈ Z N ) ≤ F L ( M ) + 1 L .
We define v S ∈ A M, (cid:98)√ S (cid:99) ( Q S ; R n ) by v S ( i ) = (cid:40) u L ( i − Lk ) + LM k if i ∈ Lk + Q L (0) , k ∈ {− (cid:98) S −√ SL (cid:99) , . . . , (cid:98) S −√ SL (cid:99)} N M i otherwise.By the periodicity assumption and (H4) we have that F S ( M ) ≤ S N (cid:88) i ∈ Z N ∩ Q S (0) φ i ( { ( v S ) j + i } j ∈ Z N )= L N S N (cid:88) k ∈{− (cid:98) S −√ SL (cid:99) ,..., (cid:98) S −√ SL (cid:99)} N L N (cid:88) i ∈ Z N ∩ Q L (0) φ i + kL ( { ( u L ) j + i − kL } j ∈ Z N ) ≤ L N S N (cid:106) S − √ SL (cid:107) N L N (cid:88) i ∈ Z N ∩ Q L (0) φ i ( { ( u L ) j + i } j ∈ Z N )+ 1 S N (cid:88) i ∈ Q S (0) (cid:88) j,ξ ∈ Z N C j − i √ L ( | D ξ v S ( j ) | p + 1) ≤ L N S N (cid:106) S − √ SL (cid:107) N L N F L ( M ) + 1 S N (cid:88) i ∈ Q S (0) (cid:88) j,ξ ∈ Z N C j − i √ L ( | D ξ v S ( j ) | p + 1) . Now, again using the same arguments as for (52)–(54), we obtainlim sup L →∞ lim sup S →∞ S N (cid:88) i ∈ Q S (0) (cid:88) j,ξ ∈ Z N C j − i √ L ( | D ξ v S ( j ) | p + 1) = 030nd therefore, noting that lim L →∞ lim S →∞ L N S N (cid:106) S −√ SL (cid:107) N = 1, we get lim sup S →∞ F S ( M ) ≤ lim inf L →∞ F L ( M )and the claim follows. An example of interactions that can be taken into account with our type of energies arediscrete determinants. For z ∈ A ε (Ω; R n ) we define φ εi ( { z j } j ∈ Z ε (Ω i ) ) = (cid:88) ξ ,...,ξ n ∈ Z N g εξ ,...,ξ n (det( D ξ ε z (0) , . . . , D ξ n ε z (0))) + N (cid:88) n =1 | D e n ε z (0) | p , where g εξ ,...,ξ n : R → [0 , ∞ ) satisfy g εξ ,...,ξ n ( z ) ≤ C ξ ,...,ξ n ( | z | pn + 1)and C ξ ,...,ξ n > (cid:88) ξ ,...,ξ n ∈ Z N C ξ ,...,ξ n < + ∞ . (H1) follows, since φ εi does only depend on its difference quotients. Note that by Hadamard’sInequality, the Geometric-Arithmetic mean Inequality and convexity we have | det( D ξ ε z (0) , . . . , D ξ n ε z (0)) | pn ≤ (cid:12)(cid:12)(cid:12) n (cid:89) j =1 | D ξ j ε z (0) | n (cid:12)(cid:12)(cid:12) p ≤ (cid:12)(cid:12)(cid:12) n n (cid:88) j =1 | D ξ j ε z (0) | (cid:12)(cid:12)(cid:12) p ≤ n n (cid:88) j =1 | D ξ j ε z (0) | p . Recall (cid:12)(cid:12)(cid:12)(cid:12) M ( i + εξ ) − M iε | ξ | (cid:12)(cid:12)(cid:12)(cid:12) ≤ | M | and therefore | det( D ξ ε z (0) , . . . , D ξ n ε z (0)) | pn ≤ | M | p and by summing over ξ , . . . , ξ n ∈ Z N (H2) follows. (H3) follows since we have exactlythe coercivity term in the definition of φ εi and the first term is positive. For δ > z ( j ) = w ( j ) in Z ε ( Q δ ( i )) we have that φ εi ( { z j } j ∈ Z ε (Ω i ) ) ≤ φ εi ( { w j } j ∈ Z ε (Ω i ) ) + (cid:88) ξ ,...,ξ n ∈ Z N ε | ξ i | ∞ >δ C ξ ,...,ξ n n n (cid:88) j =1 | D ξ j ε z (0) | p . Hence, by choosing C ,ξε,δ = (cid:88) ξ ∈{ ξ ,...,ξ n }⊂ ( Z N ) n ε | ξ i | ∞ >δ for some i n C ξ,...,ξ n , C j,ξε,δ = 0 , j (cid:54) = 031H4) follows. Setting C ,ξε = (cid:88) ξ ∈{ ξ ,...,ξ n }⊂ ( Z N ) n d C ξ,...,ξ n , C j,ξε = 0 , j (cid:54) = 0we have that C j,ξε satisfies (13) and we have φ εi ( { z j } j ∈ Z ε (Ω i ) ) ≤ (cid:88) ξ ∈ Z N C ,ξε ( | D ξε z (0) | p + 1) . Note that for all cut-off functions ψ and for all z, w ∈ A ε (Ω; R n ) we have D ξε ( ψz + (1 − ψ ) w ) = ψ ( i ) D ξε z ( i ) + (1 − ψ ( i )) D ξε w ( i ) + D ξε ψ ( i )( z ( i + εξ ) − w ( i + εξ ))(57)and hence (H5) follows by using the convexity of | · | p , 0 ≤ ψ ≤ | D ξε ψ ( i ) | ≤ max n ∈{ ,...,N } sup k ∈ Z ε (Ω) | D e n ε ψ ( k ) | . (58)A particular example could be g εe ,e ( z ) = | z | and g εξ ,ξ ( z ) = 0 otherwise. More general ourTheorems also apply to the case where we take functions g of minors of (cid:0) D ξ ε z (0) , . . . , D ξ n z (0) (cid:1) as long as g satisfies appropriate bounds. We assume N = d = 3. Our result is applicable to show an integral representation if thepotential φ εi is the linearization of the Lennard-Jones potential, where the Lennard-Jonespotential, pictured in Fig. 1, is defined by (up to renormalization) V ( r ) = 1 r − r . For Ω ⊂ R open and smooth we define E ε : L (Ω; R ) → [0 , + ∞ ] by E ε ( u ) = (cid:40)(cid:80) i,j ∈ Z ε (Ω) ε V (cid:48)(cid:48) (cid:16) (cid:12)(cid:12) i − jε (cid:12)(cid:12) (cid:17) (cid:12)(cid:12) u i − u j ε (cid:12)(cid:12) if u ∈ A ε (Ω; R )+ ∞ otherwise.In fact heuristically E ε can be obtained by linearizing the Lennard-Jones Energy definedby E LJε ( u ) = (cid:40)(cid:80) i,j ∈ Z ε (Ω) ε V (cid:16) (cid:12)(cid:12) u i − u j ε (cid:12)(cid:12) (cid:17) if u ∈ A ε (Ω; R )+ ∞ otherwise,where the set of admissible deformations u should be close to the identity (neglecting thelinear term in the expansion by the assumption that u ( i ) = i is an equilibrium point).The term ˜ φ εi ( { u j + i } j ∈ Z ε (Ω i ) ) = (cid:88) j ∈ Z ε (Ω) V (cid:48)(cid:48) (cid:16) (cid:12)(cid:12)(cid:12)(cid:12) i − jε (cid:12)(cid:12)(cid:12)(cid:12) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) u i − u j ε (cid:12)(cid:12)(cid:12)(cid:12) n fact heuristically E " can be obtained by linearizing the Lennard-Jones Energy definedby E LJ" ( u ) = (P i,j Z " (⌦) " V ⇣ u i u j " ⌘ if u " (⌦; R )+ otherwise,where the set of admissible deformations u should be close to the identity (neglecting thelinear term in the expansion by the assumption that u ( i ) = i is an equilibrium point).The term ˜ "i ( { u j + i } j Z " (⌦ i ) ) = X j Z " (⌦) V ⇣ i j" ⌘ u i u j " may not be positive in general, due to the long-range part of the potential. We may re- rV ( r )Figure 1: The Lennard-Jones Potentialgroup the interactions of ˜ "i such that we have a positive potential satisfying the assump-tions of our main theorem. For every ⇠ Z , i Z N we choose a path i⇠ = ( i k ) || ⇠ || +1 h =1 ⇢ Z by defining i ⇠ || ⇠ || +1 = i + ⇠, i ⇠ = i, i ⇠h +1 = i ⇠h + e n ( h ) , e n ( h ) = sign( ⇠ k ) e k if 1 + k X n =1 | ⇠ n | h k X n =1 | ⇠ n | . For this path it holds | D ⇠ u ( i ) | || ⇠ || || ⇠ || X h =1 | D e n ( h )1 u ( i h ) | . For ⇠ Z \ {± e , ± e , ± e } we define f ⇠i : A ( R ; R ) ! [0 , ) by f ⇠i ( { u j } j Z ) = V ( | ⇠ | ) ⇣ | D ⇠ u ( i ) | + 3 || ⇠ || || ⇠ || X h =1 | D e n ( h ) u ( i h ) | ⌘ . and for v e , ± e , ± e } we define f vi : A ( R ; R ) ! R f vi ( { u j } j Z ) = ⇣ V (1) X j Z X ⇠ Z i = i h ⇠j ,e n ( h ) = v V ( | ⇠ | ) || ⇠ || ⌘ | D v u ( i ) | . Figure 1: The Lennard-Jones potentialmay not be positive in general, due to the long-range part of the potential. We may re-group the interactions of ˜ φ εi such that we have a positive potential satisfying the assump-tions of our main theorem. For every ξ ∈ Z , i ∈ Z N we choose a path γ iξ = ( i k ) || ξ || +1 h =1 ⊂ Z by defining i ξ || ξ || +1 = i + ξ, i ξ = i, i ξh +1 = i ξh + e n ( h ) , e n ( h ) = sign( ξ k ) e k if 1 + k − (cid:88) n =1 | ξ n | ≤ h ≤ k (cid:88) n =1 | ξ n | . For this path it holds | D ξ u ( i ) | ≤ || ξ || || ξ || (cid:88) h =1 | D e n ( h )1 u ( i h ) | . For ξ ∈ Z \ {± e , ± e , ± e } we define f ξi : A ( R ; R ) → [0 , ∞ ) by f ξi ( { u j } j ∈ Z ) = V (cid:48)(cid:48) ( | ξ | ) (cid:16) − | D ξ u ( i ) | + 3 || ξ || || ξ || (cid:88) h =1 | D e n ( h ) u ( i h ) | (cid:17) , and for v ∈ {± e , ± e , ± e } we define f vi : A ( R ; R ) → R f vi ( { u j } j ∈ Z ) = (cid:16) V (cid:48)(cid:48) (1) − (cid:88) j ∈ Z (cid:88) ξ ∈ Z i = i h ∈ γ ξj ,e n ( h ) = v V (cid:48)(cid:48) ( | ξ | ) || ξ || (cid:17) | D v u ( i ) | . Moreover, we define φ ki : A ( R ; R ) → R by φ ki ( { u j } j ∈ Z ) = (cid:88) | ξ | ∞ ≤ k f ξi ( { u j } j ∈ Z )and φ i : A ( R ; R ) → R by φ i ( { u j } j ∈ Z ) = (cid:88) ξ ∈ Z f ξi ( { u j } j ∈ Z ) . We need to check that f vi ( { u j } j ∈ Z N ) ≥ c | D v u ( i ) | (59)33or some constant c > v ∈ {± e , ± e , ± e } and that φ ki , φ i satisfy (H1)–(H3) and(H p p u ε ( j ) = u ( εj ) ε it holds (cid:88) i ∈ Z ε ( R ) φ iε ( { u εj } j ∈ Z ) = (cid:88) i ∈ Z ε ( R ) ˜ φ εi ( { u j } j ∈ Z ε ( R ) ) . By the definition of φ ki , φ i it is clear, that (H1), (H2) holds. To prove (H3) we have that φ ξi ≥ ξ ∈ Z \ {± e , ± e , ± e } and by Fubini’s Theorem we have that (cid:88) j ∈ Z (cid:88) ξ ∈ Z , | ξ | > i = i h ∈ γ ξj ,e n ( h ) = v V (cid:48)(cid:48) ( | ξ | ) || ξ || = (cid:88) ξ ∈ Z , | ξ | > N ξi,v V (cid:48)(cid:48) ( | ξ | ) || ξ || , (60)where N ξi,v = (cid:110) j ∈ Z : ∃ h ∈ { , . . . , | ξ |} such that i = j h ∈ γ ξj and e n ( h ) = v (cid:111) . Notethat for ξ ∈ Z such that (cid:104) ξ, v (cid:105) > N ξi,v ≤ || ξ || and N ξi,v = 0 otherwise.Hence, using the monotonicity of V (cid:48)(cid:48) ( r ) for r ≥ √ || ξ || ∞ ≤ || ξ || andusing the fact that { ξ ∈ Z : || ξ || ∞ = k } = 3 k − k + 1 , we obtain − (cid:88) ξ ∈ Z | ξ | > N ξi,v V (cid:48)(cid:48) ( | ξ | ) || ξ || ≤ − (cid:88) ξ ∈ Z , | ξ | > (cid:104) ξ,v (cid:105) > V (cid:48)(cid:48) ( | ξ | ) = − V (cid:48)(cid:48) ( √ − ∞ (cid:88) k =2 (cid:88) || ξ | ∞ = k V (cid:48)(cid:48) ( | ξ | )(3 k − k + 1) ≤ − V (cid:48)(cid:48) ( √ − ∞ (cid:88) k =2 V (cid:48)(cid:48) ( k )(3 k − k + 1) < V (cid:48)(cid:48) (1) . (61)Hence, we obtain (59) and with that (H3). (H p
4) and (H p
7) follow from the definition of φ ki and φ i . Setting C j,e n = V (cid:48)(cid:48) (1) if j = 0 (cid:88) ξ ∈ Z , | ξ | > j = i h ,e n ( h ) = e n V (cid:48)(cid:48) ( | ξ | ) || ξ || otherwise, (62)and C j,ξ = 0 if | ξ | >
1. Using (60) and (61) we obtain (45) and φ ki ( { ψ j z j + (1 − ψ j ) w j } j ∈ Z N ) ≤ R ki ( z, w, ψ ) , with R ki defined in (H p
5) with C j,ξ defined by (62). By the non-negativity of φ ki it follows(H p C j,e n k = 2 (cid:88) ξ ∈ Z , || ξ || ∞ >kj = i h ,e n ( h ) = e n V (cid:48)(cid:48) ( | ξ | ) || ξ || (63)34nd C j,ξk = 0 if | ξ | >
1, using (60) and (61) we obtain (44). We have that | φ k i ( { z j } j ∈ Z N ) − φ k i ( { z j } j ∈ Z N ) | = (cid:88) ξ ∈ Z ∩ ( Q k \ Q k ) f ξi ( { z j } j ∈ Z ) ≤ (cid:88) ξ ∈ Z ∩ ( Q k \ Q k ) V (cid:48)(cid:48) ( | ξ | ) 3 || ξ || || ξ || (cid:88) h =1 | D e n ( h ) z ( i ξh ) | ≤ (cid:88) n =1 (cid:88) j ∈ Z ∩ Q k (cid:88) ξ ∈ Z , || ξ || ∞ >k j = i h ,e n ( h ) ∈{± e n } V (cid:48)(cid:48) ( | ξ | ) || ξ || | D e n z ( j ) | ≤ (cid:88) j,ξ ∈ Z ∩ Q k j + ξ ∈ Z ∩ Q k C j,ξk | D ξ z ( j ) | and hence we obtain (H p E ε toa functional E : L p (Ω; R ) × A (Ω) → [0 , + ∞ ] given by E ( u, A ) = (cid:90) A f hom ( ∇ u )d x, where f hom : R × → [0 , + ∞ ) is given by f hom ( M ) = lim L →∞ L N inf (cid:110) (cid:88) i ∈ Z N ∩ Q L φ i ( { z j + i } j ∈ Z N ) : z ∈ A M,m ( Q L ; R n ) (cid:111) . The compactness theorem can be applied to the special case of pair potentials where φ εi takes only into account the pair interactions of that point with every other point j ∈ Z ε (Ω), that means it is of the form φ εi ( { z j + i } j ∈ Z ε (Ω i ) ) = (cid:88) ξ ∈ Z N i + εξ ∈ Z ε (Ω) f ξε ( i, D ξε z ( i ))with f ξε ≥ f e n ε ( i, z ) ≥ c ( | z | p −
1) for all i ∈ Z ε (Ω), z ∈ R n , ε > n ∈ { , . . . , N } .(ii) f ξε ( i, z ) ≤ c ξε ( | z | p + 1) for all i ∈ Z ε (Ω), z ∈ R n , ε > ξ ∈ R N , wherelim sup ε → (cid:88) ξ ∈ Z N c ξε < + ∞ (64) ∀ δ > ∃ M δ > ε → (cid:88) | ξ | >M δ c ξε < δ. (65)35H1) follows since for each ξ ∈ R N , i ∈ Z ε (Ω) the interaction depend only on D ξε z . (H2)follows from (64) and (ii). (H3) follows from (i). (H4) follows if we choose C i,ξε,δ = (cid:40) c ξε ε | ξ | ∞ ≥ δ, i = 00 i (cid:54) = 0 . Let z, w ∈ A ε (Ω; R n ) such that z ( j ) = w ( j ) in Z ε ( Q δ ( i )). Then, using the positivity of f ξε and (ii), we obtain φ εi ( { z j } j ∈ Z ε (Ω i ) ) = (cid:88) ξ ∈ Z N i + εξ ∈ Z ε (Ω i ) f ξε (0 , D ξε z (0)) = (cid:88) | ξ | ∞ ε ≤ δεξ ∈ Z ε (Ω i ) f ξε (0 , D ξε z (0)) + (cid:88) | ξ | ∞ ε>δεξ ∈ Z ε (Ω i ) f ξε (0 , D ξε z (0)) ≤ (cid:88) | ξ | ∞ ε ≤ δεξ ∈ Z ε (Ω i ) f ξε (0 , D ξε w (0)) + (cid:88) | ξ | ∞ ε>δεξ ∈ Z ε (Ω i ) c ξε ( | D ξε z (0) | p + 1) ≤ φ εi ( { w j } j ∈ Z ε (Ω i ) ) + (cid:88) j ∈ Z ε (Ω i ) ,ξ ∈ Z N C j,ξε,δ ( | D ξε z ( j ) | p + 1)and therefore (H4) follows. Setting C i,ξε = (cid:40) c ξε if i = 00 otherwise . we have that φ εi ( { z j } j ∈ Z ε (Ω i ) ) ≤ (cid:88) j ∈ Z ε (Ω i ) ,ξ ∈ Z N j + εξ ∈ Z ε (Ω i ) C j,ξε | D ξε z ( j ) | p . and again for a cut-off function ψ and z, w ∈ A ε (Ω; R n ) (H5) follows by using (57), theconvexity of | · | p and (58). References [1] R. Alicandro, A. Braides and M. Cicalese. Continuum limits of discrete thin filmswith superlinear growth densities. Calculus of Variations and Partial DifferentialEquations (2008): 267-297.[2] R. Alicandro and M. Cicalese. A general integral representation result for continuumlimits of discrete energies with superlinear growth. SIAM journal on mathematicalanalysis (2004): 1-37.[3] R. Alicandro and M. Cicalese. Variational analysis of the asymptotics of the XYmodel. Archive for rational mechanics and analysis (2009): 501-536.[4] R. Alicandro, M. Cicalese and A. Gloria. Integral representation results for ener-gies defined on stochastic lattices and application to nonlinear elasticity. Archive forrational mechanics and analysis (2011): 881-943.365] R. Alicandro and M.S. Gelli. Local and nonlocal continuum limits of Ising-type en-ergies for spin systems. SIAM journal on mathematical analysis (2016): 895–931.[6] L. Ambrosio, N. Fusco and D. Pallara. Functions of Bounded Variation and FreeDiscontinuity Problems . (Oxford: Clarendon Press, 2000).[7] X. Blanc, C. Le Bris and P.L. Lions. From molecular models to continuum models.Archive for Rational Mechanics and Analysis (2002): 949–956.[8] X. Blanc, C. Le Bris and P.L. Lions. Atomistic to continuum limits for computationalmaterials science. ESAIM: Mathematical Modelling and Numerical Analysis (2007): 391-426.[9] A. Braides. Non-local variational limits of discrete systems. Communications in Con-temporary Mathematics (2000): 285-297.[10] A. Braides. Γ -convergence for Beginners . (Oxford University Press, Oxford, 2002).[11] A. Braides. A handbook of Γ-convergence. In
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