Γ-convergence and stochastic homogenisation of singularly-perturbed elliptic functionals
ΓΓ -CONVERGENCE AND STOCHASTIC HOMOGENISATION OFSINGULARLY-PERTURBED ELLIPTIC FUNCTIONALS ANNIKA BACH, ROBERTA MARZIANI, AND CATERINA IDA ZEPPIERI
Abstract.
We study the limit behaviour of singularly-perturbed elliptic functionals of the form F k ( u, v ) = (cid:90) A v f k ( x, ∇ u ) d x + 1 ε k (cid:90) A g k ( x, v, ε k ∇ v ) d x , where u is a vector-valued Sobolev function, v ∈ [0 ,
1] a phase-field variable, and ε k > ; i.e. , ε k →
0, as k → + ∞ .Under mild assumptions on the integrands f k and g k , we show that if f k grows superlinearlyin the gradient-variable, then the functionals F k Γ-converge (up to subsequences) to a brittle energy-functional ; i.e. , to a free-discontinuity functional whose surface integrand does not dependon the jump-amplitude of u . This result is achieved by providing explicit asymptotic formulasfor the bulk and surface integrands which show, in particular, that volume and surface term in F k decouple in the limit.The abstract Γ-convergence analysis is complemented by a stochastic homogenisation resultfor stationary random integrands. Keywords:
Elliptic approximation, singular perturbation, phase-field approximation, free-discontinuityproblems, Γ-convergence, deterministic and stochastic homogenisation.
MSC 2010: Introduction
Since the seminal work of Modica and Mortola [47, 48] and of Ambrosio and Tortorelli [7, 8],singularly-perturbed elliptic functionals have proven to be an effective tool to approximate free-discontinuity problems in manifold situations. Elliptic-functionals based approximations have beensuccessfully used, e.g. , for numerical simulations in imaging or brittle fracture (see, e.g. , [18,19, 20]), in cavitation problems [43, 44], and to define a notion of regular evolution in fracturemechanics [10, 41], just to mention few examples. Besides being an approximation tool, thesekinds of functionals are also commonly employed to model a range of phenomena where “diffuse”interfaces appear (see, e.g. , [50, 51, 54, 15, 16, 38, 37, 30, 31]), or as instances of gradient damagemodels (see, e.g. , [52, 34, 45]).In this paper we study the Γ-convergence, as k → + ∞ , of general elliptic functionals of the form F k ( u, v ) = (cid:90) A ψ ( v ) f k ( x, ∇ u ) d x + 1 ε k (cid:90) A g k ( x, v, ε k ∇ v ) d x , (1.1)where ε k (cid:38) A ⊂ R n is open bounded and withLipschitz boundary, u : A → R m is a vectorial function, v : A → [0 ,
1] is a phase-field variable,and ψ : [0 , → [0 ,
1] is an increasing and continuous function satisfying ψ (0) = 0, ψ (1) = 1,and ψ ( s ) > s >
0. For every k ∈ N , the integrands f k : R n × R m × n → [0 , + ∞ ) and g k : R n × [0 , × R n → [0 , + ∞ ) belong to suitable classes of functions denoted, respectively, by F and G (see Section 2.2 for their definition). The requirement ( f k ) ⊂ F and ( g k ) ⊂ G in particularensures the existence of an exponent p > k ∈ N and every x ∈ R n : c | ξ | p ≤ f k ( x, ξ ) ≤ c | ξ | p , (1.2) a r X i v : . [ m a t h . A P ] F e b A. BACH, R. MARZIANI, AND C. I. ZEPPIERI for every ξ ∈ R m × n and for some 0 < c ≤ c < + ∞ , and c (cid:0) | − v | p + | w | p (cid:1) ≤ g k ( x, v, w ) ≤ c (cid:0) | − v | p + | w | p (cid:1) , (1.3)for every v ∈ [0 ,
1] and w ∈ R n , and for some 0 < c ≤ c < + ∞ . As a consequence, the functionals F k are finite in W ,p ( A ; R m ) × W ,p ( A ) and are bounded both from below and from above byAmbrosio-Tortorelli functionals of the form AT k ( u, v ) = (cid:90) A ψ ( v ) |∇ u | p d x + (cid:90) A (cid:18) (1 − v ) p ε k + ε p − k |∇ v | p (cid:19) d x . (1.4)Therefore if ( u k , v k ) ⊂ W ,p ( A ; R m ) × W ,p ( A ) is a pair satisfying sup k F k ( u k , v k ) < + ∞ , thelower bound on F k immediately yields v k → L p ( A ), as k → + ∞ . On the other hand, |∇ u k | can blow up in the regions where v k is asymptotically small, so that one expects a limit u whichmay develop discontinuities. In [7, 8] Ambrosio and Tortorelli showed that functionals of type(1.4) provide a variational approximation, in the sense of Γ-convergence, of the free-discontinuityfunctional of Mumford-Shah type given by (cid:90) A |∇ u | p d x + c p H n − ( S u ) , (1.5)where now the variable u belongs to the space of generalised special functions of bounded variation GSBV p ( A ; R m ). As in the case of the Modica-Mortola approximation of the perimeter-functional[47, 48], the effect of the singular perturbation, ε p − k |∇ v | p , in (1.4) is that of producing a transitionlayer around the discontinuity set of u , denoted by S u . Similarly, the pre-factor c p > ψ vanishes, and the value one. Another similarity shared by the Modica-Mortola andthe Ambrosio-Tortorelli approximation is that they are substantially one-dimensional. Indeed, inboth cases the n -dimensional analysis can be carried out by resorting to an integral-geometricargument, the so-called slicing procedure, which allows to reduce the general situation to theone-dimensional case.A relevant feature of the Ambrosio-Tortorelli approximation is that the “regularised” bulk andsurface terms in (1.4) separately converge to their sharp-interface counterparts in (1.5). This kindof volume-surface decoupling can be also observed in a number of variants of (1.4). Indeed, thisis the case, e.g. , of the anisotropic functionals analysed in [36], of the phase-field approximation ofbrittle fracture in linearised elasticity in [29], of the second-order variants proposed in [26, 11], andof the finite-difference discretisation of the Ambrosio-Tortorelli functional on periodic [12] and onstochastic grids [13]. More specifically, in [36, 29, 26, 11] the volume-surface decoupling is obtainedby means of general integral-geometric arguments which can be employed thanks to the specificform of the approximating functionals. On the other hand, in [12, 13] the interplay between thesingular perturbation and the discretisation parameters makes for a subtle problem for which an adhoc proof is needed. In [12] this proof relies on an explicit geometric construction which, however,is feasible only in dimension n = 2. In fact, more refined arguments are necessary to deal with thecase n ≥
3, as shown in [13]. Namely, in [13] the limit volume-surface decoupling is achieved byresorting to a weighted co-area formula, which is reminiscent of a technique introduced by Ambrosio[5] (see also [24, 42, 27, 53]). This procedure allows to identify an asymptotically small region wherethe phase-field variable v k can be modified and set equal to zero while the corresponding u k makesa steep transition between two constant values. In this way, a pair ( u k , v k ) is obtained whose bulkenergy vanishes while the surface energy does not essentially increase.In the present paper we show that the volume-surface decoupling illustrated above takes placealso for the general functionals F k , whose integrands f k and g k combine both a k and an x depen-dence, and satisfy (1.2) and (1.3). Moreover, being the dependence on x only measurable, the caseof homogenisation is covered by our analysis as well, as shown in this paper. We remark that the INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 3 general character of the problem does not allow us to use either the slicing or the blow-up methodto establish a Γ-convergence result for F k , as it is instead customary for phase-field functionals ofAmbrosio-Tortorelli type. Our approach is close in spirit to that of [27] and combines the generallocalisation method of Γ-convergence [32, 23] with a careful local analysis which eventually allowsus to completely characterise the integrands of the Γ-limit thus proving, in particular, that volumeand surface term do not interact in the limit.The volume-surface decoupling has been extensively analysed in the case of free-discontinuityfunctionals, starting with the seminal work [5]. It has then been proven that a decoupling takesplace in the case of free-discontinuity functionals with periodically oscillating integrands [24], forscale-dependent scalar brittle energies both in the continuous [42] and in the discrete case [53], forgeneral vectorial scale-dependent free-discontinuity functionals [27, 28], and, more recently, also inthe setting of linearised elasticity [40]. The clear advantage of having such a decoupling is thatlimit volume and surface integrands can be determined independently from one another, by meansof asymptotic formulas which are then easier to handle, e.g. , computationally. Moreover in [42]it is shown that the noninteraction between volume and surface is crucial to prove the stabilityof unilateral minimality properties in the study of crack-propagation in composite materials. Thesame applies to the case of the evolution considered in [41], where this feature plays a central role inproving that the regular quasistatic evolution for the Ambrosio-Tortorelli functional converges toa quasistatic evolution for brittle fracture in the sense of [39]. These considerations also motivatethe analysis carried out in the present paper.The main result of this paper is contained in Theorem 3.1 and is a Γ-convergence and an integralrepresentation result for the Γ-limit. Namely, in Theorem 3.1 we show that if f k and g k satisfyrather mild assumptions, (see Subsection 2.2 for the complete list of hypotheses) together with(1.2) and (1.3), then (up to subsequences) the functionals F k Γ-converge to a free-discontinuityfunctional of the form F ( u ) = (cid:90) A f ∞ ( x, ∇ u ) d x + (cid:90) S u ∩ A g ∞ ( x, ν u ) d H n − , (1.6)where now u ∈ GSBV p ( A ; R m ) and ν u denotes the generalised normal to S u . We observe thatthe surface term in F is both inhomogeneous and anisotropic, however it does not depend on thejump-opening [ u ] = u + − u − ; in other words, F is a so-called brittle energy . The form of thesurface term in (1.6) is one of the effects of the volume-surface limit decoupling mentioned above,which is apparent from the asymptotic formulas defining f ∞ and g ∞ . In fact, in Theorem 3.1 wealso provide formulas for f ∞ and g ∞ . Namely, we prove that f ∞ ( x, ξ ) = lim sup ρ → lim k → + ∞ ρ n inf (cid:90) Q ρ ( x ) f k ( x, ∇ u ) d x (1.7)where the infimum in (1.7) is taken over all functions u ∈ W ,p ( Q ρ ( x ); R m ) with u ( x ) = ξx near ∂Q ρ ( x ). The surface energy density is given instead by g ∞ ( x, ν ) = lim sup ρ → lim k → + ∞ ρ n − inf 1 ε k (cid:90) Q νρ ( x ) g k ( x, v, ε k ∇ v ) d x (1.8)where the cube Q νρ ( x ) is a suitable rotation of Q ρ ( x ) and the infimum in v is taken in a u -dependentclass of functions. More precisely, the infimum in (1.8) is taken among all v ∈ W ,p ( Q νρ ( x )), with0 ≤ v ≤
1, for which there exists u ∈ W ,p ( Q νρ ( x ); R m ) such that v ∇ u = 0 a.e. in Q νρ ( x ) and( u, v ) = ( u νx ,
1) in U ∩ {| ( y − x ) · ν | > ε k } where U is a neighbourhood of ∂Q νρ ( x ), and u νx is thejump function given by u νx ( y ) = (cid:40) e if ( y − x ) · ν ≥ , y − x ) · ν < . A. BACH, R. MARZIANI, AND C. I. ZEPPIERI
In (1.8) the boundary datum ( u νx ,
1) cannot be prescribed in the vicinity of { y ∈ R n : ( y − x ) · ν = 0 } due to the discontinuity of u νx and to the fact that v must be equal to zero (and not to one)where u jumps. However, this mixed Dirichlet-Neumann boundary condition can be replaced by aDirichlet boundary condition prescribed on the whole boundary of Q νρ ( x ), up to replacing u νx witha regularised counterpart defined, e.g. , as in (l), Subsection 2.1.In view of the growth conditions (1.2) satisfied by f k and the properties of ψ , the constraint v ∇ u = 0 satisfied a.e. in Q νρ ( x ) is equivalent to (cid:90) Q νρ ( x ) ψ ( v ) f k ( x, ∇ u ) d x = 0 , which makes apparent why the bulk term in F k does not contribute to g ∞ . We notice, however,that due to the nature of the problem, the variable u must enter in the definition of g ∞ , so thatin this case a decoupling is not to be intended as in the case of free-discontinuity functionals [27].To derive the formula for f ∞ we follow a similar strategy as in [21] and use the co-area formulain the Modica-Mortola term in (1.1) to show that, in the set where v is bounded away from zero, F k behaves like a sequence of free-discontinuity functionals whose volume integrand is f k . Then,we conclude by invoking the decoupling result for free-discontinuity functionals proven in [27]. Infact, we notice that (1.7) coincides with the asymptotic formula for the limit volume integrandprovided in [27]. The proof of (1.8) is more subtle and is substantially different, e.g. , from thatin [13]. Namely, to prove (1.8) we need to modify a sequence ( u k , v k ) with bounded energy inthe cube Q νρ ( x ) to get a new sequence with zero volume energy which can be used as a test in(1.8), hence, in particular, the modification to ( u k , v k ) shall not increase the surface energy. Inthe case of the discretised Ambrosio-Tortorelli functional considered in [13], the discrete nature ofthe problem allows for a construction which is not feasible in a continuous setting. In our case,instead, we follow an argument which is close in spirit to a construction in [41]. This argumentamounts to partition the set where ∇ u k (cid:54) = 0 and to use the bound on the energy to single out a setof the partition with small measure and small volume energy. Then in this set the function v k ismodified by suitably interpolating between the value zero and two functions explicitly dependingon u k . The advantage of this interpolation is that it allows to easily estimate the increment insurface energy in terms of the volume energy and at the same time to define a test pair for (1.8).Eventually, to prove that the increment in surface energy is asymptotically negligible we need touse that p >
1. We notice that the assumption p > f k is linear inthe gradient variable ; i.e. , (1.2) holds with p = 1, then it is well known [2, 4] that the correspondingAmbrosio-Tortorelli functional Γ-converges to a free-discontinuity functional whose surface energyexplicitly depends on [ u ], this dependence being the result of a nontrivial limit volume-surfaceinteraction.Our general analysis is then applied to study the homogenisation of damage models ; i.e. , todeal with the case of integrands f k and g k of type f k ( x, ξ ) = f (cid:16) xε k , ξ (cid:17) and g k ( x, v, w ) = g (cid:16) xε k , v, w (cid:17) , (1.9)for some f ∈ F and g ∈ G . More specifically, in Theorem 3.5 we prove a homogenisation result for F k , with f k and g k as in (1.9), without requiring any spatial periodicity of the integrands, but ratherassuming the existence and spatial homogeneity of the limit of certain scaled minimisation problems(cf. (3.9) and (3.10)). Eventually, we show that the assumptions of Theorem 3.5 are satisfied, almostsurely, in the case where the integrands f and g are stationary random variables and derive thecorresponding stochastic homogenisation result, Theorem 8.4. Thanks to the decoupling result,Theorem 3.1, the stochastic homogenisation of the bulk term readily follows from [35]. On the otherhand, the treatment of the regularised surface term requires a new ad hoc analysis which shares INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 5 some similarities with that developed for random surface functionals [25, 3, 28]. We also mentionhere the recent paper [49] where the stochastic homogenisation of Modica-Mortola functionals witha stationary and ergodic gradient-perturbation is studied.To conclude we notice that our analysis also allows to deduce a Γ-convergence result for func-tionals with oscillating integrands of type (1.9) when the heterogeneity scale does not necessarilycoincide with the Ambrosio-Tortorelli parameter ε k , but is rather given by a different infinitesimalscale δ k >
0. In this case, though, the asymptotic formulas provided by Theorem 3.1 would fullycharacterise the homogenised volume energy but not the surface energy. In fact, in this case afull characterisation of the homogenised surface integrand requires a further investigation which,in particular, shall distinguish between the regimes ε k (cid:28) δ k and ε k (cid:29) δ k . A complete analysisof this type, in the spirit of [9], goes beyond the purpose of the present paper and is instead theobject of the ongoing work [14]. Outline of the paper.
This paper is organised as follows. In Section 2 we collect some notationused througout, introduce the mathematical setting and the functionals we are going to analyse,moreover we prove some preliminary results. In Section 3 we state the main results of the paper,namely, the Γ-convergence and integral representation result (Theorem 3.1), a convergence resultfor some associated minimisation problems (Theorem 3.4), and a homogenisation result withoutperiodicity assumptions (Theorem 3.5). In Section 4 we prove some properties satisfied by the limitvolume and surface integrands (Proposition 4.1 and Proposition 4.5). In Section 5 we implementthe localisation method of Γ-convergence proving, in particular, a fundamental estimate for thefunctionals F k (Proposition 5.1) and a compactness and integral representation result for theΓ-limit of F k (Theorem 5.2). In Section 6 we characterise the volume integrand of the Γ-limit(Proposition 6.1) and in Section 7 the surface integrand (Proposition 7.4), thus fully achieving theproof of the main result, Theorem 3.1. In Section 8 we prove a stochastic homogenisation result forstationary random integrands (Theorem 8.4). Eventually in the Appendix we prove two technicallemmas which are used in Section 8.2. Setting of the problem and preliminaries
In this section we collect some notation, introduce the functionals we are going to study, and provesome preliminary results.2.1.
Notation.
The present subsection is devoted to the notation we employ throughout.(a) m ≥ n ≥ R m := R m \ { } ;(b) S n − := { ν = ( ν , . . . , ν n ) ∈ R n : ν + · · · + ν n = 1 } and (cid:98) S n − ± := { ν ∈ S n − : ± ν i ( ν ) > } ,where i ( ν ) := max { i ∈ { , . . . , n } : ν i (cid:54) = 0 } ;(c) L n and and H n − denote, respectively, the Lebesgue measure and the ( n − R n ;(d) A denotes the collection of all open and bounded subsets of R n with Lipschitz boundary.If A, B ∈ A by A ⊂⊂ B we mean that A is relatively compact in B ;(e) Q denotes the open unit cube in R n with sides parallel to the coordinate axis, centred atthe the origin; for x ∈ R n and r > Q r ( x ) := rQ + x . Moreover, Q (cid:48) denotes theopen unit cube in R n − with sides parallel to the coordinate axis, centred at the origin,for every r > Q (cid:48) r := rQ (cid:48) ;(f) for every ν ∈ S n − let R ν denote an orthogonal ( n × n )-matrix such that R ν e n = ν ; wealso assume that R − ν Q = R ν Q for every ν ∈ S n − , R ν ∈ Q n × n if ν ∈ S n − ∩ Q n , and thatthe restrictions of the map ν (cid:55)→ R ν to (cid:98) S n − ± are continuous. For an explicit example of amap ν (cid:55)→ R ν satisfying all these properties we refer the reader, e.g. , to [27, Example A.1];(g) for x ∈ R n , r >
0, and ν ∈ S n − , we define Q νr ( x ) := R ν Q r (0) + x . A. BACH, R. MARZIANI, AND C. I. ZEPPIERI (h) for ξ ∈ R m × n we let u ξ be the linear function whose gradient is equal to ξ ; i.e. , u ξ ( x ) := ξx ,for every x ∈ R n ;(i) for x ∈ R n , ζ ∈ R m , and ν ∈ S n − we denote with u νx,ζ the piecewise constant functiontaking values 0 , ζ and jumping across the hyperplane Π ν ( x ) := { y ∈ R n : ( y − x ) · ν = 0 } ;i.e. , u νx,ζ ( y ) := (cid:40) ζ if ( y − x ) · ν ≥ , y − x ) · ν < , when ζ = e we simply write u νx in place of u νx,e ;(j) let u ∈ C ( R ), v ∈ C ( R ), with 0 ≤ v ≤
1, be one-dimensional functions satisfying thefollowing two properties:i. vu (cid:48) ≡ R ;ii. (u( t ) , v( t )) = ( χ (0 , + ∞ ) ( t ) ,
1) for | t | > x ∈ R n and ν ∈ S n − we set¯ u νx ( y ) := u(( y − x ) · ν ) e , ¯ v νx ( y ) := v(( y − x ) · ν ) ;(l) for x ∈ R n , ν ∈ S n − , ζ ∈ R m and ε > u νx,ζ,ε ( y ) := u (cid:0) ε ( y − x ) · ν ) ζ , ¯ v νx,ε ( y ) := v (cid:0) ε ( y − x ) · ν ) . When ζ = e we simply write ¯ u νx,ε in place of ¯ u νx,e ,ε . We notice that in particular,¯ u νx, = ¯ u νx , ¯ v νx, = ¯ v νx ;(m) for a given topological space X , B ( X ) denotes the Borel σ - algebra on X . If X = R d , with d ∈ N , d ≥ B d in place of B ( R d ). For d = 1 we write B .For every L n -measurable set A ⊂ R n we define L ( A ; R m ) as the space of all R m -valued Lebesguemeasurable functions. We endow L ( A ; R m ) with the topology of convergence in measure onbounded subsets of A and recall that this topology is both metrisable and separable.Let A ∈ A ; in this paper we deal with the functional space SBV ( A ; R m ) (resp. GSBV ( A ; R m ))of special functions of bounded variation (resp. of generalised special functions of bounded vari-ation) on A , for which we refer the reader to the monograph [6]. Here we only recall that if u ∈ SBV ( A ; R m ) then its distributional derivative can be represented as Du ( B ) = (cid:90) B ∇ u d x + (cid:90) B ∩ S u [ u ] ⊗ ν u d H n − , (2.1)for every B ∈ B n . In (2.1) ∇ u denotes the approximate gradient of u (which makes sense also for u ∈ GSBV ), S u the set of approximate discontinuity points of u , [ u ] := u + − u − where u ± are theone-sided approximate limit points of u at S u , and ν u is the measure theoretic normal to S u .Let p >
1; we also consider
SBV p ( A ; R m ) := { u ∈ SBV ( A ; R m ) : ∇ u ∈ L p ( A ; R m × n ) and H n − ( S u ) < + ∞} , and GSBV p ( A ; R m ) := { u ∈ GSBV ( A ; R m ) : ∇ u ∈ L p ( A ; R m × n ) and H n − ( S u ) < + ∞} . We recall that
GSBV p ( A ; R m ) is a vector space; moreover, if u ∈ GSBV p ( A ; R m ) then we havethat φ ( u ) ∈ SBV p ( A ; R m ) ∩ L ∞ ( A ; R m ), for every φ ∈ C c ( R m ; R m ) (see [33]).Throughout the paper C denotes a strictly positive constant which may vary from line to lineand within the same expression. INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 7
Setting of the problem.
Let p ∈ (1 , + ∞ ); let c , c , c , c , L , L be given constants suchthat 0 < c ≤ c < + ∞ , 0 < c ≤ c < + ∞ , 0 < L , L < + ∞ . Let F := F ( p, c , c , L ) denotethe collection of all functions f : R n × R m × n → [0 , + ∞ ) satisfying the following conditions:( f
1) (measurability) f is Borel measurable on R n × R m × n ;( f
2) (lower bound) for every x ∈ R n and every ξ ∈ R m × n c | ξ | p ≤ f ( x, ξ ) ;( f
3) (upper bound) for every x ∈ R n and every ξ ∈ R m × n f ( x, ξ ) ≤ c | ξ | p ;( f
4) (continuity in ξ ) for every x ∈ R n we have | f ( x, ξ ) − f ( x, ξ ) | ≤ L (cid:0) | ξ | p − + | ξ | p − (cid:1) | ξ − ξ | , for every ξ , ξ ∈ R m × n ;Moreover, G := G ( p, c , c , L ) denotes the collection of all functions g : R n × R × R n → [0 , + ∞ )satisfying the following conditions:( g
1) (measurability) g is Borel measurable on R n × R × R n ;( g
2) (lower bound) for every x ∈ R n , every v ∈ R , and every w ∈ R n c (cid:0) | − v | p + | w | p (cid:1) ≤ g ( x, v, w ) ;( g
3) (upper bound) for every x ∈ R n , every v ∈ R , and every w ∈ R n g ( x, v, w ) ≤ c ( | − v | p + | w | p ) . ( g
4) (continuity in v and w ) for every x ∈ R n we have | g ( x, v , w ) − g ( x, v , w ) | ≤ L (cid:16)(cid:0) | v | p − + | v | p − (cid:1) | v − v | + (cid:0) | w | p − + | w | p − (cid:1) | w − w | (cid:17) for every v , v ∈ R and every w , w ∈ R n ;( g
5) (monotonicity in v ) for every x ∈ R n and every w ∈ R n , g ( x, · , w ) is decreasing on ( −∞ , , + ∞ );( g
6) (minimum in w ) for every x ∈ R n and every v ∈ R it holds g ( x, v, ≤ g ( x, v, w )for every w ∈ R n . Remark . Let x ∈ R n ; we notice that gathering ( f
2) and ( f
3) readily implies that f ( x, ξ ) = 0 if and only if ξ = 0 . (2.2)Moreover, from ( g
2) and ( g
3) we deduce that g ( x, v, w ) = 0 if and only if ( v, w ) = (1 , . (2.3)Let ψ : R → [0 , + ∞ ) be continuous, decreasing on ( −∞ , , + ∞ ), such that ψ (1) = 1, and ψ ( v ) = 0 iff v = 0.For k ∈ N let ( f k ) ⊂ F and ( g k ) ⊂ G and let ( ε k ) be a decreasing sequence of strictly positivereal numbers converging to zero, as k → + ∞ . A. BACH, R. MARZIANI, AND C. I. ZEPPIERI
We consider the sequence of elliptic functionals F k : L ( R n ; R m ) × L ( R n ) × A −→ [0 , + ∞ ]defined by F k ( u, v, A ) := (cid:90) A ψ ( v ) f k ( x, ∇ u ) d x + 1 ε k (cid:90) A g k ( x, v, ε k ∇ v ) d x if ( u, v ) ∈ W ,p ( A ; R m ) × W ,p ( A ) , ≤ v ≤ , + ∞ otherwise . (2.4) Remark . Assumptions ( f f
3) and ( g g
3) imply that for every A ∈ A and every ( u, v ) ∈ W ,p ( A ; R m ) × W ,p ( A ), 0 ≤ v ≤ c (cid:90) A ψ ( v ) |∇ u | p d x + c (cid:90) A (cid:18) (1 − v ) p ε k + ε p − k |∇ v | p (cid:19) d x ≤ F k ( u, v, A ) ≤ c (cid:90) A ψ ( v ) |∇ u | p d x + c (cid:90) A (cid:18) (1 − v ) p ε k + ε p − k |∇ v | p (cid:19) d x ; (2.5)that is, up to a multiplicative constant, the functionals F k are bounded from below and fromabove by the Ambrosio-Tortorelli functionals AT k ( u, v ) := (cid:90) A ψ ( v ) |∇ u | p d x + (cid:90) A (cid:18) (1 − v ) p ε k + ε p − k |∇ v | p (cid:19) d x . Remark . For later use, we notice that the assumptions on ψ , f k and g k ensure that for every A ∈ A the functionals F k ( · , · , A ) are continuous in the strong W ,p ( A ; R m ) × W ,p ( A ) topology.For every A ∈ A , u ∈ L ( R n ; R m ) and v ∈ L ( R n ) it is also convenient to write F k ( u, v, A ) = F bk ( u, v, A ) + F sk ( v, A ) , where F bk : L ( R n ; R m ) × L ( R n ) × A −→ [0 , + ∞ ] and F sk : L ( R n ) × A −→ [0 , + ∞ ] denote thebulk and the surface part of F k , respectively ; i.e. , F bk ( u, v, A ) := (cid:90) A ψ ( v ) f k ( x, ∇ u ) d x if ( u, v ) ∈ W ,p ( A ; R m ) × W ,p ( A ) , ≤ v ≤ , + ∞ otherwise (2.6)and F sk ( v, A ) := ε k (cid:90) A g k ( x, v, ε k ∇ v ) d x if v ∈ W ,p ( A ) , ≤ v ≤ , + ∞ otherwise . (2.7)For ρ > ε k , x ∈ R n , ξ ∈ R m × n , and ν ∈ S n − we consider the two following minimisation problems m bk ( u ξ , Q ρ ( x )) := inf { F bk ( u, , Q ρ ( x )) : u ∈ W ,p ( Q ρ ( x ); R m ) , u = u ξ near ∂Q ρ ( x ) } (2.8)and m sk, N ( u νx , Q νρ ( x )) := inf { F sk ( v, Q νρ ( x )) : v ∈ A ε k ,ρ ( x, ν ) } , (2.9)where A ε k ,ρ ( x, ν ) := (cid:8) v ∈ W ,p ( Q νρ ( x )) , ≤ v ≤ ∃ u ∈ W ,p ( Q νρ ( x ); R m ) with v ∇ u = 0 a.e. in Q νρ ( x )and ( u, v ) = ( u νx ,
1) near ∂Q νρ ( x ) in {| ( y − x ) · ν | > ε k } (cid:9) . (2.10)We observe that in (2.8) by “ u = u ξ near ∂Q ρ ( x )” we mean that the boundary datum is attainedin a neighbourhood of ∂Q ρ ( x ). Whereas in (2.10) the boundary datum is prescribed only in U ∩ { y ∈ R n : | ( y − x ) · ν | > ε k } , for some neighbourhood U of ∂Q νρ ( x ). INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 9
Remark . Clearly, the class of competitors A ε k ,ρ ( x, ν ) is nonempty. Indeed, the pair (¯ u νx,ε k , ¯ v νx,ε k )defined as in (l), with ε = ε k , satisfies both(¯ u νx,ε k , ¯ v νx,ε k ) = ( u νx ,
1) in {| ( y − x ) · ν | > ε k } and ¯ v νx,ε k ∇ ¯ u νx,ε k ≡ . Thus the restriction of ¯ v νx,ε k to Q νρ ( x ) belongs to A ε k ,ρ ( x, ν ).In view of (2.2) and of the properties satisfied by ψ , we also observe that in (2.10) the constraint v ∇ u = 0 a.e. in Q νρ ( x )can be equivalently replaced by F bk ( u, v, Q νρ ( x )) = 0 . Hence, in particular, for every x ∈ R n , ζ ∈ R m , ν ∈ S n − , ρ > ε k , and k ∈ N we have F k (¯ u νx,ζ,ε k , ¯ v νx,ε k , Q νρ ( x )) = F sk (¯ v νx,ε k , Q νρ ( x )) . (2.11)Finally, for every x ∈ R n and every ξ ∈ R m × n we define f (cid:48) ( x, ξ ) := lim sup ρ → ρ n lim inf k → + ∞ m bk ( u ξ , Q ρ ( x )) , (2.12) f (cid:48)(cid:48) ( x, ξ ) := lim sup ρ → ρ n lim sup k → + ∞ m bk ( u ξ , Q ρ ( x )) , (2.13)while for every x ∈ R n and every ν ∈ S n − we set g (cid:48) ( x, ν ) := lim sup ρ → ρ n − lim inf k → + ∞ m sk, N ( u νx , Q νρ ( x )) , (2.14) g (cid:48)(cid:48) ( x, ν ) := lim sup ρ → ρ n − lim sup k → + ∞ m sk, N ( u νx , Q νρ ( x )) . (2.15)2.3. Equivalent formulas for g (cid:48) and g (cid:48)(cid:48) . For later use, in Proposition 2.6 below, we provethat g (cid:48) and g (cid:48)(cid:48) can be equivalently defined by replacing the boundary conditions in (2.14)–(2.15)with suitable Dirichlet boundary conditions on the whole boundary of Q νρ ( x ). More precisely, weconsider the minimum values defined as follows: For every x ∈ R n , ν ∈ S n − , and A ∈ A we set m sk (¯ u νx,ε k , A ) := inf { F sk ( v, A ) : v ∈ A (¯ u νx,ε k , A ) } , (2.16)where A (¯ u νx,ε k , A ) := { v ∈ W ,p ( A ) , ≤ v ≤ ∃ u ∈ W ,p ( A ; R m ) with v ∇ u = 0 a.e. in A and ( u, v ) = (¯ u νx,ε k , ¯ v νx,ε k ) near ∂A } , (2.17)with (¯ u νx,ε k , ¯ v νx,ε k ) as in (l). Remark . Let A ∈ A be such that A = A (cid:48) × I with A (cid:48) ⊂ R n − open and bounded and I ⊂ R open interval. Let ν ∈ S n − and set A ν := R ν A , with R ν as in (f). For every k ∈ N we have (cid:90) A ν (cid:32) (1 − ¯ v νx,ε k ( y )) p ε k + ε p − k |∇ ¯ v νx,ε k ( y ) | p (cid:33) d y ≤ (cid:90) A (cid:48) (cid:90) R (cid:0) (1 − v( t )) p + | v (cid:48) ( t ) | p (cid:1) d t d y (cid:48) ≤ C v L n − ( A (cid:48) ) , (2.18)where C v := (cid:90) R (cid:0) (1 − v( t )) p + | v (cid:48) ( t ) | p (cid:1) d t = (cid:90) − (cid:0) (1 − v( t )) p + | v (cid:48) ( t ) | p (cid:1) d t < + ∞ . In particular from ( g m sk (¯ u νx,ε k , A ν ) ≤ F sk (¯ v νx,ε k , A ν ) ≤ c C v L n − ( A (cid:48) ) . (2.19) We are now in a position to prove the following equivalent formulation for g (cid:48) and g (cid:48)(cid:48) . We observethat the most delicate part in the proof of this result is to show that a suitable Dirichlet boundarydatum can be prescribed on the whole ∂Q νρ ( x ) while keeping the nonconvex constraint v ∇ u = 0a.e. in Q νρ ( x ). Proposition 2.6.
For every ρ > ε k , x ∈ R n , and ν ∈ S n − let m sk (¯ u νx,ε k , Q νρ ( x )) be as in (2.16) with A = Q νρ ( x ) . Then we have g (cid:48) ( x, ν ) = lim sup ρ → ρ n − lim inf k → + ∞ m sk (¯ u νx,ε k , Q νρ ( x )) ,g (cid:48)(cid:48) ( x, ν ) = lim sup ρ → ρ n − lim sup k → + ∞ m sk (¯ u νx,ε k , Q νρ ( x )) , where g (cid:48) , g (cid:48)(cid:48) are as in (2.14) and (2.15) , respectively.Proof. We only prove the equality for g (cid:48) , the proof of the equality for g (cid:48)(cid:48) being analogous. Let x ∈ R n and ν ∈ S n − be fixed and set g (cid:48) ( x, ν ) := lim sup ρ → ρ n − lim inf k → + ∞ m sk (¯ u νx,ε k , Q νρ ( x )) . In view of Remark 2.4 we readily have g (cid:48) ( x, ν ) ≤ g (cid:48) ( x, ν ) , (2.20)for every x ∈ R n and every ν ∈ S n − . To prove the opposite inequality, let ρ > α ∈ (0 ,
1) befixed and let ¯ k ∈ N be such that ε k < αρ , for every k ≥ ¯ k . Let moreover v k ∈ A ε k ,ρ ( x, ν ) be suchthat F sk ( v k , Q νρ ( x )) ≤ m sk, N ( u νx , Q νρ ( x )) + ρ n . (2.21)Then, there exists u k ∈ W ,p ( Q νρ ( x ); R m ) satisfying v k ∇ u k = 0 a.e. in Q νρ ( x ) and( u k , v k ) = ( u νx ,
1) = (¯ u νx,ε k , ¯ v νx,ε k ) in U k ∩ {| ( y − x ) · ν | > ε k } , (2.22)where U k is a neighbourhood of ∂Q νρ ( x ).Starting from ( u k , v k ) ∈ W ,p ( Q νρ ( x ); R m ) × W ,p ( Q νρ ( x )) we are now going to define a new pair( (cid:101) u k , (cid:101) v k ) ∈ W ,p ( Q ν (1+ α ) ρ ( x ); R m ) × W ,p ( Q ν (1+ α ) ρ ( x )) with (cid:101) v k ∈ A (¯ u νx,ε k , Q ν (1+ α ) ρ ( x )). Moreover,we will do this in a way such that F sk ( (cid:101) v k , Q ν (1+ α ) ρ ( x )) will be bounded from above by F sk ( v k , Q νρ ( x ))and hence, thanks to (2.21), by m sk, N ( u νx , Q νρ ( x )).To this end, let β k ∈ (0 , ε k ) ⊂ (0 , ρ ) be such that Q νρ ( x ) \ Q νρ − β k ( x ) ⊂ U k and set R k := R ν (cid:16)(cid:0) Q (cid:48) ρ \ Q (cid:48) ρ − β k (cid:1) × ( − ε k , ε k ) (cid:17) + x , where R ν is as in (f). By construction we have that (cid:0) Q νρ ( x ) \ Q νρ − β k ( x ) (cid:1) \ R k ⊂ U k ∩ {| ( y − x ) · ν | > ε k } (2.23)(see Figure 1). Now let ϕ k ∈ C c ( Q (cid:48) ρ ) be a cut-off function between Q (cid:48) ρ − β k and Q (cid:48) ρ ; i.e. , 0 ≤ ϕ k ≤ ϕ k ≡ Q (cid:48) ρ − β k . Eventually, for y = ( y (cid:48) , y n ) ∈ Q ν (1+ α ) ρ ( x ) we define the pair ( (cid:101) u k , (cid:101) v k ) bysetting (cid:101) u k ( y ) := ϕ k (cid:0) ( R Tν ( y − x )) (cid:48) (cid:1) u k ( y ) + (cid:16) − ϕ k (cid:0) ( R Tν ( y − x )) (cid:48) (cid:1)(cid:17) ¯ u νx,ε k ( y )and (cid:101) v k ( y ) := (cid:40) min { v k ( y ) , d k ( y ) } in Q νρ ( x ) , min { ¯ v νx,ε k ( y ) , d k ( y ) } in Q ν (1+ α ) ρ ( x ) \ Q νρ ( x ) , where d k ( y ) := ε k dist( y, R k ). Clearly (cid:101) u k ∈ W ,p ( Q ν (1+ α ) ρ ( x ); R m ), moreover, thanks to (2.22) the INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 11 ρ ( + α ) ρ x ρ − β k Π ν ( x ) ν ε k Figure 1.
The cubes Q νρ − β k ( x ), Q νρ ( x ), Q ν (1+ α ) ρ ( x ) and in gray the sets R k (dark gray) and { d k < } (light gray). function (cid:101) v k belongs to W ,p ( Q ν (1+ α ) ρ ( x )). Furthermore, it holds (cid:101) v k ∇ (cid:101) u k = 0 a.e. in Q ν (1+ α ) ρ ( x ).Indeed, (cid:101) v k |∇ (cid:101) u k | ≤ v k |∇ u k | = 0 a.e. in Q νρ − β k ( x ); similarly in Q ν (1+ α ) ρ ( x ) \ Q νρ ( x ) there holds (cid:101) v k |∇ (cid:101) u k | ≤ ¯ v νx,ε k |∇ ¯ u νx,ε k | = 0. Finally, thanks to (2.22) and (2.23), in Q νρ ( x ) \ Q νρ − β k ( x ) \ R k wehave ∇ (cid:101) u k = ∇ ¯ u νx,ε k = 0, while, by definition, (cid:101) v k = 0 in R k .Since moreover d k > Q ν (1+ α ) ρ ( x ) \ Q νρ +2 ε k ( x ), we also have ( (cid:101) u k , (cid:101) v k ) = (¯ u νx,ε k , ¯ v νx,ε k ) in Q ν (1+ α ) ρ ( x ) \ Q νρ +2 ε k ( x ), and hence (cid:101) v k ∈ A (¯ u νx,ε k , Q ν (1+ α ) ρ ( x )) for k ≥ ¯ k .Then, to conclude it only remains to estimate F sk ( (cid:101) v k , Q ν (1+ α ) ρ ( x )). To this end, we start noticingthat (cid:101) v k = v k in Q νρ ( x ) ∩ { d k > } , (cid:101) v k = ¯ v νx,ε k in ( Q ν (1+ α ) ρ ( x ) \ Q νρ ( x )) ∩ { d k > } , while in Q νρ ( x ) ∩ { d k < } we have1 ε k g k ( y, (cid:101) v k , ε k ∇ (cid:101) v k ) ≤ ε k g k ( y, v k , ε k ∇ v k ) + 1 ε k g k ( x, d k , ε k ∇ d k ) ≤ ε k g k ( y, v k , ε k ∇ v k ) + 2 c ε k , where the last estimate follows from ( g Q ν (1+ α ) ρ ( x ) \ Q νρ ( x )) ∩ { d k < } we have1 ε k g k ( y, (cid:101) v k , ε k ∇ (cid:101) v k ) ≤ ε k g k ( y, ¯ v νx,ε k , ε k ∇ ¯ v νx,ε k ) + 2 c ε k . Thus, from (2.3) and (2.19) we infer F sk ( (cid:101) v k , Q ν (1+ α ) ρ ( x )) ≤ F sk ( v k , Q νρ ( x )) + F sk (¯ v νx,ε k , Q ν (1+ α ) ρ ( x ) \ Q νρ ( x )) + 2 c ε k L n ( { d k < } ) ≤ F sk ( v k , Q νρ ( x )) + c C v L n − ( Q (cid:48) (1+ α ) ρ \ Q (cid:48) ρ ) + C ( β k + ε k ) ρ n − . (2.24)Then, thanks to (2.21), dividing both sides of (2.24) by ((1 + α ) ρ ) n − , since β k < ε k we obtain m sk (¯ u νx,ε k , Q ν (1+ α ) ρ ( x ))((1 + α ) ρ ) n − ≤ α ) n − (cid:18) m sk, N ( u νx , Q νρ ( x )) ρ n − + ρ + c C v ((1+ α ) n − − Cε k ρ (cid:19) . (2.25)Eventually, from (2.25) passing first to the liminf as k → + ∞ and then to the limsup as ρ → α ) n − g (cid:48) ( x, ν ) ≤ g (cid:48) ( x, ν ) + c C v ((1 + α ) n − − , which, together with (2.20), thanks to the arbitrariness of α > g (cid:48) ( x, ν ) = g (cid:48) ( x, ν ). (cid:3) Statements of the main results
In this section we state the main results of this paper, namely, a Γ-convergence and integralrepresentation result (Theorem 3.1), a converge result for some associated minimisation problems(Theorem 3.4), and a homogenisation result without periodicity assumptions (Theorem 3.5).3.1. Γ -convergence.
The following result asserts that, up to subsequences, the functionals F k Γ-converge to an integral functional of free-discontinuity type. Furthermore, it provides asymptoticcell formulas for the volume and surface limit integrands. These asymptotic cell formulas show, inparticular, that volume and surface term decouple in the limit.
Theorem 3.1 (Γ-convergence) . Let ( f k ) ⊂ F , ( g k ) ⊂ G and let F k be the functionals as in (2.4) .Then there exists a subsequence, not relabelled, such that for every A ∈ A the functionals F k ( · , · , A )Γ -converge in L ( R n ; R m ) × L ( R n ) to F ( · , · , A ) with F : L ( R n ; R m ) × L ( R n ) × A −→ [0 , + ∞ ] given by F ( u, v, A ) := (cid:90) A f ∞ ( x, ∇ u ) d x + (cid:90) S u ∩ A g ∞ ( x, ν u ) d H n − if u ∈ GSBV p ( A ; R m ) ,v = 1 a.e. in A , + ∞ otherwise , where f ∞ : R n × R m × n → [0 , + ∞ ) and g ∞ : R n × S n − → [0 , + ∞ ) are Borel functions. Moreover,it holds:i. for a.e. x ∈ R n and for every ξ ∈ R m × n f ∞ ( x, ξ ) = f (cid:48) ( x, ξ ) = f (cid:48)(cid:48) ( x, ξ ) , with f (cid:48) , f (cid:48)(cid:48) as in (2.12) and (2.13) , respectively;ii. for every x ∈ R n and every ν ∈ S n − g ∞ ( x, ν ) = g (cid:48) ( x, ν ) = g (cid:48)(cid:48) ( x, ν ) , with g (cid:48) , g (cid:48)(cid:48) as in (2.14) and (2.15) , respectively.Remark . The choice of considering functionals F k which are finite when the variable v satisfiesthe bounds 0 ≤ v ≤ g
5) and ( g
6) and of the properties of ψ , the functionals F k decrease under the transformation v → min { max { v, } , } . Hence a Γ-convergence result for functionals F k defined on functions v with values in R can be easily deduced from Theorem 3.1.The proof of Theorem 3.1 will be achieved in four main steps which are addressed in Sections 4,5, 6, and 7, respectively. Firstly, we show that the functions f (cid:48) , f (cid:48)(cid:48) , g (cid:48) , and g (cid:48)(cid:48) satisfy a number ofproperties and, in particular, they are Borel measurable (see Proposition 4.1 and Proposition 4.5).In the second step, we prove the existence of a sequence ( k j ), with k j → + ∞ as j → + ∞ , suchthat for every A ∈ A the corresponding functionals F k j ( · , · , A ) Γ-converge to a free-discontinuityfunctional which is finite in GSBV p ( A ; R m ) × { } and is of the form (cid:90) A ˆ f ( x, ∇ u ) d x + (cid:90) S u ˆ g ( x, [ u ] , ν u ) d H n − , (3.1)for some Borel functions ˆ f and ˆ g (see Theorem 5.2).In the third step we identify ˆ f by showing that it is equal both to f (cid:48) and f (cid:48)(cid:48) (see Proposition6.1). Eventually, in the final step we identify ˆ g by proving that it coincides with both g (cid:48) and g (cid:48)(cid:48) (see Proposition 7.4). The representation result for ˆ g implies, in particular, that the surface termin (3.1) does not depend on [ u ]. INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 13
The following result is an immediate consequence of Theorem 3.1 and of the Urysohn propertyof Γ-convergence [32, Proposition 8.3].
Corollary 3.3.
Let ( f k ) ⊂ F , ( g k ) ⊂ G and let F k be the functionals as in (2.4) . Let f (cid:48) , f (cid:48)(cid:48) be asin (2.12) and (2.13) , respectively, and g (cid:48) , g (cid:48)(cid:48) be as in (2.14) and (2.15) , respectively. Assume that f (cid:48) ( x, ξ ) = f (cid:48)(cid:48) ( x, ξ ) =: f ∞ ( x, ξ ) , for a.e. x ∈ R n and for every ξ ∈ R m × n and g (cid:48) ( x, ν ) = g (cid:48)(cid:48) ( x, ν ) =: g ∞ ( x, ν ) , for every x ∈ R n and every ν ∈ S n − , for some Borel functions f ∞ : R n × R m × n → [0 , + ∞ ) and g ∞ : R n × S n − → [0 , + ∞ ) . Then, forevery A ∈ A the functionals F k ( · , · , A ) Γ -converge in L ( R n ; R m ) × L ( R n ) to F ( · , · , A ) with F : L ( R n ; R m ) × L ( R n ) × A −→ [0 , + ∞ ] given by F ( u, v, A ) := (cid:90) A f ∞ ( x, ∇ u ) d x + (cid:90) S u ∩ A g ∞ ( x, ν u ) d H n − if u ∈ GSBV p ( A ; R m ) ,v = 1 a.e. in A , + ∞ otherwise . Convergence of minimisation problems.
In view of Theorem 3.1 and Corollary 3.3 weare in a position to prove the following result on the convergence of certain minimisation problemsassociated with F k . Other minimisation problems can be treated similarly. Theorem 3.4 (Convergence of minimisation problems) . Assume that the hypotheses of Corollary3.3 are satisfied. Let q ≥ , let h ∈ L q ( A ; R m ) , and set M k := inf (cid:26) F k ( u, v, A ) + (cid:90) A | u − h | q d x : ( u, v ) ∈ L ( R n ; R m ) × L ( R n ) (cid:27) . Then M k → M as k → + ∞ where M := min (cid:26) F ( u, , A ) + (cid:90) A | u − h | q d x : u ∈ GSBV p ( A ; R m ) ∩ L q ( A ; R m ) (cid:27) . (3.2) Moreover if ( u k , v k ) ⊂ L ( R n ; R m ) × L ( R n ) is such that lim k → + ∞ (cid:18) F k ( u k , v k , A ) + (cid:90) A | u k − h | q d x − M k (cid:19) = 0 , (3.3) then v k → in L p ( A ) and there exists a subsequence of ( u k ) which converges in L q ( A ; R m ) to asolution of (3.2) .Proof. Let ( u k , v k ) ⊂ L ( R n ; R m ) × L ( R n ) be as in (3.3). Then the convergence v k → L p ( A )readily follows by ( g F k together with [36,Lemma 4.1] give the existence of a subsequence ( u k j ) ⊂ ( u k ) with u k j → u in L ( A ; R m ), for some u ∈ GSBV p ( A ; R m ). Eventually, the convergence M k → M , the improved convergence u k j → u in L q ( A ; R m ), and the fact that u is a solution to (3.2) follow arguing as in [34, Theorem 7.1], nowinvoking Corollary 3.3. (cid:3) Homogenisation.
In this subsection we prove a general homogenisation theorem withoutassuming any spatial periodicity of the integrands f k and g k . This theorem will be crucial to provethe stochastic homogenisation result Theorem 8.4.In order to state the homogenisation result, we need to introduce some further notation.For f ∈ F , g ∈ G , A ∈ A , and u ∈ W ,p ( A ; R m ) set F b ( u, A ) := (cid:90) A f ( x, ∇ u ) d x, (3.4) while for v ∈ W ,p ( A ) with 0 ≤ v ≤ F s ( v, A ) := (cid:90) A g ( x, v, ∇ v ) d x. (3.5)Let A ∈ A ; for x ∈ R n and ξ ∈ R m × n we define m b ( u ξ , A ) := inf { F b ( u, A ) : u ∈ W ,p ( A ; R m ) , u = u ξ near ∂A } , (3.6)while for z ∈ R n , ν ∈ S n − , and ¯ u νz as in (k) (with x = z ) we set m s (¯ u νz , A ) := inf { F s ( v, A ) : v ∈ A (¯ u νz , A ) } , (3.7)where A (¯ u νz , A ) is as in (2.17), with ¯ u νx,ε k and ¯ v νx,ε k replaced, respectively, by ¯ u νz and ¯ v νz (that is, x = z and ε k = 1).We are now ready to state the homogenisation result; the latter corresponds to the choice f k ( x, ξ ) := f (cid:16) xε k , ξ (cid:17) and g k ( x, v, w ) := g (cid:16) xε k , v, w (cid:17) , (3.8)with f ∈ F and g ∈ G . We stress again that we will not assume any spatial periodicity of f and g . Theorem 3.5 (Deterministic homogenisation) . Let f ∈ F and g ∈ G . Let also m b ( u ξ , Q r ( rx )) beas in (3.6) with A = Q r ( rx ) and m s (¯ u νrx , Q νr ( rx )) be as in (3.7) with z = rx and A = Q νr ( rx ) .Assume that for every x ∈ R n , ξ ∈ R m × n , ν ∈ S n − the two following limits lim r → + ∞ m b ( u ξ , Q r ( rx )) r n =: f hom ( ξ ) , (3.9)lim r → + ∞ m s (¯ u νrx , Q νr ( rx )) r n − =: g hom ( ν ) , (3.10) exist and are independent of x . Then, for every A ∈ A the functionals F k ( · , · , A ) defined in (2.4) with f k and g k as in (3.8) Γ -converge in L ( R n ; R m ) × L ( R n ) to the functional F hom ( · , · , A ) , with F hom : L ( R n ; R m ) × L ( R n ) × A −→ [0 , + ∞ ] given by F hom ( u, v, A ) := (cid:90) A f hom ( ∇ u ) d x + (cid:90) S u ∩ A g hom ( ν u ) d H n − if u ∈ GSBV p ( A ; R m ) ,v = 1 a.e. in A , + ∞ otherwise . Proof.
Let f (cid:48) , f (cid:48)(cid:48) be as in (2.12), (2.13), respectively, and g (cid:48) , g (cid:48)(cid:48) be as in (2.14), (2.15), respectively.By virtue of Corollary 3.3 it suffices to show that f hom ( ξ ) = f (cid:48) ( x, ξ ) = f (cid:48)(cid:48) ( x, ξ ) , g hom ( ν ) = g (cid:48) ( x, ν ) = g (cid:48)(cid:48) ( x, ν ) , (3.11)for every x ∈ R n , ξ ∈ R m × n , and ν ∈ S n − .We start by proving the first two equalities in (3.11). To this end, fix x ∈ R n , ξ ∈ R m × n , ρ > k ∈ N . Take u ∈ W ,p ( Q ρ ( x ); R m ) and let u k ∈ W ,p ( Q ρεk ( xε k ); R m ) be defined as u k ( y ) := ε k u ( ε k y ). We notice that u = u ξ near ∂Q ρ ( x ) if and only if u k = u ξ near ∂Q ρεk ( xε k ).Moreover, a change of variables gives F bk ( u, , Q ρ ( x )) = ε nk F b (cid:0) u k , Q ρεk (cid:0) xε k (cid:1)(cid:1) , from which, setting r k := ρε k we immediately deduce m bk ( u ξ , Q ρ ( x )) = ε nk m b (cid:0) u ξ , Q ρεk (cid:0) xε k (cid:1)(cid:1) = ρ n r nk m b (cid:0) u ξ , Q r k (cid:0) r k xρ (cid:1)(cid:1) , and thus (3.9) applied with x replaced by x/ρ yieldslim k → + ∞ m bε k ( u ξ , Q ρ ( x )) ρ n = lim k → + ∞ m b (cid:0) u ξ , Q r k (cid:0) r k xρ (cid:1)(cid:1) r nk = f hom ( ξ ) . INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 15
Eventually, by (2.12) and (2.13) we get f (cid:48) ( x, ξ ) = f (cid:48)(cid:48) ( x, ξ ) = f hom ( ξ ), for every x ∈ R n , ξ ∈ R m × n .We now prove the second two equalities in (3.11). To this end, for fixed x ∈ R n , ν ∈ S n − , ρ > k ∈ N , let v ∈ A (¯ u νx,ε k , Q νρ ( x )); then there exists u ∈ W ,p ( Q νρ ( x ); R m ) such that v ∇ u = 0 a.e. in Q νρ ( x ) and ( u, v ) = (¯ u νx,ε k , ¯ v νx,ε k ) near ∂Q νρ ( x ) . (3.12)Define ( u k , v k ) ∈ W ,p ( Q ν ρεk ( xε k ); R m ) × W ,p ( Q ν ρεk ( xε k )) as u k ( y ) := u k ( ε k y ), v k ( y ) := v ( ε k y ).Then (3.12) is equivalent to v k ∇ u k = 0 a.e. in Q ν ρεk ( xε k ) and ( u k , v k ) = (¯ u ν xεk , ¯ v ν xεk ) near ∂Q ν ρεk ( xε k ) , that is, v k ∈ A (¯ u ν xεk , Q ν ρεk ( xε k )). Further, by a change of variables we immediately obtain that F sk ( v, Q νρ ( x )) = ε n − k F s (cid:0) v k , Q ν ρεk (cid:0) xε k (cid:1)(cid:1) . Hence, again setting r k := ρε k , we infer m sk (¯ u νx,ε k , Q νρ ( x )) = ε n − k m s (cid:0) ¯ u ν xεk , Q ν ρεk (cid:0) xε k (cid:1)(cid:1) = ρ n − r n − k m s (cid:0) ¯ u νr k xρ , Q r k (cid:0) r k xρ (cid:1)(cid:1) . Hence invoking (3.10) applied with x replaced by x/ρ we getlim k → + ∞ m sk (¯ u νx,ε k , Q νρ ( x )) ρ n − = lim k → + ∞ m s (cid:0) ¯ u νr k xρ , Q νr k (cid:0) r k xρ (cid:1)(cid:1) r n − k = g hom ( ν ) . Eventually, Proposition 2.6 gives g (cid:48) ( x, ν ) = g (cid:48)(cid:48) ( x, ν ) = g hom ( ν ), for every x ∈ R n , ν ∈ S n − andthus the claim. (cid:3) Properties of f (cid:48) , f (cid:48)(cid:48) , g (cid:48) , g (cid:48)(cid:48) This section is devoted to prove some properties satisfied by the functions f (cid:48) , f (cid:48)(cid:48) , g (cid:48) , and g (cid:48)(cid:48) definedin (2.12), (2.13), (2.14), and (2.15), respectively. Proposition 4.1.
Let ( f k ) ⊂ F ; then the functions f (cid:48) and f (cid:48)(cid:48) defined, respectively, as in (2.12) and (2.13) satisfy properties ( f )-( f ). Moreover they also satisfy ( f ), albeit with a differentconstant (cid:101) L > .Proof. The proof readily follows from [27, Lemma A.6]. (cid:3)
Remark . As shown in [27, Lemma A.6], hypothesis ( f
4) in the definition of the class F canbe weaken to obtain a larger class of volume integrands (cid:101) F which is closed, in the sense that if( f k ) ⊂ (cid:101) F then f (cid:48) , f (cid:48)(cid:48) ∈ (cid:101) F .Before proving the corresponding result for g (cid:48) , g (cid:48)(cid:48) we need to prove the two following technicallemmas. Lemma 4.3.
Let ρ, δ, ε k > with ρ > δ > ε k . Set m s,δk (¯ u νx,ε k , Q νρ ( x )) := inf { F sk ( v, Q νρ ( x )) : v ∈ A δ (¯ u νx,ε k , Q νρ ( x )) } , where A δ (¯ u νx,ε k , Q νρ ( x )) := { v ∈ W ,p ( Q νρ ( x )) , ≤ v ≤ ∃ u ∈ W ,p ( Q νρ ( x ); R m ) with v ∇ u = 0 a.e. in Q νρ ( x ) and ( u, v ) = (¯ u νx,ε k , ¯ v νx,ε k ) in Q νρ ( x ) \ Q νρ − δ ( x ) } , and (¯ u νx,ε k , ¯ v νx,ε k ) are as in (l). Moreover, let g (cid:48) ρ , g (cid:48)(cid:48) ρ : R n × S n − → [0 , + ∞ ] be the functions definedas g (cid:48) ρ ( x, ν ) := inf δ> lim inf k → + ∞ m s,δk (¯ u νx,ε k , Q νρ ( x )) = lim δ → lim inf k → + ∞ m s,δk (¯ u νx,ε k , Q νρ ( x )) , (4.1) g (cid:48)(cid:48) ρ ( x, ν ) := inf δ> lim sup k → + ∞ m s,δk (¯ u νx,ε k , Q νρ ( x )) = lim δ → lim sup k → + ∞ m s,δk (¯ u νx,ε k , Q νρ ( x )) . Then the restrictions of g (cid:48) ρ , g (cid:48)(cid:48) ρ to the sets R n × (cid:98) S n − and R n × (cid:98) S n − − are upper semicontinuous.Proof. We only show that the restriction of the function g (cid:48) ρ to the set R n × (cid:98) S n − is upper semi-continuous, the other proofs being analogous.Let ρ > x ∈ R n , and ν ∈ (cid:98) S n − be fixed. Let δ (cid:48) > δ , since A δ (cid:48) (¯ u νx,ε k , Q νρ ( x )) ⊂ A δ (¯ u νx,ε k , Q νρ ( x )),we immediately obtain that in (4.1) the infimum in δ > δ →
0, whichin particular exists. Thus, for fixed η > δ η > k → + ∞ m s,δk (¯ u νx,ε k , Q νρ ( x )) ≤ g (cid:48) ρ ( x, ν ) + η, (4.2)for every δ ∈ (0 , δ η ). Fix δ ∈ (0 , δ η ), hence by (4.2) we getlim inf k → + ∞ m s, δ k (¯ u νx,ε k , Q νρ ( x )) ≤ g (cid:48) ρ ( x, ν ) + η . (4.3)Now let v k ∈ A δ (¯ u νx,ε k , Q νρ ( x )) be such that F sk ( v k , Q νρ ( x )) ≤ m s, δ k ( u νx , Q νρ ( x )) + η , (4.4)hence, in particular, v k ∇ u k = 0 a.e. in Q νρ ( x ), for some u k ∈ W ,p ( Q νρ ( x ); R m ); further( u k , v k ) = (¯ u νx,ε k , ¯ v νx,ε k ) in (cid:0) Q νρ ( x ) \ Q νρ − δ ( x ) (cid:1) . (4.5)Now let ( x j , ν j ) ⊂ R n × (cid:98) S n − be a sequence converging to ( x, ν ). Thanks to the continuity of themap ν (cid:55)→ R ν there exists ˆ = ˆ ( δ ) ∈ N such that for every j ≥ ˆ we have Q νρ − δ ( x ) ⊂ Q ν j ρ − δ ( x j ) and Q ν j ρ − δ ( x j ) ⊂ Q νρ − δ ( x ) . (4.6)We now modify ( u k , v k ) to obtain a new pair ( (cid:101) u k , (cid:101) v k ) ∈ W ,p ( Q ν j ρ ( x j ); R m ) × W ,p ( Q ν j ρ ( x j )) with (cid:101) v k ∈ A δ (¯ u ν j x j ,ε k , Q ν j ρ ( x j )) for any j ≥ ˆ and any δ ∈ (0 , δ ). To this end, set h k,j := ε k + | x − x j | + √ n ρ | ν − ν j | and R k,j := R ν (cid:16)(cid:0) Q (cid:48) ρ − δ +2 ε k \ Q (cid:48) ρ − δ (cid:1) × (cid:0) − h k,j , h k,j (cid:1)(cid:17) + x , (see Figure 2). Upon possibly increasing ˆ , we can assume that R k,j ⊂ Q νρ − δ +2 ε k ( x ) \ Q νρ − δ ( x ),for every j ≥ ˆ . Thus, setting d k ( y ) := ε k dist( y, R k,j ) and using (4.6), for ε k < δ we also have { d k < } ⊂ Q νρ − δ ( x ) \ Q νρ − δ ( x ) ⊂ Q ν j ρ − δ ( x j ) \ Q ν j ρ − δ ( x j ) . (4.7)Now let ϕ k ∈ C ∞ c ( Q (cid:48) ρ − δ +2 ε k ) be a cut-off between Q (cid:48) ρ − δ and Q (cid:48) ρ − δ +2 ε k ; i.e. , 0 ≤ ϕ k ≤ ϕ k = 1 in Q (cid:48) ρ − δ . For y ∈ Q ν j ρ ( x j ) we set (cid:101) u k ( y ) := ϕ k (cid:0) ( R Tν ( y − x )) (cid:48) (cid:1) u k ( y ) + (cid:16) − ϕ k (cid:0) R Tν ( y − x )) (cid:48) (cid:1)(cid:17) ¯ u ν j x j ,ε k ( y )and (cid:101) v k ( y ) := (cid:40) min (cid:8) v k ( y ) , d k ( y ) (cid:9) in Q νρ − δ ( x ) , min (cid:8) ¯ v ν j x j ,ε k ( y ) , d k ( y ) (cid:9) in Q ν j ρ ( x j ) \ Q νρ − δ ( x ) . INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 17 ρρ − δε νν j xx j Figure 2.
The cubes Q νρ ( x ), Q νρ − δ ( x ), Q ν j ρ ( x j ), and in gray the sets R k,j (dark gray) and { d k < } (light gray). By construction, we have (cid:101) u k = u k , (cid:101) v k ≤ v k in Q νρ − δ ( x ) and (cid:101) u k = ¯ u ν j x j ,ε k , (cid:101) v k ≤ ¯ v ν j x j ,ε k in Q ν j ρ ( x j ) \ Q νρ − δ +2 ε k ( x ); therefore, in particular, it holds0 ≤ (cid:101) v k |∇ (cid:101) u k | ≤ v k |∇ u k | = 0 a.e. in Q νρ − δ ( x )0 ≤ (cid:101) v k |∇ (cid:101) u k | ≤ ¯ v ν j x j ,ε k |∇ ¯ u ν j x j ,ε k | = 0 a.e. in Q ν j ρ ( x j ) \ Q νρ − δ +2 ε k ( x ) . (4.8)Moreover, for y ∈ (cid:16) Q νρ − δ +2 ε k ( x ) \ Q νρ − δ ( x ) (cid:17) \ R k,j we have | ( y − x ) · ν | = | R Tν ( y − x ) · e n | > h k,j > ε k . (4.9)Then, by applying the triangular inequality twice, also noticing that Q νρ ( x ) ⊂ B √ n ρ ( x ), we obtain | ( y − x j ) · ν j | > h k,j − | x − x j | − | y − x || ν − ν j | ≥ ε k . (4.10)In view of (4.5), gathering (4.9) and (4.10) we infer that u k = ¯ u νx,ε k = ¯ u ν j x j ,ε k , v k = ¯ v νx,ε k = v ν j x j ,ε k on (cid:16) Q νρ − δ +2 ε k ( x ) \ Q νρ − δ ( x ) (cid:17) \ R k,j . Hence (cid:101) v k ∈ W ,p ( Q ν j ρ ( x j )) and ∇ (cid:101) u k = 0 in (cid:0) Q νρ − δ +2 ε k ( x ) \ Q νρ − δ ( x ) (cid:1) \ R k,j . Since moreover (cid:101) v k = 0 in R k,j , we immediately obtain that (cid:101) v k ∇ (cid:101) u k = 0 in Q νρ − δ +2 ε k ( x ) \ Q νρ − δ ( x ), which inview of (4.8) yields (cid:101) v k ∇ (cid:101) u k = 0 a.e. in Q ν j ρ ( x j ) . Eventually, since in Q ν j ρ ( x j ) \ Q νρ − δ ( x ) we have (cid:101) v k = ¯ v ν j x j ,ε k if d k ≥
1, from (4.7) we deducethat ( (cid:101) u k , (cid:101) v k ) = (¯ u ν j x j ,ε k , ¯ v ν j x j ,ε k ) in Q ν j ρ ( x j ) \ Q ν j ρ − δ ( x j ), hence also in Q ν j ρ ( x j ) \ Q ν j ρ − δ ( x j ) for every δ ∈ (0 , δ ). In particular, (cid:101) v k ∈ A δ (¯ u ν j x j ,ε k , Q ν j ρ ( x j )) for any δ ∈ (0 , δ ). Now it only remains toestimate F sk ( (cid:101) v k , Q ν j ρ ( x j )). Arguing as in Proposition 2.6, we get F sk ( (cid:101) v k , Q ν j ρ ( x j )) ≤ F sk ( v k , Q νρ ( x )) + F sk (¯ v ν j x j ,ε k , Q ν j ρ ( x j ) \ Q νρ − δ ( x )) + 2 c ε k L n ( { d k < } ) . (4.11) Moreover, the second inclusion in (4.6) together with (2.3) and (2.19) implies that F sk (¯ v ν j x j ,ε k , Q ν j ρ ( x j ) \ Q νρ − δ ( x )) ≤ F sk (¯ v ν j x j ,ε k , Q ν j ρ ( x j ) \ Q ν j ρ − δ ( x j )) ≤ c C v L n − (cid:16) Q (cid:48) ρ \ Q (cid:48) ρ − δ (cid:1)(cid:17) ≤ Cδ ρ n − , (4.12)also 2 c ε k L n (cid:0) { d k < } (cid:1) ≤ c ε k L n − (cid:0) Q (cid:48) ρ − δ +4 ε k \ Q (cid:48) ρ − δ − ε k (cid:1) ( h k,j + ε k ) ≤ Ch k,j ( ρ − δ ) n − + O ( ε k ) , (4.13)as k → + ∞ .Since (cid:101) v k ∈ A δ (¯ u ν j x j ,ε k , Q ν j ρ ( x j )) for any δ ∈ (0 , δ ), by combining (4.4) and (4.11)-(4.13), we get m s,δk (¯ u ν j x j ,ε k , Q ν j ρ ( x j )) ≤ m s, δ k (¯ u νx,ε k , Q νρ ( x )) + η + C ( δ + h k,j ) ρ n − + O ( ε k ) , (4.14)as k → + ∞ . Using (4.3) and taking in (4.14) first the liminf as k → + ∞ , then the limit as δ → j → + ∞ giveslim sup j → + ∞ g (cid:48) ρ ( x j , ν j ) ≤ g (cid:48) ρ ( x, ν ) + 2 η + Cδ ρ n − , since lim j lim k h k,j = 0. Therefore, letting δ → g (cid:48) ρ restricted to (cid:98) S n − × R n follows by the arbitrariness of η > (cid:3) Lemma 4.4.
Let g (cid:48) and g (cid:48)(cid:48) be as in (2.14) and (2.15) , respectively, and let g (cid:48) ρ , g (cid:48)(cid:48) ρ be as in (4.1) .Then for every x ∈ R n and every ν ∈ S n − it holds g (cid:48) ( x, ν ) = lim sup ρ → ρ n − g (cid:48) ρ ( x, ν ) and g (cid:48)(cid:48) ( x, ν ) = lim sup ρ → ρ n − g (cid:48)(cid:48) ρ ( x, ν ) . Proof.
We prove the statement only for g (cid:48) , the proof for g (cid:48)(cid:48) being analogous. We notice that since A δ (¯ u νx,ε k , Q νρ ( x )) ⊂ A (¯ u νx,ε k , Q νρ ( x )) for every δ ∈ (0 , ρ ), we clearly have g (cid:48) ( x, ν ) ≤ lim sup ρ → ρ n − g (cid:48) ρ ( x, ν ) , therefore to conclude we just need to prove the opposite inequality. This can be done by means ofan easy extension argument as follows. For fixed ρ > x ∈ R n , and ν ∈ S n − and for every k ∈ N such that ε k ∈ (0 , ρ ) let v k ∈ A (¯ u νx,ε k , Q νρ ( x )) satisfy F sk ( v k , Q νρ ( x )) ≤ m sk (¯ u νx,ε k , Q νρ ( x )) + ρ n , (4.15)and let u k ∈ W ,p ( Q νρ ( x ); R m ) be the corresponding u -variable. Let α > u k , v k ) we can extend the pair ( u k , v k ) to Q ν (1+ α ) ρ ( x ) bysetting ( u k , v k ) := (¯ u νx,ε k , ¯ v νx,ε k ) in Q ν (1+ α ) ρ ( x ) \ Q νρ ( x ). Then (2.3) and (2.19) yield F sk ( v k , Q ν (1+ α ) ρ ( x )) ≤ F sk ( v k , Q νρ ( x )) + F sk (cid:0) ¯ v νx,ε k , ( Q ν (1+ α ) ρ ( x ) \ Q νρ ( x ) (cid:1) ≤ F sk ( v k , Q νρ ( x )) + c C v ((1 + α ) n − − ρ n − . (4.16)Moreover, for δ ∈ (0 , αρ ) we have v k ∈ A δ (¯ u νx,ε k , Q ν (1+ α ) ρ ( x )), thus (4.15) and (4.16) giveinf δ> lim inf k → + ∞ m s,δk (¯ u νx,ε k , Q ν (1+ α ) ρ ( x )) ≤ m sk (¯ u νx,ε k , Q νρ ( x )) + ρ n + c C v ((1 + α ) n − − ρ n − . INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 19
Hence, dividing the above inequality by ((1 + α ) ρ ) n − and taking the limsup as ρ →
0, thanks toProposition 2.6 we obtain(1 + α ) n − lim sup ρ → ρ n − g (cid:48) ρ ( x, ν ) ≤ g (cid:48) ( x, ν ) + c C v ((1 + α ) n − − , thus we conclude by the arbitrariness of α > (cid:3) We are now ready to state and prove the following proposition which establishes the propertiessatisfied by g (cid:48) and g (cid:48)(cid:48) . Proposition 4.5.
Let ( g k ) ⊂ G ; then the functions g (cid:48) and g (cid:48)(cid:48) defined, respectively, as in (2.14) and (2.15) are Borel measurable and satisfy the following two properties:(1) (symmetry) for every x ∈ R n and every ν ∈ S n − it holds g (cid:48) ( x, ν ) = g (cid:48) ( x, − ν ) , g (cid:48)(cid:48) ( x, ν ) = g (cid:48)(cid:48) ( x, − ν ); (4.17) (2) (boundedness) for every x ∈ R n and every ν ∈ S n − it holds c c p ≤ g (cid:48) ( x, ν ) ≤ c c p , c c p ≤ g (cid:48)(cid:48) ( x, ν ) ≤ c c p , (4.18) where c p := 2( p − − pp .Proof. We prove the statement only for g (cid:48) , the proof for g (cid:48)(cid:48) being analogous.We divide the proof into three steps. Step 1: g (cid:48) is Borel measurable. Let ρ > g (cid:48) ρ be the function defined in (4.1). Arguing asin the proof of Lemma 4.4 we deduce that the function ρ → g (cid:48) ρ ( x, ν ) − c C v ρ n − is nonincreasingon (0 , + ∞ ). From this it follows thatlim ρ (cid:48) → ρ − g (cid:48) ρ (cid:48) ( x, ν ) ≥ g (cid:48) ρ ( x, ν ) ≥ lim ρ (cid:48) → ρ + g (cid:48) ρ (cid:48) ( x, ν ) , for every x ∈ R n , ν ∈ S n − , and every ρ >
0. Thus, if D is a countable dense subset of (0 , + ∞ )we have lim sup ρ → ρ n − g (cid:48) ρ ( x, ν ) = lim sup ρ → ρ ∈ D ρ n − g (cid:48) ρ ( x, ν )and hence by Lemma 4.4 we get g (cid:48) ( x, ν ) = lim sup ρ → ρ ∈ D ρ n − g (cid:48) ρ ( x, ν ) . Therefore the Borel measurablility of g (cid:48) follows by Lemma 4.3 which guarantees, in particular, thatthe function ( x, ν ) (cid:55)→ g (cid:48) ρ ( x, ν ) is Borel measurable for every ρ > Step 2: g (cid:48) is symmetric in ν . Property (4.17) immediately follows from the definition of g (cid:48) andfrom the fact that u νx = − u − νx + e a.e. and Q νρ ( x ) = Q − νρ ( x ) (see (f)), which implies in particularthat v ∈ A ε k ,ρ ( x, ν ) with corresponding u ∈ W ,p ( Q νρ ( x ); R m ) if and only if v ∈ A ε k ,ρ ( x, − ν ) withcorresponding w := − u + e ∈ W ,p ( Q − νρ ( x ); R m ). Step 3: g (cid:48) is bounded. To prove that g (cid:48) satisfies the bounds in (4.18) we start by observing thatthanks to ( g
2) and ( g
3) we have c (cid:90) Q νρ ( x ) (cid:18) (1 − v ) p ε k + ε p − k |∇ v | p (cid:19) d x ≤ F sk ( v, Q νρ ( x )) ≤ c (cid:90) Q νρ ( x ) (cid:18) (1 − v ) p ε k + ε p − k |∇ v | p (cid:19) d x, for every v ∈ W ,p ( Q νρ ( x )) with 0 ≤ v ≤ Q νρ ( x ). Therefore to establish (4.18) it is enoughto show that lim k → + ∞ m k,ρ ( x, ν ) = c p ρ n − , (4.19)for every x ∈ R n and ρ >
0, where m k,ρ ( x, ν ) := min (cid:26) (cid:90) Q νρ ( x ) (cid:18) (1 − v ) p ε k + ε p − k |∇ v | p (cid:19) d y : v ∈ A ε k ,ρ ( x, ν ) (cid:27) . Let x ∈ R n , ν ∈ S n − , ρ >
0, and let k ∈ N be such that 2 ε k < ρ . By the homogeneity androtation invariance of the Ambrosio-Tortorelli functional we have m k,ρ ( x, ν ) = m ε k ,ρ (0 , e n ) , Let η > u k , v k ) ⊂ W ,p ( Q ρ (0); R m ) × W ,p ( Q ρ (0)) satisfying v k ∇ u k = 0 a.e. in Q ρ (0), ( u k , v k ) = ( u e n ,
1) in {| y n | > ε k T η } for some T η > k → + ∞ (cid:90) Q ρ (0) (cid:18) (1 − v k ) p ε k + ε p − k |∇ v k | p (cid:19) d x = ( c p + η ) ρ n − . (4.20)Then, using a similar argument as in the proof of Proposition 2.6, we can modify v k to obtain afunction (cid:101) v k ∈ W ,p ( Q ρ (0)) satisfying (cid:101) v k = 1 in Q ρ (0) \ Q ρ − ε k (0) ∩ {| y n | > ε k } and (cid:90) Q ρ (0) (cid:18) (1 − (cid:101) v k ) p ε k + ε p − k |∇ (cid:101) v k | p (cid:19) d x ≤ (cid:90) Q ρ (0) (cid:18) (1 − v k ) p ε k + ε p − k |∇ v k | p (cid:19) d x + Cε k ρ n − . (4.21)In particular, since (cid:101) v k ∈ A ε k ,ρ (0 , e n ), gathering (4.20)–(4.21), by the arbitrariness of η > k → + ∞ m k,ρ (0 , e n ) ≤ c p ρ n − . (4.22)We now turn to the proof of the lower bound. By the Fubini Theorem we have (cid:90) Q ρ (0) (cid:18) (1 − v ) p ε k + ε p − k |∇ v | p (cid:19) d x = (cid:90) Q (cid:48) ρ (cid:90) ρ − ρ (cid:18) (1 − v ( x (cid:48) , x n )) p ε k + ε p − k |∇ v ( x (cid:48) , x n ) | p (cid:19) d x n d x (cid:48) ≥ (cid:90) Q (cid:48) ρ (cid:90) ρ − ρ (cid:18) (1 − v ( x (cid:48) , x n )) p ε k + ε p − k (cid:12)(cid:12)(cid:12) ∂v ( x (cid:48) , x n ) ∂x n (cid:12)(cid:12)(cid:12) p (cid:19) d x n d x (cid:48) . (4.23)Then, if v ∈ A ε k ,ρ (0 , e n ) the corresponding u coincides with u e n in a neighbourhood of ∂ ± Q ρ (0) := (cid:110) ( x (cid:48) , x n ) ∈ R n − × R : x (cid:48) ∈ Q (cid:48) ρ , x n = ± ρ (cid:111) . Since it must hold that v ∇ u = 0 a.e. in Q ρ (0), then almost every straight line intersecting ∂ ± Q ρ (0)and parallel to e n also intersects the level set { v = 0 } . Indeed, for L n − -a.e. x (cid:48) ∈ Q (cid:48) ρ the pair( u x (cid:48) ( t ) , v x (cid:48) ( t )) := (cid:0) u ( x (cid:48) , t ) · e , v ( x (cid:48) , t ) (cid:1) belongs to W ,p ( − ρ , ρ ) × W ,p ( − ρ , ρ ) and satisfies v x (cid:48) ( t ) u (cid:48) x (cid:48) ( t ) = v ( x (cid:48) , t ) ∂u ( x (cid:48) , t ) ∂x n · e = 0 for L -a.e. t ∈ (cid:16) − ρ , ρ (cid:17) , (4.24)as well as v x (cid:48) ( ± ρ ) = 1, u x (cid:48) ( − ρ ) = 0, and u x (cid:48) ( ρ ) = 1. Since u x (cid:48) ∈ W ,p ( − ρ , ρ ), the boundaryconditions satisified by u x (cid:48) imply the existence of a subset of ( − ρ , ρ ) with positive L -measure onwhich u (cid:48) x (cid:48) (cid:54) = 0, hence v x (cid:48) = 0 in view of (4.24). In particular, for L n − -a.e. x (cid:48) ∈ Q (cid:48) ρ there exists s ∈ ( − ρ , ρ ) such that v ( x (cid:48) , s ) = 0. INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 21
Therefore, the Young Inequality(1 − v ( x (cid:48) , x n )) p ε k + ε p − k (cid:12)(cid:12)(cid:12)(cid:12) ∂v ( x (cid:48) , x n ) ∂x n (cid:12)(cid:12)(cid:12)(cid:12) p ≥ (cid:16) pp − (cid:17) p − p p p (1 − v ( x (cid:48) , x n )) p − (cid:12)(cid:12)(cid:12)(cid:12) ∂v ( x (cid:48) , x n ) ∂x n (cid:12)(cid:12)(cid:12)(cid:12) , applied for L n − -a.e. x (cid:48) ∈ Q (cid:48) ρ together with the integration on ( − ρ , ρ ) give (cid:90) ρ − ρ (cid:18) (1 − v ( x (cid:48) , x n )) p ε k + ε p − k (cid:12)(cid:12)(cid:12)(cid:12) ∂v ( x (cid:48) , x n ) ∂x n (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) d x n = (cid:90) s − ρ (cid:18) (1 − v ( x (cid:48) , x n )) p ε k + ε p − k (cid:12)(cid:12)(cid:12)(cid:12) ∂v ( x (cid:48) , x n ) ∂x n (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) d x n + (cid:90) ρ s (cid:18) (1 − v ( x (cid:48) , x n )) p ε k + ε p − k (cid:12)(cid:12)(cid:12)(cid:12) ∂v ( x (cid:48) , x n ) ∂x n (cid:12)(cid:12)(cid:12)(cid:12) p (cid:19) d x n ≥ (cid:16) pp − (cid:17) p − p p p (cid:90) (1 − t ) p − d t = c p , (4.25)for L n − -a.e. x (cid:48) ∈ Q (cid:48) ρ . Thus, gathering (4.23) and (4.25) we get (cid:90) Q ρ (0) (cid:18) (1 − v ) p ε k + ε p − k |∇ v | p (cid:19) d x ≥ c p ρ n − , for every v ∈ A ε k ,ρ (0 , e n ). Passing to the infimum on v and to the liminf as k → + ∞ we getlim inf k → + ∞ m k,ρ (0 , e n ) ≥ c p ρ n − , (4.26)for every ρ >
0. Eventually, by combining (4.22) and (4.26) we get (4.19), and hence (4.18). (cid:3)
5. Γ -convergence and integral representation
In this section we show that, up to subsequences, the functionals F k Γ-converge in L ( R n ; R m ) × L ( R n ) to an integral functional of free-discontinuity type. This result is achieved by followinga standard procedure which combines the localisation method of Γ-convergence (see e.g. , [32,Chapters 14-18] or [23, Chapters 10, 11]) together with an integral-representation result in SBV [17, Theorem 1]. Though rather technical, this procedure is by now classical. For this reason herewe only detail the adaptations of the theory to our specific setting, while we refer the reader tothe literature for the more standard aspects.We start by showing that the functionals F k satisfy the so-called fundamental estimate, uni-formly in k . Proposition 5.1 (Fundamental estimate) . Let F k be as in (2.4) . Then, for every η > and forevery A, A (cid:48) , B ∈ A with A ⊂⊂ A (cid:48) there exists a constant M η > (also depending on A, A (cid:48) , B )satisfying the following property: For every k ∈ N and for every ( u, v ) ∈ W ,p ( A (cid:48) ; R m ) × W ,p ( A (cid:48) ) , ( (cid:101) u, (cid:101) v ) ∈ W ,p ( B ; R m ) × W ,p ( B ) , ≤ v, (cid:101) v ≤ , there exists a pair (ˆ u, ˆ v ) ∈ W ,p ( A ∪ B ; R m ) × W ,p ( A ∪ B ) with ≤ ˆ v ≤ such that (ˆ u, ˆ v ) = ( u, v ) a.e. in A , (ˆ u, ˆ v ) = ( (cid:101) u, (cid:101) v ) a.e. in B \ A (cid:48) and F k (ˆ u, ˆ v, A ∪ B ) ≤ (1 + η ) (cid:0) F k ( u, v, A (cid:48) ) + F k ( (cid:101) u, (cid:101) v, B ) (cid:1) + M η (cid:0) (cid:107) u − (cid:101) u (cid:107) pL p ( S ; R m ) + ε p − k (cid:1) , (5.1) where S := ( A (cid:48) \ A ) ∩ B .Proof. Fix k ∈ N , η > A, A (cid:48) , B ∈ A with A ⊂⊂ A (cid:48) . Let N ∈ N and A , . . . , A N +1 ∈ A with A ⊂⊂ A ⊂⊂ . . . ⊂⊂ A N +1 ⊂⊂ A (cid:48) . For each i = 2 , . . . , N + 1 let ϕ i be a smooth cut-off function between A i − and A i and let M := max ≤ i ≤ N +1 (cid:107)∇ ϕ i (cid:107) ∞ . Let ( u, v ) and ( (cid:101) u, (cid:101) v ) be as in the statement and consider the function w ∈ L ( R n ) defined by w := min { v, (cid:101) v } , clearly 0 ≤ w ≤
1. For i = 3 , . . . , N we define (ˆ u i , ˆ v i ) ∈ W ,p ( A ∪ B ; R m ) × W ,p ( A ∪ B ) as followsˆ u i := ϕ i u + (1 − ϕ i ) (cid:101) u and ˆ v i := ϕ i − v + (1 − ϕ i − ) w in A i − ,w in A i \ A i − ,ϕ i +1 w + (1 − ϕ i +1 ) (cid:101) v in R n \ A i . Then, setting S i := A i \ A i − and taking into account the definition of (ˆ u i , ˆ v i ) we have F k (ˆ u i , ˆ v i , A ∪ B ) ≤ F k ( u, v, A i − ) + F k ( u, ˆ v i , S i − ∩ B ) + F k (ˆ u i , w, S i ∩ B )+ F k ( (cid:101) u, ˆ v i , S i +1 ∩ B ) + F k ( (cid:101) u, (cid:101) v, B \ A i +1 ) . (5.2)We now come to estimate the three terms in (5.2) involving the sets S i − , S i , and S i +1 . Westart observing that thanks to ( f
3) and ( f w and the fact that ψ isincreasing, we have F bk ( u, ˆ v i , S i − ∩ B ) ≤ c (cid:90) S i − ∩ B ψ ( v ) |∇ u | p d x ≤ c c F bk ( u, v, S i − ∩ B ) . (5.3)Analogously, there holds F bk ( (cid:101) u, ˆ v i , S i +1 ∩ B ) ≤ c c F bk ( (cid:101) u, (cid:101) v, S i +1 ∩ B ) . (5.4)We complete the estimate of the bulk part of the energy by noticing that on S i ∩ B we have |∇ ˆ u i | p ≤ p − (cid:0) |∇ ϕ i | p | u − (cid:101) u | p + |∇ u | p + |∇ (cid:101) u | p (cid:1) ≤ p − (cid:0) M p | u − (cid:101) u | p + |∇ u | p + |∇ (cid:101) u | p (cid:1) . Integrating over S i ∩ B , using ( f
3) and ( f w , and the monotonicity of ψ , weinfer F bk (ˆ u i , w, S i ∩ B ) ≤ c (cid:90) S i ∩ B ψ ( w ) |∇ ˆ u i | p d x ≤ c p − (cid:90) S i ∩ B (cid:0) ψ ( v ) |∇ u | p + ψ ( (cid:101) v ) |∇ (cid:101) u | p (cid:1) d x + 3 p − M p c (cid:90) S i ∩ B | u − (cid:101) u | p d x ≤ p − c c (cid:0) F bk ( u, v, S i ∩ B ) + F bk ( (cid:101) u, (cid:101) v, S i ∩ B ) (cid:1) + 3 p − M p c (cid:90) S i ∩ B | u − (cid:101) u | p d x . (5.5)It remains to estimate the surface term in F k . Thanks to ( g
3) it holds F sk ( w, S i ∩ B ) ≤ c (cid:90) S i ∩ B (cid:18) (1 − w ) p ε k + ε p − k |∇ w | p (cid:19) d x . (5.6)We now want to bound the right-hand side of (5.6) in terms of F sk ( v, S i ∩ B ) + F sk ( (cid:101) v, S i ∩ B ). Tothis end we first observe that by definition of w we have |∇ w | p ≤ |∇ v | p + |∇ (cid:101) v | p and (1 − w ) p ≤ (1 − v ) p + (1 − (cid:101) v ) p . (5.7)Thus, thanks to ( g F sk ( w, S i ∩ B ) ≤ c c (cid:0) F sk ( v, S i ∩ B ) + F sk ( (cid:101) v, S i ∩ B ) (cid:1) . (5.8)Moreover, it holds F sk (ˆ v i , ( S i − ∪ S i +1 ) ∩ B ) ≤ c (cid:90) ( S i − ∪ S i +1 ) ∩ B (cid:18) (1 − ˆ v i ) p ε k + ε p − k |∇ ˆ v i | p (cid:19) d x . (5.9) INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 23
By the definition of ˆ v i and by the convexity of z (cid:55)→ (1 − z ) p for z ∈ [0 , S i − ∩ B we have(1 − ˆ v i ) p = ( ϕ i − (1 − v ) + (1 − ϕ i − )(1 − w )) p ≤ (1 − v ) p + (1 − w ) p ≤ (cid:0) (1 − v ) p + (1 − (cid:101) v ) p (cid:1) , where in the last step we used again (5.7). Similarly, there holds |∇ ˆ v i | p ≤ p − (cid:0) |∇ ϕ i | p | v − w | p + |∇ v | p + |∇ w | p (cid:1) ≤ p − (cid:0) M p | v − (cid:101) v | p + 2( |∇ v | p + |∇ (cid:101) v | p ) (cid:1) . Since analogous arguments hold on S i +1 ∩ B , from (5.9) and ( g
2) we deduce F sk (ˆ v i , ( S i − ∪ S i +1 ) ∩ B ) ≤ p − c c (cid:0) F sk ( v, ( S i − ∪ S i +1 ) ∩ B ) + F sk ( (cid:101) v, ( S i − ∪ S i +1 ) ∩ B ) (cid:1) + 3 p − M p c (cid:90) ( S i − ∪ S i +1 ) ∩ B ε p − k | v − (cid:101) v | p d x . (5.10)Now set (cid:99) M := (cid:0) (1+3 p − ) c c + (1+3 p − c c (cid:1) ; then, summing up in (5.2) over all i , gathering (5.3)–(5.5),(5.8), and (5.10), by averaging we find an index i ∗ ∈ { , . . . , N } such that F k (ˆ u i ∗ , ˆ v i ∗ , A ∪ B ) ≤ N − N (cid:88) i =3 F k (ˆ u i , ˆ v i , A ∪ B ) ≤ (cid:16) (cid:99) MN − (cid:17)(cid:0) F k ( u, v, A (cid:48) ) + F k ( (cid:101) u, (cid:101) v, B ) (cid:1) + 3 p − M p c N − (cid:90) S | u − (cid:101) u | p d x + 3 p − M p c N − (cid:90) S ε p − k | v − (cid:101) v | p d x . Thus, upon choosing N large enough so that (cid:99) MN − < η , since 0 ≤ v, (cid:101) v ≤ v, ˆ v ) := (ˆ u i ∗ , ˆ v i ∗ ) and M η := p − M p N − (cid:0) c + 2 c L n ( S ) (cid:1) . (cid:3) On account of the fundamental estimate, Proposition 5.1, we are now in a position to prove thefollowing Γ-convergence result.
Theorem 5.2.
Let F k be as in (2.4) . Then there exist a subsequence ( F k j ) of ( F k ) and afunctional F : L ( R n ; R m ) × L ( R n ) × A −→ [0 , + ∞ ] such that for every A ∈ A the functionals F k j ( · , · , A ) Γ -converge in L ( R n ; R m ) × L ( R n ) to F ( · , · , A ) . Moreover, F is given by F ( u, v, A ) := (cid:90) A ˆ f ( x, ∇ u ) d x + (cid:90) S u ∩ A ˆ g ( x, [ u ] , ν u ) d H n − if u ∈ GSBV p ( A ; R m ) , v = 1 a.e. in A , + ∞ otherwise , with ˆ f : R n × R m × n → [0 , + ∞ ) , ˆ g : R n × R m × S n − → [0 , + ∞ ) given by ˆ f ( x, ξ ) := lim sup ρ → ρ n m ( u ξ , Q ρ ( x )) , (5.11)ˆ g ( x, ν, ζ ) := lim sup ρ → ρ n − m ( u νx,ζ , Q νρ ( x )) , (5.12) for every x ∈ R n , ξ ∈ R m × n , ζ ∈ R m , and ν ∈ S n − , where for A ∈ A and ¯ u ∈ SBV p ( A ; R m ) m (¯ u, A ) := inf { F ( u, , A ) : u ∈ SBV p ( A ; R m ) , u = ¯ u near ∂A } . Proof.
Let F (cid:48) , F (cid:48)(cid:48) : L ( R n ; R m ) × L ( R n ) × A −→ [0 , + ∞ ] be the functionals defined as F (cid:48) ( · , · , A ) := Γ- lim inf k → + ∞ F k ( · , · , A ) and F (cid:48)(cid:48) ( · , · , A ) := Γ- lim sup k → + ∞ F k ( · , · , A ) . In view of Remark 2.2 we can invoke [36, Theorem 3.1] to deduce the existence of a constant
C > C (cid:16) (cid:90) A |∇ u | p d x + H n − ( S u ∩ A ) (cid:17) ≤ F (cid:48) ( u, , A ) ≤ F (cid:48)(cid:48) ( u, , A ) ≤ C (cid:16) (cid:90) A |∇ u | p d x + H n − ( S u ∩ A ) (cid:17) , (5.13)for every A ∈ A and every u ∈ GSBV p ( A ; R m ); moreover F (cid:48) ( u, v, A ) = F (cid:48)(cid:48) ( u, v, A ) = + ∞ if either u / ∈ GSBV p ( A ; R m ) or v (cid:54) = 1 . (5.14)By the general properties of Γ-convergence we know that for every A ∈ A fixed the functionals F (cid:48) ( · , · , A ) and F (cid:48)(cid:48) ( · , · , A ) are L ( R n ; R m ) × L ( R n ) lower semicontinuous [32, Proposition 6.8] andlocal [32, Proposition 16.15]. Further, the set functions F (cid:48) ( u, v, · ) and F (cid:48)(cid:48) ( u, v, · ) are increasing[32, Proposition 6.7] and F (cid:48) ( u, v, · ) is superadditive [32, Proposition 16.12].Invoking [32, Theorem 16.9] we can deduce the existence of a subsequence ( k j ), with k j → + ∞ as j → + ∞ , such that the corresponding F (cid:48) and F (cid:48)(cid:48) also satisfysup A (cid:48) ⊂⊂ A, A (cid:48) ∈A F (cid:48) ( u, v, A (cid:48) ) = sup A (cid:48) ⊂⊂ A, A (cid:48) ∈A F (cid:48)(cid:48) ( u, v, A (cid:48) ) =: F ( u, v, A ) , (5.15)for every ( u, v ) ∈ L ( R n ; R m ) × L ( R n ) and for every A ∈ A . We notice that the set function F ( u, v, · ) given by (5.15) is inner regular by definition. Moreover F satisfies the following prop-erties: the functional F ( · , · , A ) is L ( R n ; R m ) × L ( R n ) lower semicontinuous [32, Remark 15.10]and local [32, Remark 15.25], while the set function F ( u, v, · ) is increasing and superadditive [32,Remark 15.10].Thanks to the fundamental estimate Proposition 5.1 we can appeal to [32, Proposition 18.4] todeduce that F ( u, v, · ) is also a subadditive set function. Here the only difference with a standardsituation is that the reminder in (5.1) is infinitesimal with respect to the L p ( R n ; R m ) convergencein u while we are considering the Γ-convergence of F k j in L ( R n ; R m ) × L ( R n ). However, thisissue can be easily overcome by resorting to a truncation argument together with a sequentialcharacterisation of F (see e.g. , [32, Proposition 16.4 and Remark 16.5]), which holds true on SBV p ( A ; R m ) ∩ L ∞ ( A ; R m ). Hence, we can now invoke the measure-property criterion of De Giorgiand Letta (see e.g. , [32, Theorem 14.23]) to deduce that for every ( u, v ) ∈ L ( R n ; R m ) × L ( R n )the set function F ( u, v, · ) is the restriction to A of a Borel measure.Furthermore, (5.13) together with [32, Proposition 18.6] and Proposition 5.1 imply that F ( u, , A ) = F (cid:48) ( u, , A ) = F (cid:48)(cid:48) ( u, , A ) if u ∈ GSBV p ( A ; R m ) , while, gathering (5.13) and (5.14) we may also deduce that F ( u, v, A ) = F (cid:48) ( u, v, A ) = F (cid:48)(cid:48) ( u, v, A ) = + ∞ if either u / ∈ GSBV p ( A ; R m ) or v (cid:54) = 1 . As a consequence, F ( · , · , A ) coincides with the Γ-limit of F k j ( · , · , A ) on L ( R n ; R m ) × L ( R n ), forevery A ∈ A .By [17, Theorem 1] and a standard perturbation and truncation argument (see e.g. , [27, Theorem4.3]), for every A ∈ A and u ∈ GSBV p ( A ; R m ) we can represent the Γ-limit F in an integral formas F ( u, , A ) = (cid:90) A ˆ f ( x, ∇ u ) d x + (cid:90) S u ˆ g ( x, [ u ] , ν u ) d H n − , for some Borel functions ˆ f and ˆ g . Eventually, thanks to (5.13), it can be easily shown that ˆ f andˆ g are given by the same derivation formulas provided by [17, Theorem 1], that is, they coincidewith (5.11) and (5.12), respectively. (cid:3) INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 25 Identification of the volume integrand
In this section we identify the volume integrand ˆ f . Namely, we prove that ˆ f coincides with both f (cid:48) and f (cid:48)(cid:48) , given by (2.12) and (2.13), respectively. This shows, in particular, that the limit volumeintegrand ˆ f depends only on f k , and hence only on F bk . Proposition 6.1.
Let ( f k ) ⊂ F and ( g k ) ⊂ G . Let ( k j ) and ˆ f be as in Theorem 5.2. Then it holds ˆ f ( x, ξ ) = f (cid:48) ( x, ξ ) = f (cid:48)(cid:48) ( x, ξ ) , for a.e. x ∈ R n and for every ξ ∈ R m × n , where f (cid:48) and f (cid:48)(cid:48) are, respectively, as in (2.12) and (2.13) with k replaced by k j .Proof. For notational simplicity, in what follows we still let k denote the index of the sequenceprovided by Theorem 5.2.By definition f (cid:48) ≤ f (cid:48)(cid:48) , hence to prove the claim it suffices to show that f (cid:48)(cid:48) ( x, ξ ) ≤ ˆ f ( x, ξ ) ≤ f (cid:48) ( x, ξ ) , (6.1)for a.e. x ∈ R n and for every ξ ∈ R m × n . We divide the proof of (6.1) into two steps. Step 1:
In this step we show that ˆ f ( x, ξ ) ≥ f (cid:48)(cid:48) ( x, ξ ) for a.e. x ∈ R n and for every ξ ∈ R m × n .By Theorem 5.2 we have that (cid:90) A ˆ f ( x, ξ ) d x = F ( u ξ , , A ) , (6.2)for every A ∈ A and for every ξ ∈ R n .Now let x ∈ R n be arbitrary, let A ∈ A be such that x ∈ A , and let ρ > Q ρ ( x ) ⊂ A . By Γ-convergence we can find ( u k , v k ) ⊂ L ( R n ; R m ) × L ( R n ) which is a recoverysequence for F ( u ξ , , A ) ; i.e. , ( u k , v k ) ⊂ W ,p ( A ; R m ) × W ,p ( A ), 0 ≤ v ≤
1, ( u k , v k ) converges to( u ξ ,
1) in measure on bounded sets andlim k → + ∞ F k ( u k , v k , A ) = F ( u ξ , , A ) . (6.3)Moreover, by ( g
2) we also have v k → L p ( A ).We notice that ( u k , v k ) also satisfieslim k → + ∞ F k ( u k , v k , Q ρ ( x )) = F ( u ξ , , Q ρ ( x )) . (6.4)Indeed, thanks to (6.2) we have F ( u ξ , , A ) = F ( u ξ , , Q ρ ( x )) + F ( u ξ , , A \ Q ρ ( x )) . Therefore, again by Γ-convergence we getlim inf k → + ∞ F k ( u k , v k , Q ρ ( x )) ≥ F ( u ξ , , Q ρ ( x ))and lim inf k → + ∞ F k ( u k , v k , A \ Q ρ ( x )) ≥ F ( u ξ , , A \ Q ρ ( x )) . Hence (6.4) follows by (6.3).We now estimate separately the surface and bulk term in F k . We notice that by Young’sInequality we have p − p (1 − v k ) p ε k + 1 p ε p − k |∇ v k | p ≥ (1 − v k ) p − |∇ v k | . Hence, thanks to ( g F sk ( v k , Q ρ ( x )) ≥ c (cid:90) Q ρ ( x ) (1 − v k ) p − |∇ v k | d y = c (cid:90) (1 − t ) p − H n − ( ∂ ∗ E tk,ρ ) d t , where E tk,ρ := { y ∈ Q ρ ( x ) : v k ( y ) < t } . Now let η ∈ (0 ,
1) be fixed, then by the mean-valueTheorem we deduce the existence of ¯ t = ¯ t ( k, ρ, η ), ¯ t ∈ ( η,
1) such that F sk ( v k , Q ρ ( x )) ≥ c (cid:90) η (1 − t ) p − d t H n − ( ∂ ∗ E ¯ tk,ρ ) . Set w k := u k χ R n \ E ¯ tk,ρ ; since u k ∈ W ,p ( A ; R m ) and E ¯ tk,ρ is a set of finite perimeter, we have that w k ∈ GSBV p ( A ; R m ) and H n − ( S w k ) ≤ H n − ( ∂ ∗ E ¯ tk,ρ ). Moreover, since v k → L p ( A ) we havethat L n ( E ¯ tk,ρ ) → k → + ∞ so that w k → u ξ in measure on bounded sets. Eventually, since F k is increasing as set function, using the fact that ψ is increasing we obtain F k ( u k , v k , Q ρ ( x )) = F bk ( u k , v k , Q ρ ( x )) + F sk ( v k , Q ρ ( x )) ≥ ψ ( η ) (cid:90) Q ρ ( x ) \ E ¯ tk,ρ f k ( y, ∇ u k ) d y + c (cid:90) η (1 − t ) p − d t H n − ( ∂ ∗ E ¯ tk,ρ ) ≥ ψ ( η ) (cid:90) Q ρ ( x ) f k ( y, ∇ w k ) d y + c (cid:90) η (1 − t ) p − d t H n − ( S w k ) , where the last inequality follows from the definition of w k .In view of ( f f
4) the right-hand side belongs to the class of functionals considered in [27].Then, thanks to [27, Theorem 3.5 and Theorem 5.2 (b)] we havelim k → + ∞ F k ( u k , v k , Q ρ ( x )) ≥ ψ ( η ) (cid:90) Q ρ ( x ) f (cid:48)(cid:48) ( y, ξ ) d y , where f (cid:48)(cid:48) is given by (2.13).Hence appealing to (6.4) gives F ( u ξ , , Q ρ ( x )) ≥ ψ ( η ) (cid:90) Q ρ ( x ) f (cid:48)(cid:48) ( y, ξ ) d y . Then, by dividing both terms in the above inequality by ρ n and using (6.2), we obtain1 ρ n (cid:90) Q ρ ( x ) ˆ f ( y, ξ ) d y ≥ ψ ( η ) 1 ρ n (cid:90) Q ρ ( x ) f (cid:48)(cid:48) ( y, ξ ) d y . Thus invoking the Lebesgue differentiation Theorem together with the continuity in ξ of ˆ f and f (cid:48)(cid:48) (see [17] and Proposition 4.1) we deduce thatˆ f ( x, ξ ) ≥ ψ ( η ) f (cid:48)(cid:48) ( x, ξ ) , for a.e. x ∈ R n , for every ξ ∈ R m × n , and for every η ∈ (0 , ψ is continuousand ψ (1) = 1, the claim follows by letting η → Step 2:
In this step we show that ˆ f ( x, ξ ) ≤ f (cid:48) ( x, ξ ) for every x ∈ R n and for every ξ ∈ R m × n .The proof is similar to that of [27, Theorem 5.2]. However, we repeat it here for the readers’convenience.Let x ∈ R n , ξ ∈ R m × n , ρ >
0, and η > k ∈ N fixed we can find u k ∈ W ,p ( Q ρ ( x ); R m ) with u k = u ξ near ∂Q ρ ( x ) such that F bk ( u k , , Q ρ ( x )) ≤ m bk ( u ξ , Q ρ ( x )) + ηρ n . (6.5) INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 27
Combining (6.5) with ( f
2) and ( f
3) yields c (cid:107)∇ u k (cid:107) pL p ( Q ρ ( x ); R m × n ) ≤ F bk ( u k , , Q ρ ( x )) ≤ ρ n ( c | ξ | p + η ) , (6.6)where the second inequality follows by taking u ξ as a test in the definition of m bk ( u ξ , Q ρ ( x )). Letnow ( k j ) be a diverging sequence such thatlim j → + ∞ F bk j ( u k j , , Q ρ ( x )) = lim inf k → + ∞ F bk ( u k , , Q ρ ( x )) . Since u k j − u ξ ∈ W ,p ( Q ρ ( x ); R m ), the uniform bound (6.6) together with the Poincar´e Inequal-ity provide us with a further subsequence (not relabelled) and a function u ∈ W ,p ( Q ρ ( x ); R m )such that u k j (cid:42) u weakly in W ,p ( Q ρ ( x ); R m ). Then, by the Rellich Theorem u k j → u in L p ( Q ρ ( x ); R m ). We now extend u and u k to functions w, w k ∈ W ,p loc ( R n ; R m ) by setting w := (cid:40) u in Q ρ ( x ) ,u ξ in R n \ Q ρ ( x ) , w k := (cid:40) u k in Q ρ ( x ) ,u ξ in R n \ Q ρ ( x ) , respectively; clearly, w = u ξ in a neighbourhood of ∂Q (1+ η ) ρ ( x ) and w k j → w in L p loc ( R n ; R m ).Hence by Γ-convergence, by (2.3) and (6.5) we get m ( u ξ , Q (1+ η ) ρ ( x )) ≤ F ( w, , Q (1+ η ) ρ ( x )) ≤ lim j → + ∞ F k j ( w k j , , Q (1+ η ) ρ ( x )) ≤ lim j → + ∞ F bk j ( u k j , Q ρ ( x )) + c | ξ | p (cid:0) (1 + η ) n − (cid:1) ρ n ≤ lim inf k → + ∞ m bk ( u ξ , Q ρ ( x )) + ηρ n + c | ξ | p (cid:0) (1 + η ) n − (cid:1) ρ n . Eventually, dividing by ρ n , passing to the limsup as ρ →
0, and recalling the definition of ˆ f and f (cid:48) we get (1 + η ) n ˆ f ( x, ξ ) ≤ f (cid:48) ( x, ξ ) + η + c | ξ | p (cid:0) (1 + η ) n − (cid:1) , and hence the claim follows by the arbitrariness of η > (cid:3) Identification of the surface integrand
In this section we identify the surface integrand ˆ g . Namely, we show that ˆ g coincides with both g (cid:48) and g (cid:48)(cid:48) , given by (2.14) and (2.15), respectively. This shows, in particular, that the limit surfaceintegrand ˆ g is obtained by minimising only the surface term F sk . We notice, however, that in thiscase the presence the bulk term F bk affects the class of test functions over which the minimisationis performed (cf. (2.9)-(2.10)).We start by proving some preliminary lemmas. The first lemma concerns the approximation ofa minimisation problem involving the Γ-limit F . Lemma 7.1 (Approximation of minimum values) . Let ( f k ) ⊂ F and ( g k ) ⊂ G . Let ρ > ; for x ∈ R n , ζ ∈ R m , ν ∈ S n − , and k ∈ N such that ε k < ρ set m k (¯ u νx,ζ,ε k , Q νρ ( x )) := inf { F k ( u, v, Q νρ ( x )) : ( u, v ) ∈ W ,p ( Q νρ ( x ); R m ) × W ,p ( Q νρ ( x )) , ≤ v ≤ , ( u, v ) = (¯ u νx,ζ,ε k , ¯ v νx,ε k ) near ∂Q νρ ( x ) } . Let ( k j ) be as in Theorem 5.2 and ˆ g be as in (5.12) . Then for every x ∈ R n , ζ ∈ R m , and ν ∈ S n − it holds ˆ g ( x, ζ, ν ) = lim sup ρ → ρ n − lim inf j → + ∞ m k j (¯ u νx,ζ,ε kj , Q νρ ( x ))= lim sup ρ → ρ n − lim sup j → + ∞ m k j (¯ u νx,ζ,ε kj , Q νρ ( x )) . (7.1) Proof.
For notational simplicity, in what follows we still denote with k the index of the (sub)sequenceprovided by Theorem 5.2.We divide the proof into two steps. Step 1:
In this step we show thatˆ g ( x, ζ, ν ) ≤ lim sup ρ → ρ n − lim inf k → + ∞ m k (¯ u νx,ζ,ε k , Q νρ ( x )) , (7.2)for every x ∈ R n , ζ ∈ R m , and ν ∈ S n − .Let ρ > η > m k (¯ u νx,ζ,ε k , Q νρ ( x )) there exists ( u k , v k ) ⊂ W ,p ( Q νρ ( x ); R m ) × W ,p ( Q νρ ( x )) such that ( u k , v k ) = (¯ u νx,ζ,ε k , ¯ v νx,ε k ) in a neighbourhood of ∂Q νρ ( x ),and F k ( u k , v k , Q νρ ( x )) ≤ m k (¯ u νx,ζ,ε k , Q νρ ( x )) + ηρ n − . (7.3)Since the pair (¯ u νx,ζ,ε k , ¯ v νx,ε k ) is admissible for m k (¯ u νx,ζ,ε k , Q νρ ( x )), then (2.11) and (2.19) readilygive m k (¯ u νx,ζ,ε k , Q νρ ( x )) ≤ F k (¯ u νx,ζ,ε k , ¯ v νx,ε k , Q νρ ( x )) = F sk (¯ v νx,ε k , Q νρ ( x )) ≤ c C v ρ n − . (7.4)By a truncation argument (see e.g. , [23, Lemma 3.5] or [27, Lemma 4.1]) it is not restrictive toassume that sup k (cid:107) u k (cid:107) L ∞ ( Q νρ ( x ); R m ) < + ∞ . We now extend u k to a W ,p loc ( R n ; R m )-function bysetting w k := (cid:40) u k in Q νρ ( x ) , ¯ u νx,ζ,ε k in R n \ Q νρ ( x ) , then, clearly, sup k (cid:107) w k (cid:107) L ∞ ( R n ; R m ) < + ∞ . Now let ( k j ) be such thatlim j → + ∞ F k j ( u k j , v k j , Q νρ ( x )) = lim inf k → + ∞ F k ( u k , v k , Q νρ ( x )) . In view of (7.4), (2.5), and the uniform L ∞ ( R n ; R m )-bound on w k j we can invoke [36, Lemma 4.1]to deduce the existence of a subsequence (not relabelled) such that( w k j , v k j ) → ( u,
1) in L p loc ( R n ; R m ) × L p loc ( R n ) , for some u ∈ L p loc ( R n ; R m ) also belonging to SBV p ( Q ν (1+ η ) ρ ( x ); R m ). Moreover, we also have u = u νx,ζ in a neighbourhood of ∂Q ν (1+ η ) ρ ( x ), so that m ( u νx,ζ , Q ν (1+ η ) ρ ( x )) ≤ F ( u, , Q ν (1+ η ) ρ ( x )) . (7.5)Eventually, by Γ-convergence together with (7.3) we obtain F ( u, , Q ν (1+ η ) ρ ( x )) ≤ lim inf j → + ∞ F k j ( w k j , v k j , Q ν (1+ η ) ρ ( x )) ≤ lim j → + ∞ F k j ( u k j , v k j , Q νρ ( x )) + c C v (cid:0) (1 + η ) n − − ρ n − ≤ lim inf k → + ∞ m k (¯ u νx,ζ,ε k , Q νρ ( x )) + ηρ n − + c C v (cid:0) (1 + η ) n − − (cid:1) ρ n − . INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 29
Thus, using (7.5), dividing the above inequality by ρ n − and passing to the limsup as ρ → η ) n − ˆ g ( x, ζ, ν ) ≤ lim sup ρ → ρ n − lim inf k → + ∞ m k (¯ u νx,ζ,ε k , Q νρ ( x )) + η + c C v (cid:0) (1 + η ) n − − (cid:1) , hence (7.2) follows by the arbitrariness of η > Step 2:
In this step we show thatlim sup ρ → ρ n − lim sup k → + ∞ m k (¯ u νx,ζ,ε k , Q νρ ( x )) ≤ ˆ g ( x, ζ, ν ) , (7.6)for every x ∈ R n , ζ ∈ R m , ν ∈ S n − , and ρ >
0. To this end, we fix η > u ∈ SBV p ( Q νρ ( x ); R m ) with u = u νx,ζ near ∂Q νρ ( x ) and F ( u, , Q νρ ( x )) ≤ m ( u νx,ζ , Q νρ ( x )) + η . (7.7)We extend u to the whole R n by setting u = u νx,ζ in R n \ Q νρ ( x ). Then, by Γ-convergence thereexists a sequence ( u k , v k ) converging to ( u,
1) in measure on bounded sets such thatlim k → + ∞ F k ( u k , v k , Q νρ ( x )) = F ( u, , Q νρ ( x )) . (7.8)We notice, moreover, that thanks to a truncation argument (both on u and u k ) and to the bound( g u k , v k ) converges to ( u,
1) in L p loc ( R n ; R m ) × L p loc ( R n ). Wenow modify the sequence ( u k , v k ) in such a way that it satisfies the boundary conditions requiredin the definition of m k (¯ u νx,ζ,ε k , Q νρ ( x )). This will be done by resorting to the fundamental estimateProposition 5.1. Namely, we choose 0 < ρ (cid:48)(cid:48) < ρ (cid:48) < ρ such that u = u νx,ζ on Q νρ ( x ) \ Q νρ (cid:48)(cid:48) ( x ) and weapply Proposition 5.1 with A = Q νρ (cid:48)(cid:48) ( x ), A (cid:48) = Q νρ (cid:48) ( x ), B = Q νρ ( x ) \ Q νρ (cid:48)(cid:48) ( x ). In this way, we obtain asequence (ˆ u k , ˆ v k ) ⊂ W ,p loc ( R n ; R m ) × W ,p loc ( R n ) converging to ( u, v ) in L p ( Q νρ ( x ); R m ) × L p ( Q νρ ( x ))such that (ˆ u k , ˆ v k ) = ( u k , v k ) in Q νρ (cid:48)(cid:48) ( x ), (ˆ u k , ˆ v k ) = (¯ u νx,ζ,ε k , ¯ v νx,ε k ) in Q νρ ( x ) \ Q νρ (cid:48) ( x ), andlim sup k → + ∞ F k (ˆ u k , ˆ v k , Q νρ ( x )) ≤ (1 + η ) lim sup k → + ∞ (cid:16) F k ( u k , v k , Q νρ (cid:48) ( x )) + F sk (¯ v νx,ε k , Q νρ ( x ) \ Q νρ (cid:48)(cid:48) ( x )) (cid:17) ≤ (1 + η ) F ( u, , Q νρ ( x )) + c C v L n − (cid:0) Q (cid:48) ρ \ Q (cid:48) ρ (cid:48)(cid:48) (cid:1) , (7.9)where the second inequality follows from (7.8) and (2.19) together with (2.3), respectively.Eventually, since (ˆ u k , ˆ v k ) is admissible in the definition of m k (¯ u νx,ζ,ε k , Q νρ ( x )), gathering (7.7)and (7.9) we deduce thatlim sup k → + ∞ m k (¯ u νx,ζ,ε k , Q νρ ( x )) ≤ (1 + η ) (cid:0) m ( u νx,ζ , Q νρ ( x )) + η (cid:1) + C ( ρ n − − ( ρ (cid:48)(cid:48) ) n − ) . Then letting ρ (cid:48)(cid:48) → ρ , by the arbitrariness of η > k → + ∞ m k (¯ u νx,ζ,ε k , Q νρ ( x )) ≤ m ( u νx,ζ , Q νρ ( x )) , (7.10)therefore (7.6) follows by dividing both sides of (7.10) by ρ n − and passing to the limsup as ρ → (cid:3) Remark . Thanks to the ( W ,p ( A ; R m ) × W ,p ( A ))-continuity of F k ( · , · , A ) (cf. Remark 2.3),a standard convolution argument shows that (7.1) holds also true if the minimisation in m k iscarried over C ( Q νρ ( x ); R m ) × C ( Q νρ ( x )).The following lemma shows that if v is “small” in some region, then it can be replaced by afunction which is equal to zero in that region, without essentially increasing F sk . Lemma 7.3.
Let ( g k ) ⊂ G , A ∈ A , v ∈ W ,p ( A ) , and η ∈ (0 , be fixed. Let v η ∈ W ,p ( A ) bedefined as v η := min { ((1 + √ η ) v − η ) + , v } . (7.11) Then for a.e. x ∈ A we have v η ( x ) = 0 iff v ( x ) ≤ η √ η and v η ( x ) = v ( x ) iff v ( x ) ≥ √ η . (7.12) Moreover, v η satisfies F sk ( v η , A ) ≤ (1 + c η ) F sk ( v, A ) , (7.13) where c η > is independent of v and A and such that c η → as η → .Proof. A direct computation shows that ((1 + √ η ) v − η ) + ≤ v if and only if v ≤ √ η and that((1 + √ η ) v − η ) + = 0 if and only if v ≤ η √ η ; i.e. , (7.12) is satisfied. Thus it remains to showthat (7.13) holds true. To this end we introduce the sets A η := (cid:110) x ∈ A : v ( x ) ≤ η √ η (cid:111) and B η := (cid:110) x ∈ A : η √ η < v ( x ) < √ η (cid:111) , so that v η = v in A \ ( A η ∪ B η ); we have F sk ( v η , A ) = 1 ε k (cid:90) A η g k ( x, ,
0) d x + 1 ε k (cid:90) B η g k ( x, v η , ε k ∇ v η ) d x + 1 ε k (cid:90) A \ ( A η ∪ B η ) g k ( x, v, ε k ∇ v ) d x . (7.14)We start by estimating the first term on the right-hand side of (7.14). Since v ≤ η √ η < η < A η , using ( g
4) and ( g
6) we get g k ( x, , ≤ g k ( x, v,
0) + L (1 + v p − ) v ≤ g k ( x, v, ε k ∇ v ) + 2 L η ≤ g k ( x, v, ε k ∇ v ) + 2 L η (1 − η ) p (1 − v ) p , in A η , which together with ( g
2) yields1 ε k (cid:90) A η g k ( x, ,
0) d x ≤ (cid:16) L ηc (1 − η ) p (cid:17) ε k (cid:90) A η g k ( x, v, ε k ∇ v ) d x . (7.15)We now come to estimate the second term on the right-hand side of (7.14). By definition, we have v η = (1 + √ η ) v − η < v < √ η in B η , thus from ( g
4) we deduce that g k ( x, v η , ε k ∇ v η ) ≤ g k ( x, v, ε k ∇ v ) + L (cid:0) η p − (cid:1) ( η − √ ηv )+ L (cid:0) ε p − k |∇ v | p − + ε p − k |∇ v η | p − (cid:1) ε k |∇ v − ∇ v η | , (7.16)in B η . Furthermore, since v < √ η < B η , it also holds (cid:0) η p − (cid:1) ( η − √ ηv ) ≤ η ≤ η (1 − √ η ) p (1 − v ) p , (7.17)and similarly |∇ v − ∇ v η | = √ η |∇ v | ≤ √ η (1 − √ η ) p − (1 − v ) p − |∇ v | ≤ √ η (1 − √ η ) p − (cid:16) (1 − v ) p ε k + ε p − k |∇ v | p (cid:17) , (7.18)where the last estimate follows by Young’s inequality. Eventually, we also have ε pk (cid:0) |∇ v | p − + |∇ v η | p − (cid:1) |∇ v −∇ v η | = ε pk (cid:0) √ η ) p − (cid:1) √ η |∇ v | p ≤ ε pk (cid:0) p − (cid:1) √ η |∇ v | p . (7.19) INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 31
Gathering (7.16)–(7.19), dividing by ε k and using ( g
2) give1 ε k (cid:90) B η g k ( x, v η , ε k ∇ v η ) d x ≤ ε k (cid:90) B η g k ( x, v, ε k ∇ v ) d x + c η ε k (cid:90) B η g k ( x, v, ε k ∇ v ) d x , (7.20)where c η := L c max { η (1 −√ η ) p + (1 + 2 p − ) √ η , √ η (1 −√ η ) p − } . Then, since (1 − η ) p > (1 − √ η ) p for η ∈ (0 , F sk ( v η , A ) ≤ (1 + c η ) F sk ( v, A ) . Since c η → η → (cid:3) With the help of Lemma 7.1 and Lemma 7.3 we are now in a position to identify the surfaceintegrand ˆ g . Proposition 7.4.
Let ( f k ) ⊂ F and ( g k ) ⊂ G ; let ( k j ) and ˆ g be as in Theorem 5.2. Then, it holds ˆ g ( x, ζ, ν ) = g (cid:48) ( x, ν ) = g (cid:48)(cid:48) ( x, ν ) , for every x ∈ R n , ζ ∈ R m , and ν ∈ S n − , where g (cid:48) and g (cid:48)(cid:48) are, respectively, as in (2.14) and (2.15) , with k replaced by k j .Proof. For notational simplicity, in what follows we still denote with k the index of the sequenceprovided by Theorem 5.2.By definition we have g (cid:48) ≤ g (cid:48)(cid:48) ; hence to prove the claim it suffices to show that g (cid:48)(cid:48) ( x, ν ) ≤ ˆ g ( x, ζ, ν ) ≤ g (cid:48) ( x, ν ) , (7.21)for every x ∈ R n , ζ ∈ R m , and ν ∈ S n − . We divide the proof of (7.21) into two steps. Step 1:
In this step we show that ˆ g ( x, ζ, ν ) ≥ g (cid:48)(cid:48) ( x, ν ), for every x ∈ R n , ζ ∈ R m , and ν ∈ S n − .In view of Lemma 7.1 we haveˆ g ( x, ζ, ν ) = lim sup ρ → ρ n − lim sup k → + ∞ m k (¯ u νx,ζ,ε k , Q νρ ( x )) . Thanks to Remark 7.2 the minimisation in the definition of m k (¯ u νx,ζ,ε k , Q νρ ( x )) can be carried over C -pairs ( u k , v k ). Now let ρ > η ∈ (0 ,
1) be fixed and for every k such that 2 ε k < ρ let( u k , v k ) ⊂ C ( Q νρ ( x ); R m ) × C ( Q νρ ( x )) satisfy( u k , v k ) = (¯ u νx,ζ,ε k , ¯ v νx,ε k ) in U k , (7.22)where U k is a neighbourhood of ∂Q νρ ( x ) and F k ( u k , v k , Q νρ ( x )) ≤ m k (¯ u νx,ζ,ε k , Q νρ ( x )) + ηρ n − , (7.23)then, (7.4) readily gives F k ( u k , v k , Q νρ ( x )) ≤ Cρ n − . (7.24)We now modify v k in order to obtain a new function (cid:101) v k for which there exists a corresponding (cid:101) u k such that the pair ( (cid:101) u k , (cid:101) v k ) satisfies both( (cid:101) u k , (cid:101) v k ) = ( u νx ,
1) in (cid:101) U k ∩ {| ( y − x ) · ν | > ε k } , (7.25)where (cid:101) U k is a a neighbourhood of ∂Q νρ ( x ), and the constraint (cid:101) v k ∇ (cid:101) u k = 0 a.e. in Q νρ ( x ) . (7.26)In this way we have F k ( (cid:101) u k , (cid:101) v k , Q νρ ( x )) = F sk ( (cid:101) v k , Q νρ ( x )) with (cid:101) v k ∈ A ε k ,ρ ( x, ν ). The modificationas above shall be performed without essentially increasing the energy F k . To this end, set u k := ( u k , . . . , u mk ) and ζ := ( ζ , . . . , ζ m ). Since ζ ∈ R m we can find i ∈{ , . . . , m } so that ζ i (cid:54) = 0; without loss of generality we assume that ζ i >
0. We now consider theopen set S ρk := (cid:8) y ∈ Q νρ ( x ) : 0 < u ik ( y ) < ζ i (cid:9) . Moreover, let σ ∈ (0 ,
1) be fixed and consider the following partition of S ρk : S ρk = h σk − (cid:91) (cid:96) =0 S ρk,(cid:96) with S ρk,(cid:96) := (cid:8) y ∈ Q νρ ( x ) : (cid:96) ζ i h σk < u ik ( y ) ≤ ( (cid:96) + 1) ζ i h σk (cid:9) (cid:96) = 0 , . . . , h σk − ,S ρk,h σk − := (cid:8) y ∈ Q νρ ( x ) : ( h σk − ζ i h σk < u ik ( y ) < ζ i (cid:9) and h σk ∈ N to be chosen later.Then, there exists ¯ (cid:96) = ¯ (cid:96) ( k, ρ, σ ) ∈ { , . . . , h σk − } such that (cid:90) (cid:101) S ρk (cid:0) σψ ( v k ) |∇ u ik | p + (1 − σ ) (cid:1) dy ≤ h σk (cid:90) S ρk (cid:0) σψ ( v k ) |∇ u ik | p + (1 − σ ) (cid:1) d y , (7.27)where (cid:101) S ρk := S ρk, ¯ (cid:96) .Therefore, gathering ( f (cid:90) (cid:101) S ρk σψ ( v k ) |∇ u ik | p dy + (1 − σ ) L n ( (cid:101) S ρk ) ≤ ρ n − h σk C ( σ + (1 − σ ) ρ ) . (7.28)In view of (7.22) we have that (cid:101) S ρk ∩ U k ⊂ {| ( y − x ) · ν | ≤ ε k } ∩ U k . (7.29)In this way any modification to v k performed in the set (cid:101) S ρk will not affect the boundary conditions.In order to modify v k in (cid:101) S ρk , we introduce an auxiliary function ˆ v k which interpolates in a suitableway between the values 0 and 1 in (cid:101) S ρk . To this end, we set γ k := (¯ (cid:96) + ) ζ i h σk and τ k := ζ i h σk and wedefine ˆ v k ∈ W ,p ( Q νρ ( x )) as follows:ˆ v k := min (cid:26) u ik − γ k − τ k τ k , (cid:27) in { u ik ≥ γ k + τ k } , { γ k − τ k < u ik < γ k + τ k } , min (cid:26) γ k − τ k − u ik τ k , (cid:27) in { u ik ≤ γ k − τ k } , Eventually, we let v ηk ∈ W ,p ( Q νρ ( x )) be the function defined in (7.11) in Lemma 7.3, with v k inplace of v and we define (cid:101) v k ∈ W ,p ( Q νρ ( x )) as (cid:101) v k := min { v ηk , ˆ v k } . We notice that by definition ˆ v k ≡ Q νρ ( x ) \ (cid:101) S ρk , so that in particular (cid:101) v k ≡ v ηk in Q νρ ( x ) \ (cid:101) S ρk ; moreover, (cid:101) v k ≡ { γ k − τ k < u ik < γ k + τ k } . By the regularity of u ik we can find ξ k > { y ∈ Q νρ ( x ) : dist( y, { u ik > γ k } ) < ξ k } ⊂ { y ∈ Q νρ ( x ) : u ik ( y ) > γ k − τ k } . INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 33
Then, we define (cid:101) u k ∈ W ,p ( Q νρ ( x ); R m ) as (cid:101) u k ( y ) := (cid:18) − dist( y, { u ik > γ k } ) ξ k (cid:19) e if dist( y, { u ik > γ k } ) < ξ k , . Thus the pair ( (cid:101) u k , (cid:101) v k ) belongs to W ,p ( Q νρ ( x ); R m ) × W ,p ( Q νρ ( x )) and by construction satisfies(7.26). Moreover, (7.29) together with (7.22) ensures that (7.25) is satisfied. Eventually, we have1 ρ n − m sk, N ( u νx , Q νρ ( x )) ≤ ρ n − F sk ( (cid:101) v k , Q νρ ( x )) . (7.30)To conclude the proof it only remains to show that, up to a small error, F sk ( (cid:101) v k , Q νρ ( x )) is a lowerbound for F k ( u k , v k , Q νρ ( x )). To this end we consider the following partition of Q νρ ( x ): S k := (cid:8) y ∈ Q νρ ( x ) : v ηk ( y ) ≤ ˆ v k ( y ) (cid:9) , S k := (cid:8) y ∈ Q νρ ( x ) : v ηk ( y ) > ˆ v k ( y ) (cid:9) ;then, appealing to (7.13) in Lemma 7.3, we deduce F sk ( (cid:101) v k , Q νρ ( x )) = 1 ε k (cid:90) S k g k ( y, v ηk , ε k ∇ v ηk ) d y + 1 ε k (cid:90) S k g k ( y, ˆ v k , ε k ∇ ˆ v k ) d y ≤ (1 + c η ) F sk ( v k , Q νρ ( x )) + 1 ε k (cid:90) S k g k ( y, ˆ v k , ε k ∇ ˆ v k ) d y , (7.31)where c η → η →
0. Hence, it remains to estimate the second term on the right-hand sideof (7.31).To this end, we start noticing that S k ⊂ (cid:26) v k > η √ η (cid:27) . (7.32)Indeed, since v ηk > S k , by definition of v ηk we readily get (7.32).Therefore, by ( g v k ≡ Q νρ ( x ) \ (cid:101) S ρk we obtain1 ε k (cid:90) S k g k ( y, ˆ v k , ε k ∇ ˆ v k ) d y ≤ c (cid:90) S k (cid:18) (1 − ˆ v k ) p ε k + ε p − k |∇ ˆ v k | p (cid:19) d y = c (cid:90) S k ∩ (cid:101) S ρk (cid:18) (1 − ˆ v k ) p ε k + ε p − k |∇ ˆ v k | p (cid:19) d y ≤ c (cid:32) L n ( (cid:101) S ρk ) ε k + ε p − k ψ ( η √ η ) (cid:18) h σk ζ i (cid:19) p (cid:90) S k ∩ (cid:101) S ρk ψ ( v k ) |∇ u k | p d y (cid:33) , (7.33)where the last inequality follows by (7.32), the definition of ˆ v k , and the monotonicity of ψ .From (7.28) we deduce both that L n ( (cid:101) S ρk ) ≤ C ρ n − h σk (cid:18) σ − σ + ρ (cid:19) (7.34)and (cid:90) S k ∩ (cid:101) S k ψ ( v k ) |∇ u k | p d x ≤ C ρ n − h σk (cid:16) − σσ ρ (cid:17) . (7.35)Hence, gathering (7.33), (7.34), and (7.35) we obtain1 ε k (cid:90) S k g k ( y, ˆ v k , ε k ∇ ˆ v k ) d y ≤ C c ρ n − (cid:18) ε k h σk (cid:16) σ − σ + ρ (cid:17) + K η ( ε k h σk ) p − (cid:16) − σσ ρ (cid:17)(cid:19) , (7.36) with K η := 4 p ( ψ ( η √ η )( ζ i ) p ) − . Now set h σk := (cid:98) (cid:101) h σk (cid:99) where (cid:101) h σk := 1 ε k (cid:16) σ − σ (cid:17) p . (7.37)Using that h σk ≤ (cid:101) h σk ≤ h σk + 1 we infer1 ε k (cid:90) S k g k ( y, ˆ v k , ε k ∇ ˆ v k ) d y ≤ C c ρ n − (cid:18) (cid:101) h σk (cid:101) h σk − K η (cid:19)(cid:18)(cid:16) σ − σ (cid:17) p − p + ρ (cid:16) σ − σ (cid:17) − p (cid:19) . Plugging (7.36) into (7.31) gives1 ρ n − F sk ( (cid:101) v k , Q νρ ( x )) ≤ (1+ c η ) 1 ρ n − F k ( u k , v k , Q νρ ( x ))+ (cid:98) K η (cid:18)(cid:16) σ − σ (cid:17) p − p + ρ (cid:16) σ − σ (cid:17) − p (cid:19) , (7.38)where (cid:98) K η := C c (1 + K η ). We notice that c η → η →
0, while (cid:98) K η → + ∞ as η →
0. Finally,by combining (7.23), (7.30), and (7.38) we get1 ρ n − m sk, N ( u νx , Q νρ ( x )) ≤ (1+ c η ) (cid:18) ρ n − m k (¯ u νx,ζ,ε k , Q νρ ( x ))+ η (cid:19) + (cid:98) K η (cid:18)(cid:16) σ − σ (cid:17) p − p + ρ (cid:16) σ − σ (cid:17) − p (cid:19) . (7.39)Eventually, the claim follows by passing to the limit in (7.39) in the following order: first as k → + ∞ , then as ρ → σ →
0, and finally as η → Step 2:
In this step we show that ˆ g ( x, ζ, ν ) ≤ g (cid:48) ( x, ν ) , for every x ∈ R n , ζ ∈ R m and ν ∈ S n − . Thanks to (5.12), in view of Lemma 7.1 and Proposition2.6 it suffices to show that m k (¯ u νx,ζ,ε k , Q νρ ( x )) ≤ m sk (¯ u νx,ε k , Q νρ ( x )) , (7.40)for every ε >
0, where m sk (¯ u νx,ε k , Q νρ ( x )) is defined in (2.16). To prove (7.40) let v ∈ A (¯ u νx,ε k , Q νρ ( x ))with corresponding u ∈ W ,p ( Q νρ ( x ); R m ) and notice that the pair ( (cid:101) u, v ) with (cid:101) u := ( u · e ) ζ is anadmissible competitor for m k (¯ u νx,ζ,ε k , Q νρ ( x )). Therefore, we obtain m k (¯ u νx,ζ,ε k , Q νρ ( x )) ≤ F k ( (cid:101) u, v, Q νρ ( x )) = F sk ( v, Q νρ ( x )) , from which we deduce (7.40) by passing to the infimum in v ∈ A (¯ u νx,ε k , Q νρ ( x )). (cid:3) Remark . We observe that the second term in the right hand side of (7.39) is infinitesimal as ρ, σ → p >
1. We notice, moreover, that in (7.39) the presence of theexponent p in the reminder is a consequence of the p -growth of the volume integrand f k . Indeedif g k satisfied conditions ( g g
4) with p replaced by some exponent q ∈ (1 , p ], then arguing as inthe proof of Proposition 7.4, in place of (7.36) we would get1 ε k (cid:90) S k g k ( y, ˆ v k , ε k ∇ ˆ v k ) d y ≤ Cc ρ n − (cid:18) ε k h σk (cid:16) σ − σ + ρ (cid:17) + K η ( ε k h σk ) q − (cid:16) − σσ ρ (cid:17) qp (cid:16) σ − σ + ρ (cid:17) p − qp (cid:19) , with K η := 4 q ( ψ ( η √ η )( ζ i ) q ) − . Then, an easy computation shows that choosing h σk = (cid:98) (cid:101) h σk (cid:99) , with (cid:101) h σk given by (7.37), exactly yields (7.39).We are now in a position to prove the main result of this paper, namely, Theorem 3.1. INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 35
Proof of Theorem 3.1.
The proof follows by combining Proposition 4.1, Proposition 4.5, Theorem5.2, Proposition 6.1, and Proposition 7.4. (cid:3) Stochastic homogenisation
In this section we study the Γ-convergence of the functionals F k when f k and g k are randomintegrands of type f k ( ω, x, ξ ) = f (cid:16) ω, xε k , ξ (cid:17) , g k ( ω, x, v, w ) = g (cid:16) ω, xε k , v, w (cid:17) , where ω belongs to the sample space Ω of a probability space (Ω , T , P ).Before stating the main result of this section, we need to recall some useful definitions. Definition 8.1 (Group of P -preserving transformations) . Let d ∈ N , d ≥
2. A group of P -preserving transformations on (Ω , T , P ) is a family ( τ z ) z ∈ Z d of mappings τ z : Ω → Ω satisfying thefollowing properties:(1) (measurability) τ z is T -measurable for every z ∈ Z d ;(2) (invariance) P ( τ z ( E )) = P ( E ), for every E ∈ T and every z ∈ Z d ;(3) (group property) τ = id Ω and τ z + z (cid:48) = τ z ◦ τ z (cid:48) for every z, z (cid:48) ∈ Z d .If, in addition, every ( τ z ) z ∈ Z d -invariant set ( i.e. , every E ∈ T with τ z ( E ) = E for every z ∈ Z d )has probability 0 or 1, then ( τ z ) z ∈ Z d is called ergodic.Let a := ( a , . . . , a d ) , b := ( b , . . . , b d ) ∈ Z d with a i < b i for all i ∈ { , . . . , d } ; we define the d -dimensional interval [ a, b ) := { x ∈ Z d : a i ≤ x i < b i for i = 1 , . . . , d } and we set I d := { [ a, b ) : a, b ∈ Z d , a i < b i for i = 1 , . . . , d } . Definition 8.2 (Subadditive process) . A discrete subadditive process with respect to a group( τ z ) z ∈ Z d of P -preserving transformations on (Ω , T , P ) is a function µ : Ω × I d → R satisfying thefollowing properties:(1) (measurability) for every A ∈ I d the function ω (cid:55)→ µ ( ω, A ) is T -measurable;(2) (covariance) for every ω ∈ Ω, A ∈ I d , and z ∈ Z d we have µ ( ω, A + z ) = µ ( τ z ( ω ) , A );(3) (subadditivity) for every A ∈ I d and for every finite family ( A i ) i ∈ I ⊂ I d of pairwise disjointsets such that A = ∪ i ∈ I A i , we have µ ( ω, A ) ≤ (cid:88) i ∈ I µ ( ω, A i ) for every ω ∈ Ω ;(4) (boundedness) there exists c > ≤ µ ( ω, A ) ≤ c L d ( A ) for every ω ∈ Ω and A ∈ I d . Definition 8.3 (Stationarity) . Let ( τ z ) z ∈ Z n be a group of P -preserving transformations on (Ω , T , P ).We say that f : Ω × R n × R m × n → [0 , + ∞ ) is stationary with respect to ( τ z ) z ∈ Z n if f ( ω, x + z, ξ ) = f ( τ z ( ω ) , x, ξ )for every ω ∈ Ω, x ∈ R n , z ∈ Z n and ξ ∈ R m × n .Analogously, we say that g : Ω × R n × R × R n → [0 , + ∞ ) is stationary with respect to ( τ z ) z ∈ Z n if g ( ω, x + z, v, w ) = g ( τ z ( ω ) , x, v, w )for every ω ∈ Ω, x ∈ R n , z ∈ Z n , v ∈ [0 ,
1] and w ∈ R n . In all that follows we consider random integrands f : Ω × R n × R m × n → [0 , + ∞ ) satisfying( F f is ( T ⊗ B n ⊗ B m × n )-measurable;( F f ( ω, · , · ) ∈ F for every ω ∈ Ω;and random integrands g : Ω × R n × R × R n → [0 , + ∞ ) satisfying( G g is ( T ⊗ B n ⊗ B ⊗ B n )-measurable;( G g ( ω, · , · , · ) ∈ G for every ω ∈ Ω.Let f and g be random integrands satisfying ( F F
2) and ( G G F k ( ω ) : L ( R n ; R m ) × L ( R n ) × A −→ [0 , + ∞ ] givenby F k ( ω )( u, v, A ) := (cid:90) A ψ ( v ) f (cid:18) ω, xε k , ∇ u (cid:19) d x + 1 ε k (cid:90) A g (cid:18) ω, xε k , v, ε k ∇ v (cid:19) d x , (8.1)if ( u, v ) ∈ W ,p ( A ; R m ) × W ,p ( A ), 0 ≤ v ≤ ∞ otherwise. We also let F b ( ω ) be as in (3.4) and F s ( ω ) as in (3.5), with f ( · , · ) and g ( · , · , · ) replaced by f ( ω, · , · ) and g ( ω, · , · , · ), respectively. Moreover, for ω ∈ Ω and A ∈ A we set m bω ( u ξ , A ) := inf { F b ( ω )( u, A ) : u ∈ W ,p ( A ; R m ) , u = u ξ near ∂A } (8.2)and m sω (¯ u νx , A ) := inf { F s ( ω )( v, A ) : v ∈ A (¯ u νx , A ) } , (8.3)where A (¯ u νx , A ) is as in (2.17) with ¯ u νx in place of ¯ u νx,ε k ; i.e. , with ε k = 1.Eventually, we extend the definition of m bω ( u ξ , · ), m sω (¯ u νx , · ), and A (¯ u ν , · ) to any A ⊂ R n withint A ∈ A by setting m bω ( u ξ , A ) := m bω ( u ξ , int A), m sω (¯ u νx , A ) := m sω (¯ u νx , int A), and A (¯ u ν , A ) := A (¯ u ν , int A).We are now ready to state the main result of this section. Theorem 8.4 (Stochastic homogenisation) . Let f and g be random integrands satisfying ( F )-( F ) and ( G )-( G ), respectively. Assume moreover that f and g are stationary with respect to agroup ( τ z ) z ∈ Z n of P -preserving transformations on (Ω , T , P ) . For every ω ∈ Ω let F k ( ω ) be as in (8.1) and m bω , m sω be as in (8.2) and (8.3) , respectively. Then there exists Ω (cid:48) ∈ T , with P (Ω (cid:48) ) = 1 such that for every ω ∈ Ω (cid:48) , x ∈ R n , ξ ∈ R m × n , ν ∈ S n − the limits lim r → + ∞ m bω ( u ξ , Q r ( rx )) r n = lim r → + ∞ m bω ( u ξ , Q r (0)) r n =: f hom ( ω, ξ ) , (8.4)lim r → + ∞ m sω (¯ u νrx , Q νr ( rx )) r n − = lim r → + ∞ m sω (¯ u ν , Q νr (0)) r n − =: g hom ( ω, ν ) (8.5) exist and are independent of x . The function f hom : Ω × R m × n → [0 , + ∞ ) is ( T ⊗B m × n ) -measurableand g hom : Ω × S n − → [0 , + ∞ ) is ( T ⊗ B ( S n − )) -measurable.Moreover, for every ω ∈ Ω (cid:48) and for every A ∈ A the functionals F k ( ω )( · , · , A ) Γ -converge in L ( R n ; R m ) × L ( R n ) to the functional F hom ( ω )( · , · , A ) with F hom ( ω ) : L ( R n ; R m ) × L ( R n ) ×A −→ [0 , + ∞ ] given by F hom ( ω )( u, v, A ) := (cid:90) A f hom ( ω, ∇ u ) d x + (cid:90) S u ∩ A g hom ( ω, ν u ) d H n − if u ∈ GSBV p ( A ; R m ) ,v = 1 a.e. in A , + ∞ otherwise . INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 37
If, in addition, ( τ z ) z ∈ Z n is ergodic, then f hom and g hom are independent of ω and f hom ( ξ ) = lim r → + ∞ r n (cid:90) Ω m bω ( u ξ , Q r (0)) d P ( ω ) ,g hom ( ν ) = lim r → + ∞ r n − (cid:90) Ω m sω (¯ u ν , Q νr (0)) d P ( ω ) , thus, F hom is deterministic. The almost sure Γ-convergence result in Theorem 8.4 is an immediate consequence of Theo-rem 3.5 once we show the existence of a T -measurable set Ω (cid:48) ⊂ Ω, with P (Ω (cid:48) ) = 1, such that forevery ω ∈ Ω (cid:48) the limits in (8.4)-(8.5) exist and are independent of x . Therefore, the rest of thissection is devoted to prove the existence of such a set.8.1. Homogenisation formulas.
In this subsection we prove that conditions ( F F G G
2) together with the stationarity of the random integrands f and g ensure that the assump-tions of Theorem 3.5 are satisfied almost surely.The proof of the following result is based on the pointwise Subbaditive Ergodic Theorem [1,Theorem 2.4] applied to the function ( ω, I ) (cid:55)→ m bω ( u ξ , I ), which defines a subadditive process onΩ × I n , as shown in [46, Proposition 3.2]. Proposition 8.5 (Homogenised volume integrand) . Let f satisfy ( F )-( F ) and assume that itis stationary with respect to a group ( τ z ) z ∈ Z n of P -preserving transformations on (Ω , T , P ) . For ω ∈ Ω let m bω be as in (8.2) . Then there exists Ω (cid:48) ∈ T , with P (Ω (cid:48) ) = 1 and a ( T ⊗ B m × n ) -measurable function f hom : Ω × R m × n → [0 , + ∞ ) such that lim r → + ∞ m bω ( u ξ , Q r ( rx )) r n = lim r → + ∞ m bω ( u ξ , Q r (0)) r n = f hom ( ω, ξ ) . for every ω ∈ Ω (cid:48) , x ∈ R n and every ξ ∈ R m × n . If, in addition, ( τ z ) z ∈ Z n is ergodic, then f hom isindependent of ω and given by f hom ( ξ ) = lim r → + ∞ r n (cid:90) Ω m bω ( u ξ , Q r (0)) d P ( ω ) . Proof.
The proof follows by [46, Proposition 3.2] arguing as in [35, Theorem 1] (see also [46,Corollary 3.3]). (cid:3)
We now deal with the existence of the homogenised surface integrand g hom . Unlike the caseof f hom , the existence and x -homogeneity of the limit in (8.5) cannot be deduced by a directapplication of the Subadditive Ergodic Theorem [1, Theorem 2.4]. In fact, due to the x -dependentboundary datum appearing in m sω (¯ u νrx , Q νr ( rx )) (cf. (8.3)), the proof of the x -homogeneity of g hom is rather delicate and follows by ad hoc arguments, which are typical of surface functionals [3, 28].The proof of the existence of g hom will be carried out in several step. In a first step we provethat when x = 0 the minimisation problem (8.3) defines a subadditive process on Ω × I n − . Todo so we follow the same procedure as in [28, Section 5] (see also [3, 22]). Namely, given ν ∈ S n − we let R ν be an orthogonal matrix as in (f). Then { R ν e i : i = 1 , . . . , n − } is an orthonormalbasis for Π ν , further R ν ∈ Q n × n , if ν ∈ S n − ∩ Q n . Now let M ν > M ν R ν ∈ Z n × n ; therefore M ν R ν ( z (cid:48) , ∈ Π ν ∩ Z n for every z (cid:48) ∈ Z n − .Let I ∈ I n − ; i.e. , I = [ a, b ) with a, b ∈ Z n − . Starting from I we define the n -dimensionalinterval I ν as I ν := M ν R ν (cid:0) I × [ − c, c ) (cid:1) where c := 12 max i =1 ,...,n − ( b i − a i ) . (8.6) Correspondingly, for fixed ν ∈ S n − ∩ Q n we define the function µ ν : Ω × I n − (cid:55)→ R as µ ν ( ω, I ) := 1 M n − ν m sω (¯ u ν , I ν ) , (8.7)where m sω (¯ u ν , I ν ) is as in (8.3) with x = 0 and A = I ν .The following result asserts that µ ν defines a subadditive process on Ω × I n − . Proposition 8.6.
Let g satisfy ( G )-( G ) and assume that it is stationary with respect to a group ( τ z ) z ∈ Z n of P -preserving transformations on (Ω , T , P ) . Let ν ∈ S n − ∩ Q n and let µ ν : Ω ×I n − (cid:55)→ R be as in (8.7) . Then there exists a group of P -preserving transformations ( τ νz (cid:48) ) z (cid:48) ∈ Z n − on (Ω , T , P ) such that µ ν is a subadditive process on (Ω , T , P ) with respect to ( τ νz (cid:48) ) z (cid:48) ∈ Z n − . Moreover, it holds ≤ µ ν ( ω, I ) ≤ c C v L n − ( I ) , (8.8) for P -a.e. ω ∈ Ω and for every I ∈ I n − .Proof. Let ν ∈ S n − ∩ Q n be fixed; below we show that µ ν satisfies conditions (1)–(4) in Definition8.2, for some group of P -preserving transformations ( τ νz (cid:48) ) z (cid:48) ∈ Z n − .We divide the proof into four step, each of them corresponding to one of the four conditions inDefinition 8.2. Step 1: measurability.
Let I ∈ I n − and let I ν ⊂ R n be as in (8.6). Let v ∈ W ,p ( I ν ) be fixed.In view of ( G
1) the Fubini Theorem ensures that the map ω (cid:55)→ F s ( ω )( v, I ν ) is T -measurable. Onthe other hand the space W ,p ( I ν ) is separable, hence the set of functions A (¯ u ν , I ν ) = { v ∈ W ,p ( I ν ) , ≤ v ≤ , v = ¯ v ν near ∂I ν and ∃ u ∈ W ,p ( I ν ; R m ) ,u = ¯ u ν near ∂I ν , such that v ∇ u = 0 a.e. in I ν } defines a separable metric space, when endowed with the distance induced by the W ,p ( I ν )-norm.Then, by the continuity of F s ( ω )( · , I ν ) with respect to the strong W ,p ( I ν )-topology, the infimumin the definition of m sω (¯ u ν , I ν ) can be equivalently expressed as an infimum on a countable subsetof A (¯ u ν , I ν ), thus ensuring the T -measurability of ω (cid:55)→ m sω (¯ u ν , I ν ) and consequently that of ω (cid:55)→ µ ν ( ω, I ), as desired. Step 2: covariance.
Let z (cid:48) ∈ Z n − be fixed. Let I ∈ I n − ; by (8.6) we have( I + z (cid:48) ) ν = I ν + M ν R ν ( z (cid:48) ,
0) = I ν + z (cid:48) ν , where z (cid:48) ν := M ν R ν ( z (cid:48) , ∈ Z n ∩ Π ν . Then, by (8.7) we get µ ν ( ω, I + z (cid:48) ) = 1 M n − ν m sω (¯ u ν , I ν + z (cid:48) ν ) . (8.9)Now let v ∈ A (¯ u ν , I ν + z (cid:48) ν ) with its corresponding u ∈ W ,p ( I (cid:48) ν + z (cid:48) ν ; R m ). Setting (cid:101) v ( x ) := v ( x + z (cid:48) ν )for x ∈ I ν , a change of variables together with the stationarity of g yield F s ( ω )( v, int ( I ν + z (cid:48) ν )) = (cid:90) I ν + z (cid:48) ν g ( ω, x, v, ∇ v ) d x = (cid:90) I ν g ( ω, x + z (cid:48) ν , (cid:101) v, ∇ (cid:101) v ) d x = (cid:90) I ν g ( τ z (cid:48) ν ( ω ) , x, (cid:101) v, ∇ (cid:101) v ) d x = F s ( τ z (cid:48) ν ( ω ))( (cid:101) v, int I ν ) . Set ( τ νz (cid:48) ) z (cid:48) ∈ Z n − := ( τ z (cid:48) ν ) z (cid:48) ∈ Z n − ; we notice that ( τ νz (cid:48) ) z (cid:48) ∈ Z n − is well defined since z (cid:48) ν ∈ Z n and itdefines a group of P -preserving transformations on (Ω , P, T ). Then, the equality above can berewritten as F s ( ω )( v, int ( I ν + z (cid:48) ν )) = F s ( τ νz (cid:48) ( ω ))( (cid:101) v, int I ν ) . (8.10) INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 39
Moreover, if we set (cid:101) u ( x ) := u ( x + z (cid:48) ν ) for x ∈ I ν , then (cid:101) v ∇ (cid:101) u = 0 a.e. in I ν , while since z (cid:48) ν ∈ Π ν , wealso have (cid:0)(cid:101) u, (cid:101) v (cid:1) = (cid:0) ¯ u ν ( · + z (cid:48) ν ) , ¯ v ν ( · + z (cid:48) ν ) (cid:1) = (cid:0) ¯ u ν , ¯ v ν (cid:1) near ∂I ν ; i.e. , (cid:101) v ∈ A (¯ u ν , I ν ). Thus, gathering (8.9) and (8.10), by the arbitrariness of v we infer µ ν ( ω, I + z (cid:48) ) = µ ν ( τ νz (cid:48) ( ω ) , I ) , and hence the covariance of µ ν with respect to ( τ νz (cid:48) ) z (cid:48) ∈ Z n − . Step 3: subadditivity.
Let ω ∈ Ω, I ∈ I n − , and let { I , . . . , I N } ⊂ I n − be a finite familyof pairwise disjoint sets such that I = (cid:83) i I i . Let η > i = 1 , . . . , N let v i ∈ W ,p (( I i ) ν ) be admissible for m sω (¯ u ν , ( I i ) ν ) and such that F s ( ω )( v i , int ( I i ) ν ) ≤ m sω (¯ u ν , ( I i ) ν ) + η . (8.11)Therefore for every i = 1 , . . . , N there exists a corresponding u i ∈ W ,p (( I i ) ν ; R m ) such that v i ∇ u i = 0 a.e. in ( I i ) ν with ( u i , v i ) = (¯ u ν , ¯ v ν ) near ∂ ( I i ) ν . We define v := (cid:40) v i in ( I i ) ν , i = 1 , . . . , N , ¯ v ν in I ν \ (cid:83) i ( I i ) ν , u := (cid:40) u i in ( I i ) ν , i = 1 , . . . , N , ¯ u ν in I ν \ (cid:83) i ( I i ) ν , so that ( u, v ) ∈ W ,p ( I ν ) × W ,p ( I ν ; R m ) and v ∇ u = 0 a.e. in I ν with ( u, v ) = (¯ u ν , ¯ v ν ) near ∂I ν . Therefore v ∈ A (¯ u ν , I ν ). Furthermore, we have F s ( ω )( v, int I ν ) = N (cid:88) i =1 F s ( ω )( v i , int ( I i ) ν ) + F s ( ω ) (cid:0) ¯ v ν , int ( I ν \ N (cid:91) i =1 ( I i ) ν ) (cid:1) . (8.12)Since M ν > c ≥ in (8.6), it follows that { y ∈ I ν : | y · ν | ≤ } ⊂ (cid:83) i ( I i ) ν . Thus, ¯ v ν ≡ I ν \ (cid:83) i ( I i ) ν which, thanks to (2.3), gives F s ( ω ) (cid:0) ¯ v ν , int ( I ν \ N (cid:91) i =1 ( I i ) ν ) (cid:1) = 0 . Eventually, gathering (8.11)–(8.12) we obtain m sω (¯ u ν , I ν ) ≤ F s ( ω )( v, int I ν ) = N (cid:88) i =1 F s ( ω )( v i , int ( I i ) ν ) ≤ N (cid:88) i =1 m sω (¯ u ν , ( I i ) ν ) + N η , thus the subadditivity of µ ν follows from (8.7) and the arbitrariness of η > Step 4: boundedness.
Let ω ∈ Ω and I ∈ I n − . Then (8.6) and (2.19) readily imply0 ≤ µ ν ( ω, I ) = 1 M n − ν m sω (¯ u ν , I ν ) ≤ c C v L n − ( I ) . (cid:3) Having at hand Proposition 8.6, with the help of Lemma A.1 and Lemma A.2 we now prove thefollowing result, which establishes the almost sure existence of the limit defining g hom when x = 0. Proposition 8.7 (Homogenised surface integrand for x = 0) . Let g satisfy ( G )-( G ) and assumethat it is stationary with respect to a group ( τ z ) z ∈ Z n of P -preserving transformations on (Ω , T , P ) .For ω ∈ Ω let m sω be as in (8.3) . Then there exist (cid:101) Ω ∈ T with P ( (cid:101) Ω) = 1 and a ( T ⊗ B ( S n − )) -measurable function g hom : Ω × S n − → [0 , + ∞ ) such that lim r → + ∞ m sω (¯ u ν , Q νr (0)) r n − = g hom ( ω, ν ) (8.13) for every ω ∈ (cid:101) Ω and every ν ∈ S n − . Moreover, (cid:101) Ω and g hom are ( τ z ) z ∈ Z n -translation invariant;i.e., τ z ( (cid:101) Ω) = (cid:101) Ω for every z ∈ Z n and g hom ( τ z ( ω ) , ν ) = g hom ( ω, ν ) , (8.14) for every z ∈ Z n , for every ω ∈ (cid:101) Ω , and every ν ∈ S n − . Therefore, if ( τ z ) z ∈ Z n is ergodic then g hom is independent of ω and given by g hom ( ν ) = lim r → + ∞ r n − (cid:90) Ω m sω (¯ u ν , Q νr (0)) d P ( ω ) . (8.15) Proof.
We divide the proof into three steps.
Step 1: existence of the limit for ν ∈ S n − ∩ Q n . Let ν ∈ S n − ∩ Q n be fixed. Thanks toProposition 8.6 we can apply the Subadditive Ergodic Theorem [1, Theorem 2.4] to the subadditiveprocess µ ν defined on (Ω , T , P ) by (8.7). Then choosing I = [ − , n − we get the existence of aset Ω ν ∈ T , with P (Ω ν ) = 1, and of a T -measurable function g ν : Ω → [0 , + ∞ ) such thatlim j → + ∞ µ ν ( ω, jI )(2 j ) n − = g ν ( ω ) , for every ω ∈ Ω ν . By (8.7), since I ν = 2 M ν Q ν (0) this yields g ν ( ω ) = lim j → + ∞ m sω (¯ u ν , j M ν Q ν (0))( j M ν ) n − , (8.16)for every ω ∈ Ω ν . Let ( r j ) be a sequence of strictly positive real numbers with r j → + ∞ , as j → + ∞ , and consider the two sequences of integers defined as follows: r − j := 2 M ν (cid:16)(cid:106) r j M ν (cid:107) − (cid:17) and r + j := 2 M ν (cid:16)(cid:106) r j M ν (cid:107) + 2 (cid:17) . Let j ∈ N be such that r j > M ν ), and thus r − j >
4. We clearly have Q νr − j +2 (0) ⊂⊂ Q νr j (0) ⊂⊂ Q νr j +2 (0) ⊂⊂ Q νr + j (0) , hence we can apply Lemma A.1 with x = (cid:101) x = 0 first choosing r = r − j and (cid:101) r = r j to get m sω (¯ u ν , Q νr j (0)) r n − j ≤ m sω (¯ u ν , Q νr − j (0))( r − j ) n − + L ( r j − r − j + 1) r j , (8.17)and then r = r j and (cid:101) r = r + j to obtain m sω (¯ u ν , Q νr j (0)) r n − j ≥ m sω (¯ u ν , Q νr + j (0))( r + j ) n − − L ( r + j − r j + 1) r j . (8.18)Clearly, r + j − r j ≤ M ν and r j − r − j ≤ M ν , thus, if ω ∈ Ω ν , thanks to (8.16), passing to the limsupin as j → + ∞ in (8.17) gives lim sup j → + ∞ m sω (¯ u ν , Q νr j (0)) r n − j ≤ g ν ( ω ) , (8.19) INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 41 while passing to the liminf in as j → + ∞ in (8.18) yieldslim inf j → + ∞ m sω (¯ u ν , Q νr j (0)) r n − j ≥ g ν ( ω ) . (8.20)Hence gathering (8.19) and (8.20) gives for every ω ∈ Ω ν the existence of the limit along ( r j )together with the equality lim j → + ∞ m sω (¯ u ν , Q νr j (0)) r n − j = g ν ( ω ) . Therefore setting (cid:101)
Ω := (cid:92) ν ∈ S n − ∩ Q n Ω ν , clearly gives (cid:101) Ω ∈ T as well as P ( (cid:101) Ω) = 1; moreover we have g ν ( ω ) = lim r → + ∞ m sω (¯ u ν , Q νr (0)) r n − for every ω ∈ (cid:101) Ω and every ν ∈ S n − ∩ Q n . Step 2: existence of the limit for ν ∈ S n − \ Q n . Consider the two functions g, g : (cid:101) Ω × S n − → [0 , + ∞ ] defined as g ( ω, ν ) := lim inf r → + ∞ m sω (¯ u ν , Q νr (0)) r n − , g ( ω, ν ) := lim sup r → + ∞ m sω (¯ u ν , Q νr (0)) r n − . By the previous step we have that g ( ω, ν ) = g ( ω, ν ) = g ν ( ω ), for every ω ∈ (cid:101) Ω and for every ν ∈ S n − ∩ Q n . Therefore, if we show that the restrictions of the functions ν (cid:55)→ g ( ω, ν ) and ν (cid:55)→ g ( ω, ν ) to the sets (cid:98) S n − ± are continuous, by the density of (cid:98) S n − ± ∩ Q n in (cid:98) S n − ± we can readilydeduce that g ( ω, ν ) = g ( ω, ν ) = g ν ( ω ) for every ω ∈ (cid:101) Ω and every ν ∈ S n − , and thus the claim.Then it remains to show that g ( ω, · ) and g ( ω, · ) are continuous in (cid:98) S n − ± . We only prove that g ( ω, · ) is continuous in (cid:98) S n − , the other proofs being analogous. To this end, let ν ∈ (cid:98) S n − , ( ν j ) ⊂ (cid:98) S n − be such that ν j → ν , as j → + ∞ . Then, for every α ∈ (0 , ) there exists j α ∈ N such thatthe assumptions of Lemma A.2 are satisfied for every j ≥ j α . Thus, choosing x = 0 and (cid:101) ν = ν j in(A.7) we obtain m sω (¯ u ν j , Q ν j (1+ α ) r (0)) − c α r n − ≤ m sω (¯ u ν , Q νr (0)) ≤ m sω (¯ u ν j , Q ν j (1 − α ) r (0)) + c α r n − , where c α →
0, as α →
0. Therefore, by definition of g passing to the limsup as r → + ∞ we getthe two following inequalities (1 + α ) n − g ( ω, ν j ) ≤ g ( ω, ν ) + c α , (8.21)(1 − α ) n − g ( ω, ν j ) ≥ g ( ω, ν ) − c α . (8.22)Passing to the limsup as j → + ∞ in (8.21) and to the liminf as j → + ∞ in (8.22) and eventuallyletting α → j → + ∞ g ( ω, ν j ) ≤ g ( ω, ν ) ≤ lim inf j → + ∞ g ( ω, ν j ) , and hence the claim.For every ω ∈ Ω and ν ∈ S n − set g hom ( ω, ν ) := (cid:40) g ( ω, ν ) if ω ∈ (cid:101) Ω ,c c p if ω ∈ Ω \ (cid:101) Ω , (8.23) then, for every ω ∈ (cid:101) Ω and every ν ∈ S n − we have g hom ( ω, ν ) = lim r → + ∞ m sω (¯ u ν , Q νr (0)) r n − . (8.24)Moreover ω (cid:55)→ g ( ω, ν ) is T -measurable in (cid:101) Ω, for every ν ∈ S n − and ν (cid:55)→ g ( ω, ν ) is continuous in (cid:98) S n − ± , for every ω ∈ (cid:101) Ω . Therefore the restriction of g to (cid:101) Ω × (cid:98) S n − ± is measurable with respect to the σ -algebra induced in (cid:101) Ω × (cid:98) S n − ± by T ⊗ B ( S n − ) thus, finally, g hom is ( T ⊗ B ( S n − ))-measurable on Ω × S n − . Step 3: ( τ z ) z ∈ Z n -translation invariance. Let z ∈ Z n , ω ∈ (cid:101) Ω, and ν ∈ S n − be fixed. Let r > v ∈ A (¯ u ν , Q νr (0)) with F s ( ω )( v, Q νr (0)) ≤ m sω (¯ u ν , Q νr (0)) + 1 . (8.25)Set (cid:101) v ( y ) := v ( y + z ); by the stationarity of g we have F s ( ω )( v, Q νr (0)) = F s ( τ z ( ω ))( (cid:101) v, Q νr ( − z )) , further, since (cid:101) v ∈ A (¯ u ν − z , Q νr ( − z )), by (8.25) we immediately get m sτ z ( ω ) (¯ u ν − z , Q νr ( − z )) ≤ m sω (¯ u ν , Q νr (0)) + 1 . (8.26)Now let r, (cid:101) r be such that (cid:101) r > r and Q νr +2 ( − z ) ⊂⊂ Q ν (cid:101) r (0) and dist(0 , Π ν ( − z )) ≤ r , therefore applying Lemma A.1 with x = − z and (cid:101) x = 0 we may deduce the existence of a constant L > m sτ z ( ω ) (¯ u ν , Q ν (cid:101) r (0)) ≤ m sτ z ( ω ) (¯ u ν − z , Q νr ( − z )) + L (cid:0) | z | + | r − (cid:101) r | + 1 (cid:1) ( (cid:101) r ) n − . (8.27)Hence, gathering (8.26) and (8.27) we obtain m sτ z ( ω ) (¯ u ν , Q ν (cid:101) r (0)) (cid:101) r n − ≤ m sω (¯ u ν , Q νr (0)) + 1 r n − + L (cid:0) | z | + | r − (cid:101) r | + 1 (cid:1)(cid:101) r . (8.28)An analogous argument, now replacing ω with τ z ( ω ) and z with − z , yields m sω (¯ u ν , Q ν (cid:101) r (0)) (cid:101) r n − ≤ m sτ z ( ω ) (¯ u ν , Q νr (0)) + 1 r n − + L (cid:0) | z | + | r − (cid:101) r | + 1 (cid:1)(cid:101) r . (8.29)Taking in (8.28) the limsup as (cid:101) r → + ∞ and the limit as r → + ∞ giveslim sup (cid:101) r → + ∞ m sτ z ( ω ) (¯ u ν , Q ν (cid:101) r (0)) (cid:101) r n − ≤ g hom ( ω, ν ) , (8.30)while taking in (8.29) the limit as (cid:101) r → + ∞ and the liminf as r → + ∞ entails g hom ( ω, ν ) ≤ lim inf r → + ∞ m sτ z ( ω ) (¯ u ν , Q νr (0)) r n − . (8.31)Eventually, by combining (8.30) and (8.31) we both deduce that τ z ( ω ) ∈ (cid:101) Ω and that g hom ( τ z ( ω ) , ν ) = g hom ( ω, ν ) , for every z ∈ Z , ω ∈ (cid:101) Ω, and ν ∈ S n − . Then, we observe that thanks to the group propertiesof ( τ z ) z ∈ Z n we also have that ω ∈ τ z ( (cid:101) Ω), for every z ∈ Z n . Indeed we have ω = τ z ( τ − z ( ω )) and τ − z ( ω ) ∈ (cid:101) Ω. INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 43
If ( τ z ) z ∈ Z n is ergodic, then the independence of ω of the function of g hom is a direct consequenceof (8.14) (cf. [28, Corollary 6.3]). Furthermore, (8.15) can be obtained by integrating (8.24) on Ωand using the Dominated Convergence Theorem, thanks to (8.8) (see also (8.7)). (cid:3) The following result is of crucial importance in our analysis as it extends Proposition 8.7 tothe case of an arbitrary x ∈ R n . More precisely, Proposition 8.8 below establishes the existenceof the limit in (8.13) when x = 0 is replaced by any x ∈ R n ; moreover it shows that this limit is x -independent, and hence it coincides with (8.13).The proof of the following proposition can be obtained arguing exactly as in [28, Theorem 6.1](see also [3, Theorem 5.5]), now appealing to Proposition 8.7, Lemma A.1, and Lemma A.2. Forthis reason we omit its proof here. Proposition 8.8 (Homogenised surface integrand) . Let g satisfy ( G )-( G ) and assume that itis stationary with respect to a group ( τ z ) z ∈ Z n of P -preserving transformations on (Ω , T , P ) . For ω ∈ Ω let m sω be as in (8.3) . Then there exists Ω (cid:48) ∈ T with P (Ω (cid:48) ) = 1 such that lim r → + ∞ m sω (¯ u νrx , Q νr ( rx )) r n − = g hom ( ω, ν ) (8.32) for every ω ∈ Ω (cid:48) , every x ∈ R n , and every ν ∈ S n − , where g hom is given by (8.13) . In particular,the limit in (8.32) is independent of x . Moreover, if ( τ z ) z ∈ Z n is ergodic, then g hom is independentof ω and given by (8.15) . We conclude this section with the proof of Theorem 8.4.
Proof of Theorem 8.4.
The proof follows by Theorem 3.5 now invoking Proposition 8.5 and Propo-sition 8.8. (cid:3)
Acknowledgments
The authors wish to thank Manuel Friedrich and Matthias Ruf for fruitful discussions. The workof R. Marziani and C. I. Zeppieri was supported by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) project 3160408400 and under the Germany Excellence StrategyEXC 2044-390685587, Mathematics M¨unster: Dynamics–Geometry–Structure. The work of A.Bach was supported by the DFG Collaborative Research Center TRR 109, “Discretization inGeometry and Dynamics”.
Appendix
In this last section we state and prove two technical lemmas which are used in Subsection 8.1.For A ∈ A , x ∈ R n , and ν ∈ S n − , in what follows m s (¯ u νx , A ) denotes the infimum value givenby (3.7). Lemma A.1.
Let g ∈ G ; let ν ∈ S n − , x, (cid:101) x ∈ R n , and (cid:101) r > r > be such that (i) Q νr +2 ( x ) ⊂⊂ Q ν (cid:101) r ( (cid:101) x ) , (ii) dist( (cid:101) x, Π ν ( x )) ≤ r . Then there exists a constant
L > (independent of ν, x, (cid:101) x, r, (cid:101) r ) such that m s (¯ u ν (cid:101) x , Q ν (cid:101) r ( (cid:101) x )) ≤ m s (¯ u νx , Q νr ( x )) + L (cid:0) | x − (cid:101) x | + | r − (cid:101) r | + 1 (cid:1)(cid:101) r n − . (A.1) Proof.
To prove (A.1) we are going to perform a construction which is similar to that used inProposition 2.6.Let ν ∈ S n − , η > v ∈ A (¯ u νx , Q νr ( x )) with F s ( v, Q νr ( x )) ≤ m s (¯ u νx , Q νr ( x )) + η . (A.2) Let u ∈ W ,p ( Q νr ( x ); R m ) correspond to the v as above ; i.e. , v ∇ u = 0 a.e. in Q νr ( x ) and ( u, v ) =(¯ u νx , ¯ v νx ) a.e. in U , where U is a neighbourhood of ∂Q νr ( x ). Let moreover β ∈ (0 ,
1) be such that Q νr ( x ) \ Q νr − β ( x ) ⊂ U . Let (cid:101) x ∈ R n and (cid:101) r > r > R := R ν (cid:18)(cid:0) Q (cid:48) r \ Q (cid:48) r − β (cid:1) × (cid:16) − − | ( x − (cid:101) x ) · ν | , | ( x − (cid:101) x ) · ν | (cid:17)(cid:19) + x + ( (cid:101) x − x ) · ν ν , where R ν be as in (f) (see Figure 3a). Then let ϕ ∈ C ∞ c ( Q (cid:48) r ) be a cut-off function between Q (cid:48) r − β and Q (cid:48) r ; i.e. , 0 ≤ ϕ ≤
1, and ϕ ≡ Q (cid:48) r − β . Eventually, for y = ( y (cid:48) , y n ) ∈ Q ν (cid:101) r ( (cid:101) x ) we define thepair ( (cid:101) u, (cid:101) v ) by setting (cid:101) u ( y ) := ϕ (( R Tν ( y − x )) (cid:48) ) u ( y ) + (1 − ϕ (( R Tν ( y − x )) (cid:48) ))¯ u ν (cid:101) x ( y ) , (cid:101) v ( y ) := (cid:40) min { v ( y ) , d ( y ) } in Q νr ( x ) , min { ¯ v ν (cid:101) x ( y ) , d ( y ) } in Q ν (cid:101) r ( (cid:101) x ) \ Q νr ( x ) , where d ( y ) := dist( y, R ). Clearly, (cid:101) u ∈ W ,p ( Q ν (cid:101) r ( (cid:101) x ); R m ), moreover, by construction we have | ( y − x ) · ν | > | ( y − (cid:101) x ) · ν | > y ∈ ( Q νr ( x ) \ Q νr − β ( x )) \ R .
Hence, the boundary conditions satisfied by v imply that v = ¯ v νx = ¯ v ν (cid:101) x = 1 in ( Q νr ( x ) \ Q νr − β ( x )) \ R , thus (cid:101) v ∈ W ,p ( Q ν (cid:101) r ( (cid:101) x )).Thanks to (ii) and to the fact that r > R ⊂ Q νr ( x ) so that { d < } ⊂ Q νr +2 ( x ).Therefore, in view of (i) we get that (cid:101) v = ¯ v ν (cid:101) x in a neighbourhood of ∂Q ν (cid:101) r ( (cid:101) x ), further (cid:101) u = ¯ u ν (cid:101) x in Q ν (cid:101) r ( (cid:101) x ) \ Q νr ( x ). Then, to show that (cid:101) v is admissible for m s (¯ u ν (cid:101) x , Q ν (cid:101) r ( (cid:101) x )) it only remains to check that (cid:101) v ∇ (cid:101) u = 0 a.e. in Q ν (cid:101) r ( (cid:101) x ). Clearly, (cid:101) v ∇ (cid:101) u = 0 a.e. in R . On the other hand, in Q νr ( x ) \ R we have (cid:101) v |∇ (cid:101) u | ≤ v |∇ u | = 0, while in (cid:0) Q ν (cid:101) r ( (cid:101) x ) \ Q νr ( x ) (cid:1) \ R we get (cid:101) v |∇ (cid:101) u | ≤ ¯ v ν (cid:101) x |∇ ¯ u ν (cid:101) x | = 0, and hence the claim.Eventually, arguing as in Proposition 2.6 we obtain F s (cid:0)(cid:101) v, Q ν (cid:101) r ( (cid:101) x ) (cid:1) ≤ F s ( v, Q νr ( x )) + F s (cid:0) ¯ v ν (cid:101) x , Q ν (cid:101) r ( (cid:101) x ) \ Q νr ( x ) (cid:1) + 2 c L n ( { d < } ) . (A.3)Thanks to (2.3) and (2.19) we have F s (cid:0) ¯ v ν (cid:101) x , Q ν (cid:101) r ( (cid:101) x ) \ Q νr ( x ) (cid:1) ≤ c C v L n − (cid:0) Q (cid:48) (cid:101) r \ Q (cid:48) r (cid:1) ≤ C | r − (cid:101) r | (cid:101) r n − , (A.4)moreover L n ( { d < } ) ≤ C ( | x − (cid:101) x | + 1)( β + 1) r n − . (A.5)We observe that both in (A.4) and (A.5) the positive constant C > ν . Therefore, gathering (A.2) and (A.3) we finally obtain m s (¯ u ν (cid:101) x , Q ν (cid:101) r ( (cid:101) x )) ≤ F s (cid:0)(cid:101) v, Q ν (cid:101) r ( (cid:101) x ) (cid:1) ≤ m s (¯ u νx , Q νr ( x )) + L (cid:0) | x − (cid:101) x | + | r − (cid:101) r | + 1 (cid:1)(cid:101) r n − + η , for some L > x, (cid:101) x, r, (cid:101) r, ν , thus (A.1) follows by the arbitrariness of η > (cid:3) INGULARLY-PERTURBED ELLIPTIC FUNCTIONALS 45 r x (cid:101) x (cid:101) rr - β Π ν ( x ) Π ν ( (cid:101) x ) (a) The sets Q νr − β ( x ), Q νr ( x ), Q ν (cid:101) r ( (cid:101) x ) and in graythe sets R (dark gray) and { d < } (light gray). r (1+ α ) rr - β αr rx Π (cid:101) ν ( rx )Π ν ( rx ) (b) The sets Q νr − β ( rx ), Q νr ( rx ), Q (cid:101) ν (1+ α ) r ( rx ) andin gray the sets R (dark gray) and { d < } (lightgray). Figure 3.
The sets used in the construction of ( (cid:101) u, (cid:101) v ) in Lemma A.1 and Lemma A.2. Lemma A.2.
Let g ∈ G ; let α ∈ (0 , ) and ν, (cid:101) ν ∈ S n − be such that max ≤ i ≤ n − | R ν e i − R (cid:101) ν e i | + | ν − (cid:101) ν | < α √ n , (A.6) where R ν and R (cid:101) ν are orthogonal ( n × n ) -matrices as in (f ). Then there exists a constant c α > (independent of ν, (cid:101) ν ), with c α → as α → , such that for every x ∈ R n and every r > we have m s (cid:0) ¯ u (cid:101) νrx , Q (cid:101) ν (1+ α ) r ( rx ) (cid:1) − c α r n − ≤ m s (cid:0) ¯ u νrx , Q νr ( rx ) (cid:1) ≤ m s (cid:0) ¯ u (cid:101) νrx , Q (cid:101) ν (1 − α ) r ( rx ) (cid:1) + c α r n − . (A.7) Proof.
We only prove that m s (cid:0) ¯ u (cid:101) νrx , Q (cid:101) ν (1+ α ) r ( rx ) (cid:1) − c α r n − ≤ m s (cid:0) ¯ u νrx , Q νr ( rx ) (cid:1) , (A.8)for some c α >
0, with c α →
0, as α →
0; the proof of the other inequality is analogous.Let x ∈ R n , r >
2, and set r ± α := (1 ± α ) r . We notice that condition (A.6) readily implies that Q (cid:101) νr − α ( rx ) ⊂⊂ Q νr ( rx ) ⊂⊂ Q (cid:101) νr + α ( rx ) . (A.9)Moreover, we let r be so large that Q νr +2 ( rx ) ⊂⊂ Q (cid:101) νr + α ( rx ). This allows us to use a similarargument as in the proofs of Proposition 2.6 and Lemma A.1, which we repeat here for the readers’convenience.Let η > v ∈ W ,p ( Q νr ( rx )) be a test function for m s (¯ u νrx , Q νr ( rx )) satisfying F s ( v, Q νr ( rx )) ≤ m s (¯ u νrx , Q νr ( rx )) + η. (A.10)Therefore we know that there exist u ∈ W ,p ( Q νr ( rx ); R m ) and a neighbourhood U of ∂Q νr ( rx )such that ( u, v ) = (¯ u νrx , ¯ v νrx ) in U and v ∇ u = 0 a.e. in Q νr ( rx ) . (A.11)Now let β ∈ (0 ,
1) be such that Q νr ( rx ) \ Q νr − β ( rx ) ⊂ U and set R := R ν (cid:16) Q (cid:48) r \ Q (cid:48) r − β × ( − − αr, αr ) (cid:17) + rx , where R ν is as in (f) (see Figure 3b). Let ϕ ∈ C ∞ c ( Q (cid:48) r ) be a cut-off function between Q (cid:48) r − β and Q (cid:48) r ; we define (cid:101) u ( y ) := ϕ (cid:0)(cid:0) R Tν ( y − rx ) (cid:1) (cid:48) (cid:1) u ( y ) + (cid:16) − ϕ (cid:0)(cid:0) R Tν ( y − rx ) (cid:1) (cid:48) (cid:1)(cid:17) ¯ u (cid:101) νrx ( y ) , (cid:101) v ( y ) := min { v ( y ) , d ( y ) } in Q νr ( rx ) , min { ¯ v (cid:101) νrx ( y ) , d ( y ) } in Q νr + α ( rx ) \ Q νr ( rx ) , where d ( y ) := dist( y, R ). We now observe that in view of (A.6) and (A.11), by definition of R weget v = ¯ v νrx = ¯ v (cid:101) νrx = 1 in ( Q νr ( rx ) \ Q r − β ( rx )) \ R, so that (cid:101) v ∈ W ,p (cid:0) Q (cid:101) νr + α ( rx ) (cid:1) . Since Q νr +2 ( rx ) ⊂⊂ Q (cid:101) νr + α ( rx ), arguing as in Lemma A.1 it is easy tocheck that the pair ( (cid:101) u, (cid:101) v ) ∈ W ,p (cid:0) Q (cid:101) νr + α ( rx ); R m (cid:1) × W ,p (cid:0) Q (cid:101) νr + α ( rx ) (cid:1) also satisfies( (cid:101) u, (cid:101) v ) = (¯ u (cid:101) νrx , ¯ v (cid:101) νrx ) near ∂Q (cid:101) νr + α ( rx ) and (cid:101) v ∇ (cid:101) u = 0 a.e. in Q (cid:101) νr + α ( rx ) . Therefore, (cid:101) v is admissible for m s (¯ u (cid:101) νrx , Q (cid:101) νr + α ( rx )). Then, using the same arguments as in Proposi-tion 2.6 we deduce F s ( (cid:101) v, Q (cid:101) νr + α ( rx )) ≤ F s ( v, Q νr ( rx )) + F s (¯ v (cid:101) νrx , Q (cid:101) νr + α ( rx ) \ Q νr ( rx )) + 2 c L n ( { d < } ) ≤ F s ( v, Q νr ( rx )) + F s (¯ v (cid:101) νrx , Q (cid:101) νr + α ( rx ) \ Q (cid:101) νr − α ( rx )) + 2 c L n ( { d < } ) , (A.12)where to obtain the second inequality we used the first inclusion in (A.9). Furthermore by (2.3)and (2.19) we have F s (¯ v (cid:101) νrx , Q (cid:101) νr + α ( rx ) \ Q (cid:101) νr − α ( rx )) ≤ c C v L n − (cid:0) Q (cid:48) r + α ( rx ) \ Q (cid:48) r − α ( rx ) (cid:1) = c C v ((1 + α ) n − − (1 − α ) n − ) r n − (A.13)whereas L n ( { d < } ) ≤ C ( β + 1) r n − αr ≤ Cαr n − , (A.14)for some C > x, ν , and r . Thus, gathering (A.10), (A.12), (A.13), and (A.14),thanks to (A.11) we infer m s (¯ u (cid:101) νrx , Q (cid:101) νr + α ( rx )) ≤ m s (¯ u νrx , Q νr ( rx )) + c α r n − + η , where c α := c C v (cid:0) (1 + α ) n − − (1 − α ) n − (cid:1) + Cα . Eventually, (A.8) follows by the arbitrarinessof η > (cid:3) References [1]
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Zentrum Mathematik (M7) TU M¨unchen, Germany
Email address , Annika Bach: [email protected]
Angewandte Mathematik, WWU M¨unster, Germany
Email address , Roberta Marziani: [email protected]
Angewandte Mathematik, WWU M¨unster, Germany
Email address , Caterina Zeppieri:, Caterina Zeppieri: