Two-scale convergence of elliptic spectral problems with indefinite density function in perforated domains
aa r X i v : . [ m a t h . A P ] A ug T WO - SCALE C ONVE RGENCE OF P E RI ODIC E L L I PTIC S PE CT RAL P ROB LEMS WI T H I NDE FI NITE D E NSI TY F UNCT ION I N P E RFORATED D OMAI NS H ERMANN D OUANLA ∗ Department of Mathematical SciencesChalmers University of TechnologyGothenburg, SE-41296, Sweden
Abstract
Spectral asymptotics of linear periodic elliptic operators with indefinite (sign-changing)density function is investigated in perforated domains with the two-scale convergencemethod. The limiting behavior of positive and negative eigencouples depends cruciallyon whether the average of the weight over the solid part is positive, negative or equalto zero. We prove concise homogenization results in all three cases.
AMS Subject Classification:
Keywords : Homogenization, eigenvalue problems, perforated domains, indefinite weightfunction, two-scale convergence.
Many nonlinear problems lead, after linearization, to elliptic eigenvalue problems with anindefinite density function (see e.g., the survey paper by de Figueiredo[11] and the work ofHess and Kato[13, 14]). A vast literature in engineering, physics and applied mathematicsdeals with such problems arising, for instance, in the study of transport theory, reaction-diffusion equations and fluid dynamics. In 1904, Holmgren[16] considered the Dirichletproblem D u + lr ( x , y ) u =
0, on a fixed bounded open set W ⊂ R when r is continuousand changes sign; he proved the existence of a double sequence of real eigenvalues of finitemultiplicity (one nonnegative and converging to + ¥ , the other one negative and tending to − ¥ ) which can be characterized by the minimax principle. This result has been extendedto higher dimensions, noncontinuous weight and coefficients in many papers including forexample [3, 4, 22]. Asymptotic analysis of the eigenvalues has been visited by many math-ematicians and is still a hot topic in mathematical analysis. Generally speaking, spectralasymptotics is a two folded research area. On the one hand it deals with asymptotic for-mulas (estimates) and asymptotic distribution of the eigenvalues. On the other hand it is ∗ E-mail address: [email protected]
Hermann Douanlaconcerned with homogenization of eigenvalues of oscillating operators on possibly varyingdomains such as perforated ones. This paper falls within the second framework, homoge-nization theory.Let W be a bounded domain in R Nx (the numerical space of variables x = ( x , ..., x N ) , withinteger N ≥
2) and let T ⊂ Y = ( , ) N be a compact subset of Y in R Ny . Unless otherwisespecified we assume that W and T have C boundaries ¶W and ¶ T , respectively. For e > W e as follows. we put t e = { k ∈ Z N : e ( k + T ) ⊂ W } , T e = [ k ∈ t e e ( k + T ) and W e = W \ T e . In this setup, T is the reference hole whereas e ( k + T ) is a hole of size e and T e is thecollection of the holes of the perforated domain W e . The family T e is made up with a finitenumber of holes since W is bounded. In the sequel, Y ∗ stands for Y \ T and n = ( n i ) denotesthe outer unit normal vector to ¶ T with respect to Y ∗ .We are interested in the spectral asymptotics (as e →
0) of the linear elliptic eigenvalueproblem − N (cid:229) i , j = ¶¶ x j (cid:18) a i j ( x e ) ¶ u e ¶ x i (cid:19) = r ( x e ) l e u e in W e N (cid:229) i , j = a i j ( x e ) ¶ u e ¶ x j n i ( x e ) = ¶ T e u e = ¶W , (1.1)where a i j ∈ L ¥ ( R Ny ) (1 ≤ i , j ≤ N ), with the symmetry condition a ji = a i j , the Y -periodicityhypothesis: for every k ∈ Z N one has a i j ( y + k ) = a i j ( y ) almost everywhere in y ∈ R Ny , andfinally the (uniform) ellipticity condition: there exists a > N (cid:229) i , j = a i j ( y ) x j x i ≥ a | x | (1.2)for all x ∈ R N and for almost all y ∈ R Ny , where | x | = | x | + · · · + | x N | . The density func-tion r ∈ L ¥ ( R Ny ) is Y -periodic and changes sign on Y ∗ , that is, both the set { y ∈ Y ∗ , r ( y ) < } and { y ∈ Y ∗ , r ( y ) > } are of positive Lebesgue measure. This hypothesis makes theproblem under consideration nonstandard. As stated above, it is well known (see [16, 22])that under the preceding hypotheses, for each e > < l , + e ≤ l , + e ≤ · · · ≤ l n , + e ≤ . . . , lim n → + ¥ l n , + e = + ¥ and 0 > l , − e ≥ l , − e ≥ · · · ≥ l n , − e ≥ . . . , lim n → + ¥ l n , − e = − ¥ . pectral Asymptotics in Porous Media 3The asymptotic behavior of the eigencouples depends crucially on whether the average of r over Y ∗ , M Y ∗ ( r ) = R Y ∗ r ( y ) dy , is positive, negative or equal to zero. All three cases arecarefully investigated in this paper.The homogenization of spectral problems has been widely explored. In a fixed domain,homogenization of spectral problems with point-wise positive density function goes backto Kesavan [18, 19]. In perforated domains, spectral asymptotics was first considered byRauch and Taylor[28, 29] but the first homogenization result in that direction pertains toVanninathan[31]. Since then a lot has been written on spectral asymptotics in perforatedmedia, we mention the works [17, 27, 30] and the references therein to cite a few. Homog-enization of elliptic operators with sing-changing density function in a fixed domain hasbeen investigated by Nazarov et al. [21, 22, 23] via a combination of formal asymptotic ex-pansion and Tartar’s energy method. Recently, the Two-scale convergence method has beenutilized to handle the homogenization process for some eigenvalue problems with constantdensity function[9, 10] and sign-changing density function[8].In this paper we investigate in periodically perforated domains the spectral asymptoticsof periodic elliptic linear differential operators of order two in divergence form with asing-changing density function. We obtain accurate and concise homogenization resultsin all three cases: M Y ∗ ( r ) > M Y ∗ ( r ) = M Y ∗ ( r ) < M Y ∗ ( r ) > k ≥ l k , + e converges as e → k th eigenvalue ofthe limit spectral problem on W , corresponding extended eigenfunctions converge alongsubsequences. As regards the ”negative” eigencouples, l k , − e converges to − ¥ at the rate e and the corresponding eigenfunctions oscillate rapidly. We use a factorization technique([23, 31]) to prove convergence of { l k , − e − e l − } - where ( l − , q − ) is the first negativeeigencouple to a local spectral problem - to the k th eigenvalue of a limit spectral problemwhich is different from that obtained for positive eigenvalues. As regards eigenfunctions,extensions of { u k , − e ( q − ) e } e ∈ E - where ( q − ) e ( x ) = q − ( x e ) - converge along subsequences to the k th eigenfunctions of the limit problem. In the case when M Y ∗ ( r ) = l k , ± e converges to ± ¥ at the rate e and the limit spectral problem generates a quadratic operator pencil. Weprove that el k , ± e converges to the ( k , ± ) th eigenvalue of the limit operator, extended eigen-functions converge along subsequences as well. The case when M Y ∗ ( r ) < M Y ∗ ( r ) >
0, just replace r with − r . The reader may consider the reiterationprocedure in multiscale periodically perforated domains to have some fun.Unless otherwise specified, vector spaces throughout are considered over R , and scalarfunctions are assumed to take real values. We will make use of the following notations. Let F ( R N ) be a given function space. We denote by F per ( Y ) the space of functions in F loc ( R N ) (when it makes sense) that are Y -periodic, and by F per ( Y ) / R the space of those functions u ∈ F per ( Y ) with R Y u ( y ) dy =
0. We denote by H per ( Y ∗ ) the space of functions in H ( Y ∗ ) assuming same values on the opposite faces of Y and H per ( Y ∗ ) / R stands for the subset of H per ( Y ∗ ) made up of functions u ∈ H per ( Y ∗ ) verifying R Y ∗ u ( y ) dy =
0. Finally, the letter E denotes throughout a family of strictly positive real numbers ( < e < ) admitting 0as accumulation point. The numerical space R N and its open sets are provided with the Hermann DouanlaLebesgue measure denoted by dx = dx ... dx N . The usual gradient operator will be denotedby D . The rest of the paper is organized as follows. Section 2 deals with some preliminaryresults while homogenization processes are considered in Section 3. We first recall the definition and the main compactness theorems of the two-scale conver-gence method. Throughout this section, W is a smooth open bounded set in R Nx (integer N ≥
2) and Y = ( , ) N is the unit cube. Definition 2.1.
A sequence ( u e ) e ∈ E ⊂ L ( W ) is said to two-scale converge in L ( W ) to some u ∈ L ( W × Y ) if as E ∋ e → Z W u e ( x ) f ( x , x e ) dx → ZZ W × Y u ( x , y ) f ( x , y ) dxdy (2.1)for all f ∈ L ( W ; C per ( Y )) . Notation.
We express this by writing u e s −→ u in L ( W ) .The following compactness theorems (see [1, 24, 26]) are cornerstones of the two-scaleconvergence method. Theorem 2.2.
Let ( u e ) e ∈ E be a bounded sequence in L ( W ) . Then a subsequence E ′ canbe extracted from E such that as E ′ ∋ e → , the sequence ( u e ) e ∈ E ′ two-scale converges inL ( W ) to some u ∈ L ( W × Y ) . Theorem 2.3.
Let ( u e ) e ∈ E be a bounded sequence in H ( W ) . Then a subsequence E ′ canbe extracted from E such that as E ′ ∋ e → u e → u in H ( W ) -weak (2.2) u e → u in L ( W ) (2.3) ¶ u e ¶ x j s −→ ¶ u ¶ x j + ¶ u ¶ y j in L ( W ) ( ≤ j ≤ N ) (2.4) where u ∈ H ( W ) and u ∈ L ( W ; H per ( Y )) . Moreover, as E ′ ∋ e → we have Z W u e ( x ) e y ( x , x e ) dx → ZZ W × Y u ( x , y ) y ( x , y ) dx dy (2.5) for y ∈ D ( W ) ⊗ ( L per ( Y ) / R ) .Proof. The first part (2.2)-(2.4) is classical (see [1, 24]). The second part, (2.5), was provedin [26] in the general framework of deterministic homogenization but as it is of great im-portance in this paper and for the sake of completeness, we provide its proof in the periodicsetting. Let y = ( j , q ) ∈ D ( W ) × ( L per ( Y ) / R ) . By the mean value zero condition over Y for q we conclude that there exists a unique solution J ∈ H per ( Y ) / R to ( D y J = q in Y J ∈ H per ( Y ) / R . pectral Asymptotics in Porous Media 5Put f = D y J . We get Z W u e ( x ) e y ( x , x e ) dx = Z W u e ( x ) e j ( x ) q ( x e ) dx = Z W u e ( x ) j ( x ) div x f ( x e ) dx = − Z W D x ( u e ( x ) j ( x )) · f ( x e ) dx A limit passage ( e → ) using (2.4) yieldslim e → Z W u e ( x ) e y ( x , x e ) dx = − ZZ W × Y [ D x u ( x ) + D y u ( x , y )] j ( x ) · f ( y ) dydx = − ZZ W × Y D y u ( x , y ) j ( x ) · f ( y ) dydx = ZZ W × Y u ( x , y ) j ( x ) div y f ( y ) dydx = ZZ W × Y u ( x , y ) y ( x , y ) dydx . This completes the proof.
Remark . In Theorem 2.3 the function u is unique up to an additive function of variable x . We need to fix its choice according to our future needs. To do this, we introduce thefollowing space H , ∗ per ( Y ) = { u ∈ H per ( Y ) : Z Y ∗ u ( y ) dy = } . This defines a closed subspace of H per ( Y ) as it is the kernel of the bounded linear functional u R Y ∗ u ( y ) dy defined on H per ( Y ) . It is to be noted that for u ∈ H , ∗ per ( Y ) , its restriction to Y ∗ (which will still be denoted by u in the sequel) belongs to H per ( Y ∗ ) / R .We will use the following version of Theorem 2.3. Theorem 2.5.
Let ( u e ) e ∈ E be a bounded sequence in H ( W ) . Then a subsequence E ′ canbe extracted from E such that as E ′ ∋ e → u e → u in H ( W ) -weak (2.6) u e → u in L ( W ) (2.7) ¶ u e ¶ x j s −→ ¶ u ¶ x j + ¶ u ¶ y j in L ( W ) ( ≤ j ≤ N ) (2.8) where u ∈ H ( W ) and u ∈ L ( W ; H , ∗ per ( Y )) . Moreover, as E ′ ∋ e → we have Z W u e ( x ) e y ( x , x e ) dx → ZZ W × Y u ( x , y ) y ( x , y ) dx dy (2.9) for y ∈ D ( W ) ⊗ ( L per ( Y ) / R ) .Proof. Let e u ∈ L ( W ; H per ( Y )) be such that Theorem 2.3 holds with e u in place of u . Put u ( x , y ) = e u ( x , y ) − | Y ∗ | Z Y ∗ e u ( x , y ) dy ( x , y ) ∈ W × Y , where | Y ∗ | stands for the Lebesgue measure of Y ∗ . Then u ∈ L ( W ; H , ∗ per ( Y )) and moreover D y u = D y e u so that (2.8) holds. Hermann DouanlaWe now gather some preliminary results we will need in our homogenization processes.We introduce the characteristic function c G of G = R Ny \ Q with Q = [ k ∈ Z N ( k + T ) . It follows from the closeness of T that Q is closed in R Ny so that G is an open subset of R Ny .Next, let e ∈ E be arbitrarily fixed and define V e = { u ∈ H ( W e ) : u = ¶W } . We equip V e with the H ( W e ) -norm which makes it a Hilbert space. We recall the followingclassical extension result [7]. Proposition 2.6.
For each e ∈ E there exists an operator P e of V e into H ( W ) with thefollowing properties: • P e sends continuously and linearly V e into H ( W ) . • ( P e v ) | W e = v for all v ∈ V e . • k D ( P e v ) k L ( W ) N ≤ c k Dv k L ( W e ) N for all v ∈ V e , where c is a constant independent of e . In the sequel, we will explicitly write the just-defined extension operator everywhereneeded but we will abuse notations on the local extension operator (see [7] for its definition):the extension to Y of u ∈ H per ( Y ∗ ) / R will still be denoted by u (this extension is an elementof H , ∗ per ( Y ) ).Now, let Q e = W \ ( eQ ) . This is an open set in R N and W e \ Q e is the intersectionof W with the collection of the holes crossing the boundary ¶W . We have the followingresult which implies that the holes crossing the boundary ¶W are of no effects as regards thehomogenization processes since they are in arbitrary narrow stripe along the boundary. Lemma 2.7. [25] Let K ⊂ W be a compact set independent of e . There is some e > suchthat W e \ Q e ⊂ W \ K for any < e ≤ e . Next, we introduce the space F = H ( W ) × L (cid:0) W ; H , ∗ per ( Y ) (cid:1) . Endowed with the following norm k v k F = k D x v + D y v k L ( W × Y ) ( v = ( v , v ) ∈ F ) , F is a Hilbert space admitting F ¥ = D ( W ) × [ D ( W ) ⊗ C ¥ , ∗ per ( Y )] (where C ¥ , ∗ per ( Y ) = { u ∈ C ¥ per ( Y ) : R Y ∗ u ( y ) dy = } ) as a dense subspace. This being so, for ( u , v ) ∈ F × F , let a W ( u , v ) = N (cid:229) i , j = ZZ W × Y ∗ a i j ( y ) (cid:18) ¶ u ¶ x j + ¶ u ¶ y j (cid:19) (cid:18) ¶ v ¶ x i + ¶ v ¶ y i (cid:19) dxdy . This define a symmetric, continuous bilinear form on F × F . We will need the followingresults whose proof can be found in [9].pectral Asymptotics in Porous Media 7 Lemma 2.8.
Fix F = ( y , y ) ∈ F ¥ and define F e : W → R ( e > ) by F e ( x ) = y ( x ) + ey ( x , x e ) ( x ∈ W ) . If ( u e ) e ∈ E ⊂ H ( W ) is such that ¶ u e ¶ x i s −→ ¶ u ¶ x i + ¶ u ¶ y i in L ( W ) ( ≤ i ≤ N ) as E ∋ e → for some u = ( u , u ) ∈ F , thena e ( u e , F e ) → a W ( u , F ) as E ∋ e → , where a e ( u e , F e ) = N (cid:229) i , j = Z W e a i j ( x e ) ¶ u e ¶ x j ¶F e ¶ x i dx . We now construct and point out the main properties of the so-called homogenized co-efficients. We put a ( u , v ) = N (cid:229) i , j = Z Y ∗ a i j ( y ) ¶ u ¶ y j ¶ v ¶ y i dy , (2.10) l j ( v ) = N (cid:229) k = Z Y ∗ a k j ( y ) ¶ v ¶ y k dy ( ≤ j ≤ N ) and l ( v ) = Z Y ∗ r ( y ) v ( y ) dy for u , v ∈ H per ( Y ∗ ) / R . Equipped with the norm k u k H per ( Y ∗ ) / R = k D y u k L ( Y ∗ ) N ( u ∈ H per ( Y ∗ ) / R ) , (2.11) H per ( Y ∗ ) / R is a Hilbert space. Proposition 2.9.
Let ≤ j ≤ N. The local variational problemsu ∈ H per ( Y ∗ ) / R and a ( u , v ) = l j ( v ) for all v ∈ H per ( Y ∗ ) / R (2.12) and u ∈ H per ( Y ∗ ) / R and a ( u , v ) = l ( v ) for all v ∈ H per ( Y ∗ ) / R (2.13) admit each a unique solution, assuming for (2.13) that M Y ∗ ( r ) = . Let 1 ≤ i , j ≤ N . The homogenized coefficients read q i j = Z Y ∗ a i j ( y ) dy − N (cid:229) l = Z Y ∗ a il ( y ) ¶c j ¶ y l ( y ) dy (2.14) Hermann Douanlawhere c j ( ≤ j ≤ N ) is the solution to (2.12). We recall that q ji = q i j ( ≤ i , j ≤ N ) andthere exists a constant a > N (cid:229) i , j = q i j x j x i ≥ a | x | for all x ∈ R N (see e.g., [2]).We now say a few words on the existence result for (1.1). The weak formulation of(1.1) reads: Find ( l e , u e ) ∈ C × V e , ( u e =
0) such that a e ( u e , v ) = l e ( r e u e , v ) W e , v ∈ V e , (2.15)where ( r e u e , v ) W e = Z W e r e u e vdx . Since r e changes sign, the classical results on the spectrum of semi-bounded self-adjointoperators with compact resolvent do not apply. To handle this, we follow the ideas in [23].The bilinear form ( r e u , v ) W e defines a bounded linear operator K e : V e → V e such that ( r e u , v ) W e = a e ( K e u , v ) ( u , v ∈ V e ) . The operator K e is symmetric and its domains D ( K e ) coincides with the whole V e , thus it isself-adjoint. Recall that the gradient norm is equivalent to the H ( W e ) -norm on V e . Lookingat K e u as the solution to the boundary value problem − div ( a ( x e ) D x ( K e u )) = r e u in W e a ( x e ) D x K e u · n ( x e ) = ¶ T e K e u ( x ) = ¶W , (2.16)we get a constant C e > k K e u k V e ≤ C e k u k L ( W e ) . As V e is compactly embeddedin L ( W e ) (indeed, H ( W e ) ֒ → L ( W e ) is compact as ¶W e is C ), the operator K e is compact.We can rewrite (2.15) as follows K e u e = µ e u e , µ e = l e . Notice that (see e.g., [5]) in the case r ≥ Y , the operator K e is positive and its spectrum s ( K e ) lives in [ , k K e k ] and µ e = s e ( K e ) . The essentialspectrum of a self-adjoint operator L is by definition s e ( L ) = s ¥ p ( L ) ∪ s c ( L ) , where s ¥ p ( L ) is the set of eigenvalues of infinite multiplicity and s c ( L ) is the continuous spectrum. Thespectrum of K e is described by the following proposition whose proof is omitted sincesimilar to that of [23, Lemma 1]. Lemma 2.10.
Let r ∈ L ¥ per ( Y ) be such that the sets { y ∈ Y ∗ : r ( y ) < } and { y ∈ Y ∗ : r ( y ) > } are both of positive Lebesgue measure. Then for any e > , we have s ( K e ) ⊂ pectral Asymptotics in Porous Media 9 [ −k K e k , k K e k ] and µ = is the only element of the essential spectrum s e ( K e ) . Moreover,the discrete spectrum of K e consists of two infinite sequencesµ , + e ≥ µ , + e ≥ · · · ≥ µ k , + e ≥ · · · → + , µ , − e ≤ µ , − e ≤ · · · ≤ µ k , − e ≤ · · · → − . Corollary 2.11.
The hypotheses are those of Lemma 2.10. Problem (1.1) has a discrete setof eigenvalues consisting of two sequences < l , + e ≤ l , + e ≤ · · · ≤ l k , + e ≤ · · · → + ¥ , > l , + e ≥ l , − e ≥ · · · ≥ l k , − e ≥ · · · → − ¥ . We are now in a position to state the main results of this paper.
In this section we state and prove homogenization results for both cases M Y ∗ ( r ) > M Y ∗ ( r ) =
0. The homogenization results in the case when M Y ∗ ( r ) < M Y ∗ ( r ) > r with − r . We start with the less technical case. M Y ∗ ( r ) > We start with the homogenization result for the positive part of the spectrum ( l k , + e , u k , + e ) e ∈ E . We assume (this is not a restriction) that the corresponding eigenfunctions are orthonormal-ized as follows Z W e r ( x e ) u k , + e u l , + e dx = d k , l k , l = , , · · · (3.1)The homogenization results states as Theorem 3.1.
We assume that W and T have C boundaries. For each k ≥ and each e ∈ E, let ( l k , + e , u k , + e ) be the k th positive eigencouple to (1.1) with M Y ∗ ( r ) > and (3.1).Then, there exists a subsequence E ′ of E such that l k , + e → l k in R as E ∋ e → P e u k , + e → u k in H ( W ) -weak as E ′ ∋ e → P e u k , + e → u k in L ( W ) as E ′ ∋ e → ¶ P e u k , + e ¶ x j s −→ ¶ u k ¶ x j + ¶ u k ¶ y j in L ( W ) as E ′ ∋ e → ( ≤ j ≤ N ) (3.5)0 Hermann Douanla where ( l k , u k ) ∈ R × H ( W ) is the k th eigencouple to the spectral problem − N (cid:229) i , j = ¶¶ x i (cid:18) M Y ∗ ( r ) q i j ¶ u ¶ x j (cid:19) = l u in W u = on ¶W Z W | u | dx = M Y ∗ ( r ) , (3.6) u k ∈ L ( W ; H , ∗ per ( Y )) and where the coefficients { q i j } ≤ i , j ≤ N are defined by (2.14). More-over, for almost every x ∈ W the following hold true: (i) The restriction to Y ∗ of u k ( x ) is the solution to the variational problem u k ( x ) ∈ H per ( Y ∗ ) / R a ( u k ( x ) , v ) = − N (cid:229) i , j = ¶ u k ¶ x j Z Y ∗ a i j ( y ) ¶ v ¶ y i dy ∀ v ∈ H per ( Y ∗ ) / R , (3.7) the bilinear form a ( · , · ) being defined by (2.10); (ii) We have u k ( x , y ) = − N (cid:229) j = ¶ u k ¶ x j ( x ) c j ( y ) a.e. in ( x , y ) ∈ W × Y ∗ , (3.8) where c j is the solution to the cell problem (2.12).Proof. We present only the outlines since this proof is similar but less technical to that ofthe case M Y ∗ ( r ) = k ≥
1. By means of the minimax principle, as in [31], one easily proves the existenceof a constant C independent of e such that l k , + e < C . Clearly, for fixed E ∋ e > u k , + e liesin V e , and N (cid:229) i , j = Z W e a i j ( x e ) ¶ u k , + e ¶ x j ¶ v ¶ x i dx = l k , + e Z W e r ( x e ) u k , + e v dx (3.9)for any v ∈ V e . Bear in mind that R W e r ( x e )( u k , + e ) dx = v = u k , + e in (3.9). Theboundedness of the sequence ( l k , + e ) e ∈ E and the ellipticity assumption (1.2) imply at once bymeans of Proposition 2.6 that the sequence ( P e u k , + e ) e ∈ E is bounded in H ( W ) . Theorem 2.5applies and gives us u k = ( u k , u k ) ∈ F such that for some l k ∈ R and some subsequence E ′ ⊂ E we have (3.2)-(3.5), where (3.4) is a direct consequence of (3.3) by the Rellich-Kondrachov theorem. For fixed e ∈ E ′ , let F e be as in Lemma 2.8. Multiplying bothsides of the first equality in (1.1) by F e and integrating over W e leads us to the variational e -problem N (cid:229) i , j = Z W e a i j ( x e ) ¶ P e u k , + e ¶ x j ¶F e ¶ x i dx = l k , + e Z W e ( P e u k , + e ) r ( x e ) F e dx . (3.10)pectral Asymptotics in Porous Media 11Sending e ∈ E ′ to 0, keeping (3.2)-(3.5) and Lemma 2.8 in mind, we obtain N (cid:229) i , j = ZZ W × Y ∗ a i j ( y ) (cid:18) ¶ u k ¶ x j + ¶ u k ¶ y j (cid:19) (cid:18) ¶y ¶ x i + ¶y ¶ y i (cid:19) dxdy = l k ZZ W × Y ∗ u k y ( x ) r ( y ) dxdy . Therefore, ( l k , u k ) ∈ R × F solves the following global homogenized spectral problem : Find ( l , u ) ∈ C × F such that N (cid:229) i , j = ZZ W × Y ∗ a i j ( y ) (cid:18) ¶ u ¶ x j + ¶ u ¶ y j (cid:19) (cid:18) ¶y ¶ x i + ¶y ¶ y i (cid:19) dxdy = l M Y ∗ ( r ) Z W u y dx for all F ∈ F , (3.11)which leads to the macroscopic and microscopic problems (3.6)-(3.7) without any majordifficulty.As regards the normalization condition in (3.6), we use the decomposition W e = Q e ∪ ( W e \ Q e ) and the equality Q e = W ∩ e G . On the one hand, when E ′ ∋ e → Z Q e r ( x e )( P e u k , + e )( P e u l , + e ) dx → M Y ∗ ( r ) Z W u k u l dx , k , l = , , · · · since Z Q e r ( x e )( P e u k , + e )( P e u l , + e ) dx = Z W c G ( x e ) r ( x e )( P e u k , + e )( P e u l , + e ) dx and ( P e u k , + e ) c e G r e ⇀ M Y ∗ ( r ) u k in L ( W ) -weak and P e u l , + e → u l in L ( W ) -strong as E ′ ∋ e →
0. On the other hand, the same line of reasoning as in the proof of [10, Proposition 3.6]leads to lim E ′ ∋ e → Z W e \ Q e r ( x e )( P e u k , + e )( P e u l , + e ) dx = { u k , + } ¥ k = is an orthogonal basis in L ( W ) . Remark . • The eigenfunctions { u k } ¥ k = are orthonormalized by Z W u k u l dx = d k , l M Y ∗ ( r ) k , l = , , , · · ·• If l k is simple (this is the case for l ), then by Theorem 3.1, l k , + e is also simple, forsmall e , and we can choose the eigenfunctions u k , + e such that the convergence results(3.3)-(3.5) hold for the whole sequence E . In this case, the following corrector typeresult holds: lim E ∋ e → (cid:13)(cid:13)(cid:13) D x ( P e u k , + e ( · )) − D x u k ( · ) − D y u k ( · , . e ) (cid:13)(cid:13)(cid:13) L ( W ) N = . • Replacing r with − r in (1.1), Theorem 3.1 also applies to the negative part of thespectrum in the case M Y ∗ ( r ) < We now investigate the negative part of the spectrum ( l k , − e , u k , − e ) e ∈ E . Before we can do thiswe need a few preliminaries and stronger regularity hypotheses on T , r and the coefficients ( a i j ) Ni , j = . We assume in this subsection that ¶ T is C , d and r and the coefficients ( a i j ) Ni , j = are d -H ¨older continuous (0 < d < Find ( l , q ) ∈ C × H per ( Y ∗ ) − N (cid:229) i , j = ¶¶ y j (cid:18) a i j ( y ) ¶q¶ y i (cid:19) = lr ( y ) q in Y ∗ N (cid:229) i , j = a i j ( y ) ¶q¶ y j n i = ¶ T (3.13)and possesses a spectrum with similar properties to that of (1.1), two infinite (positive andnegative) sequences. We recall that (3.13) admits a unique nontrivial eigenvalue having aneigenfunction with definite sign, the first negative one, since we have M Y ∗ ( r ) > ( l − , q − ) , the first negative eigencouple to(3.13). After proper sign choice we assume that q − > ∈ Y ∗ . (3.14)We also recall that q − is d -H ¨older continuous(see e.g., [12]), hence can be extended to a Y -periodic function living in L ¥ ( R Ny ) still denoted by q − . Notice that we have Z Y ∗ r ( y )( q − ( y )) dy < , (3.15)as is easily seen from the variational equality ( keep the ellipticity hypothesis (1.2) in mind) N (cid:229) i , j = Z Y ∗ a i j ¶q − ¶ y j ¶q − ¶ y i dy = l − Z Y ∗ r ( y )( q − ( y )) dy . Bear in mind that problem (3.13) induces by a scaling argument the following equalities: − N (cid:229) i , j = ¶¶ x j (cid:18) a i j ( x e ) ¶q e ¶ x i (cid:19) = e lr ( x e ) q ( x e ) in Q e N (cid:229) i , j = a i j ( x e ) ¶q e ¶ x j n i ( x e ) = ¶ Q e , (3.16)where q e ( x ) = q ( x e ) . However, q e is not zero on ¶W . We now introduce the followingspectral problem (with an indefinite density function) Find ( x e , v e ) ∈ C × V e − N (cid:229) i , j = ¶¶ x j (cid:18)e a i j ( x e ) ¶ v e ¶ x i (cid:19) = x e e r ( x e ) v e ( x ) in W e N (cid:229) i , j = e a i j ( x e ) ¶ v e ¶ x j n i ( x e ) = ¶ T e v e ( x ) = ¶W , (3.17)pectral Asymptotics in Porous Media 13with new spectral eigencouple ( x e , v e ) ∈ C × V e , where e a i j ( y ) = ( q − ) ( y ) a i j ( y ) and e r ( y ) =( q − ) ( y ) r ( y ) . Notice that e a i j ∈ L ¥ per ( Y ) and e r ∈ L ¥ per ( Y ) . As 0 < c − ≤ q − ( y ) ≤ c + < + ¥ ( c − , c + ∈ R ), the operator on the left hand side of (3.17) is uniformly elliptic and Theorem3.1 applies to the negative part of the spectrum of (3.17) (see (3.15) and Remark 3.2). Theeffective spectral problem for (3.17) reads − N (cid:229) i , j = ¶¶ x j (cid:18)e q i j ¶ v ¶ x i (cid:19) = x M Y ∗ ( e r ) v in W v = ¶W Z W | v | dx = − M Y ∗ ( e r ) . (3.18)The effective coefficients { e q i j } ≤ i , j ≤ N being defined as expected, i.e., e q i j = Z Y ∗ e a i j ( y ) dy − N (cid:229) l = Z Y ∗ e a il ( y ) ¶ e c j ¶ y l ( y ) dy , (3.19)with e c l ∈ H per ( Y ∗ ) / R ( l = , ..., N ) being the solution to the following local problem e c l ∈ H per ( Y ∗ ) / R N (cid:229) i , j = Z Y ∗ e a i j ( y ) ¶ e c l ¶ y j ¶ v ¶ y i dy = N (cid:229) i = Z Y ∗ e a il ( y ) ¶ v ¶ y i dy for all v ∈ H per ( Y ∗ ) / R . (3.20)We will use the following notation in the sequel: e a ( u , v ) = N (cid:229) i , j = Z Y ∗ e a i j ( y ) ¶ u ¶ y j ¶ v ¶ y i dy (cid:0) u , v ∈ H per ( Y ∗ ) / R (cid:1) . (3.21)Notice that the spectrum of (3.18) is as follows0 > x > x ≥ x ≥ · · · ≥ x j ≥ · · · → − ¥ as j → ¥ . Making use of (3.16), the same line of reasoning as in [31, Lemma 6.1] shows that thenegative spectral parameters of problems (1.1) and (3.17) verify: u k , − e = ( q − ) e v k , − e ( e ∈ E , k = , · · · ) and l k , − e = e l − + x k , − e + o ( ) , ( e ∈ E , k = , · · · ) . The presence of the term o ( ) is due to integrals over W e \ Q e , like the one in (3.12), whichconverge to zero with e , remember that (3.16) holds in Q e but not W e . As will be seen below,the sequence ( x k , − e ) e ∈ E is bounded in R . In another words, l k , − e is of order 1 / e and tendsto − ¥ as e goes to zero. It is now clear why the limiting behavior of negative eigencouplesis not straightforward as that of positive ones.4 Hermann DouanlaThe suitable orthonormalization condition for (3.17) is the one the reader is expecting: Z W e e r ( x e ) v k , − e v l , − e dx = − d k , l k , l = , , · · · (3.22)We now state the homogenization theorem for the negative part of the spectrum of (1.1). Theorem 3.3.
We assume that ¶ T is C , d and r and the coefficients ( a i j ) Ni , j = are d -H¨oldercontinuous ( < d < ). For each k ≥ and each e ∈ E, let ( l k , − e , u k , − e ) be the k th negativeeigencouple to (1.1) with M Y ∗ ( r ) > and (3.22). Then, there exists a subsequence E ′ of Esuch that l k , − e − l − e → x k in R as E ∋ e → P e v k , − e → v k in H ( W ) -weak as E ′ ∋ e → P e v k , − e → v k in L ( W ) as E ′ ∋ e → ¶ P e v k , − e ¶ x j s −→ ¶ v k ¶ x j + ¶ v k ¶ y j in L ( W ) as E ′ ∋ e → ( ≤ j ≤ N ) (3.26) where ( x k , v k ) ∈ R × H ( W ) is the k th eigencouple to the spectral problem − N (cid:229) i , j = ¶¶ x i (cid:18) M Y ∗ ( e r ) e q i j ¶ v ¶ x j (cid:19) = x v in W v = on ¶W Z W | v | dx = − M Y ∗ ( e r ) , (3.27) v k ∈ L ( W ; H , ∗ per ( Y )) and where the coefficients { e q i j } ≤ i , j ≤ N are defined by (3.19). Moreover,for almost every x ∈ W the following hold true: (i) The restriction to Y ∗ of v k ( x ) is the solution to the variational problem v k ( x ) ∈ H per ( Y ∗ ) / R e a ( v k ( x ) , u ) = − N (cid:229) i , j = ¶ v k ¶ x j Z Y ∗ e a i j ( y ) ¶ u ¶ y i dy ∀ u ∈ H per ( Y ∗ ) / R , (3.28) the bilinear form e a ( · , · ) being defined by (3.21); (ii) We have v k ( x , y ) = − N (cid:229) j = ¶ v k ¶ x j ( x ) e c j ( y ) a.e. in ( x , y ) ∈ W × Y ∗ , (3.29) where e c j is the solution to the cell problem (3.20).Remark . • The eigenfunctions { v k } ¥ k = are orthonormalized by Z W v k v l dx = − d k , l M Y ∗ ( e r ) k , l = , , , · · · pectral Asymptotics in Porous Media 15 • If x k is simple (this is the case for x ), then by Theorem 3.3, l k , − e is also simple, forsmall e , and we can choose the ‘eigenfunctions’ v k , − e such that the convergence results(3.24)-(3.26) hold for the whole sequence E . In this case, the following corrector typeresult holds: lim E ∋ e → (cid:13)(cid:13)(cid:13) D x ( P e v k , − e ( · )) − D x v k ( · ) − D y v k ( · , . e ) (cid:13)(cid:13)(cid:13) L ( W ) N = . • Replacing r with − r in (1.1), Theorem 3.3 adapts to the positive part of the spectrumin the case M Y ∗ ( r ) < M Y ∗ ( r ) = We prove a homogenization result for both the positive part and the negative part of thespectrum simultaneously. As will be clear in the proof of Theorem 3.5 below, we assumein this case that the eigenfunctions are orthonormalized as follows Z W e r ( x e ) u k , ± e u l , ± e dx = ± ed k , l k , l = , , · · · (3.30)Let c be the solution to ( . ) and put n = N (cid:229) i , j = Z Y ∗ a i j ( y ) ¶c ¶ y j ¶c ¶ y i dy . (3.31)Indeed, the right hand side of (3.31) is positive. We now recall that the following spectralproblem for a quadratic operator pencil with respect to n , − N (cid:229) i , j = ¶¶ x j (cid:18) q i j ¶ u ¶ x i (cid:19) = l n u in W u = ¶W , (3.32)has a spectrum consisting of two infinite sequences0 < l , + < l , + ≤ · · · ≤ l k , + ≤ . . . , lim k → + ¥ l k , + = + ¥ and 0 > l , − > l , − ≥ · · · ≥ l k , − ≥ . . . , lim k → + ¥ l k , − = − ¥ . with l k , + = − l k , − k = , , · · · and with the corresponding eigenfunctions u k , + = u k , − .We note by passing that l , + and l , − are simple. We are now in a position to state thehomogenization result in the present case. Theorem 3.5.
We assume that W and T have C boundaries. For each k ≥ and each e ∈ E,let ( l k , ± e , u k , ± e ) be the ( k , ± ) th eigencouple to (1.1) with M Y ∗ ( r ) = and (3.30). Then, there exists a subsequence E ′ of E such that el k , ± e → l k , ± in R as E ∋ e → P e u k , ± e → u k , ± in H ( W ) -weak as E ′ ∋ e → P e u k , ± e → u k , ± in L ( W ) as E ′ ∋ e → ¶ P e u k , ± e ¶ x j s −→ ¶ u k , ± ¶ x j + ¶ u k , ± ¶ y j in L ( W ) as E ′ ∋ e → ( ≤ j ≤ N ) (3.36) where ( l k , ± , u k , ± ) ∈ R × H ( W ) is the ( k , ± ) th eigencouple to the following spectral problemfor a quadratic operator pencil with respect to n , − N (cid:229) i , j = ¶¶ x i (cid:18) q i j ¶ u ¶ x j (cid:19) = l n u in W u = on ¶W , (3.37) u k , ± ∈ L ( W ; H , ∗ per ( Y )) and where the coefficients { q i j } ≤ i , j ≤ N are defined by (2.14). Wehave the following normalization condition Z W | u k , ± | dx = ± l k , ± n k = , , · · · (3.38) Moreover, for almost every x ∈ W the following hold true: (i) The restriction to Y ∗ of u k , ± ( x ) is the solution to the variational problem u k , ± ( x ) ∈ H per ( Y ∗ ) / R a ( u k , ± ( x ) , v ) = l k , ± u k , ± ( x ) Z Y ∗ r ( y ) v ( y ) dy − N (cid:229) i , j = ¶ u k , ± ¶ x j ( x ) Z Y ∗ a i j ( y ) ¶ v ¶ y i dy ∀ v ∈ H per ( Y ∗ ) / R , (3.39) the bilinear form a ( · , · ) being defined by (2.10); (ii) We haveu k , ± ( x , y ) = l k , ± u k , ± ( x ) c ( y ) − N (cid:229) j = ¶ u k , ± ¶ x j ( x ) c j ( y ) a.e. in ( x , y ) ∈ W × Y ∗ , (3.40) where c j ( ≤ j ≤ N ) and c are the solutions to the cell problems (2.12) and (2.13),respectively.Proof. Fix k ≥
1, using the minimax principle, as in [31], we get a constant C independentof e such that | el k , ± e | < C . We have u k , ± e ∈ V e and N (cid:229) i , j = Z W e a i j ( x e ) ¶ u k , ± e ¶ x j ¶ v ¶ x i dx = ( el k , ± e ) e Z W e r ( x e ) u k , ± e v dx (3.41)for any v ∈ V e . Bear in mind that R W e r ( x e )( u k , ± e ) dx = ± e and choose v = u k , ± e in (3.41).The boundedness of the sequence ( el k , ± e ) e ∈ E and the ellipticity assumption (1.2) implypectral Asymptotics in Porous Media 17at once by means of Proposition 2.6 that the sequence ( P e u k , ± e ) e ∈ E is bounded in H ( W ) .Theorem 2.5 applies and gives us u k , ± = ( u k , ± , u k , ± ) ∈ F such that for some l k , ± ∈ R andsome subsequence E ′ ⊂ E we have (3.33)-(3.36), where (3.35) is a direct consequence of(3.34) by the Rellich-Kondrachov theorem. For fixed e ∈ E ′ , let F e be as in Lemma 2.8.Multiplying both sides of the first equality in (1.1) by F e and integrating over W e leads usto the variational e -problem N (cid:229) i , j = Z W e a i j ( x e ) ¶ P e u k , ± e ¶ x j ¶F e ¶ x i dx = ( el k , ± e ) e Z W e ( P e u k , ± e ) r ( x e ) F e dx . Sending e ∈ E ′ to 0, keeping (3.33)-(3.36) and Lemma 2.8 in mind, we obtain a W ( u k , ± , F ) = l k , ± ZZ W × Y ∗ (cid:16) u k , ± ( x , y ) y ( x ) r ( y ) + u k , ± y ( x , y ) r ( y ) (cid:17) dxdy (3.42)The right-hand side follows as explained below. Using the decomposition W e = Q e ∪ ( W e \ Q e ) and the equality Q e = W ∩ e G we arrive at1 e Z W e ( P e u k , ± e ) r ( x e ) F e dx = e Z W ( P e u k , ± e ) y ( x ) r ( x e ) c G ( x e ) dx + Z W ( P e u k , ± e ) y ( x , x e ) r ( x e ) c G ( x e ) dx + o ( ) . On the one hand we havelim E ′ ∋ e → Z W ( P e u k , ± e ) y ( x , x e ) r ( x e ) c G ( x e ) dx = ZZ W × Y u k , ± y ( x , y ) r ( y ) c G ( y ) dxdy . On the other hand, owing to (2.9) of Theorem 2.5, the following holds:lim E ′ ∋ e → e Z W ( P e u k , ± e ) y ( x ) r ( x e ) c G ( x e ) dx = ZZ W × Y u k , ± ( x , y ) y ( x ) r ( y ) c G ( y ) dxdy . Indeed rc G ∈ L per ( Y ) / R as we clearly have R Y r ( y ) c G ( y ) dy = R Y ∗ r ( y ) dy =
0. We havejust proved that ( l k , ± , u k , ± ) ∈ R × F solves the following global homogenized spectralproblem : Find ( l , u ) ∈ C × F such that a W ( u , F ) = l ZZ W × Y ∗ ( u ( x , y ) y ( x ) r ( y ) + u y ( x , y ) r ( y )) dxdy for all F ∈ F . (3.43)To prove (i), choose F = ( y , y ) in (3.42) such that y = y = j ⊗ v , where j ∈ D ( W ) and v ∈ H per ( Y ∗ ) / R to get Z W j ( x ) " N (cid:229) i , j = Z Y ∗ a i j ( y ) ¶ u k , ± ¶ x j + ¶ u k , ± ¶ y j ! ¶ v ¶ y i dy dx = Z W j ( x ) (cid:20) l k , ± u k , ± ( x ) Z Y ∗ v ( y ) r ( y ) dy (cid:21) dx j , we have a.e. in W N (cid:229) i , j = Z Y ∗ a i j ( y ) ¶ u k , ± ¶ x j + ¶ u k , ± ¶ y j ! ¶ v ¶ y i dy = l k , ± u k , ± ( x ) Z Y ∗ v ( y ) r ( y ) dy for any v in H per ( Y ∗ ) / R , which is nothing but (3.39).Fix x ∈ W , multiply both sides of (2.12) by − ¶ u k , ± ¶ x j ( x ) and sum over 1 ≤ j ≤ N . Addingside by side to the resulting equality that obtained after multiplying both sides of (2.13) by l k , ± u k , ± ( x ) , we realize that z ( x ) = − (cid:229) Nj = ¶ u k , ± ¶ x j ( x ) c j ( y ) + l k , ± u k , ± ( x ) c ( y ) solves (3.39).Hence u k , ± ( x , y ) = l k , ± u k , ± ( x ) c ( y ) − N (cid:229) j = ¶ u k , ± ¶ x j ( x ) c j ( y ) a.e. in ( x , y ) ∈ W × Y ∗ . (3.44)by uniqueness of the solution to the variational problem (3.39). Thus (3.40).Considering now F = ( y , y ) in (3.42) such that y ∈ D ( W ) and y = N (cid:229) i , j = ZZ W × Y ∗ a i j ( y ) ¶ u k , ± ¶ x j + ¶ u k , ± ¶ y j ! ¶y ¶ x i dxdy = l k , ± ZZ W × Y ∗ u k , ± ( x , y ) r ( y ) y ( x ) dxdy , which by means of (3.44) leads to N (cid:229) i , j = Z W q i j ¶ u k , ± ¶ x j ¶y ¶ x i dx + l k , ± N (cid:229) i , j = Z W u k , ± ( x ) ¶y ¶ x i dx (cid:18) Z Y ∗ a i j ( y ) ¶c ¶ y j ( y ) dy (cid:19) = − l k , ± N (cid:229) j = Z W ¶ u k , ± ¶ x j y ( x ) dx (cid:18) Z Y ∗ r ( y ) c j ( y ) dy (cid:19) (3.45) +( l k , ± ) Z W u k , ± ( x ) y ( x ) dx (cid:18) Z Y ∗ r ( y ) c ( y ) dy (cid:19) . Choosing c l ( ≤ l ≤ N ) as test function in (2.13) and c as test function in (2.12) weobserve that N (cid:229) j = Z Y ∗ a l j ( y ) ¶c ¶ y j ( y ) dy = Z Y ∗ r ( y ) c l ( y ) dy = a ( c l , c ) ( l = , · · · N ) . Thus, in (3.45), the second term in the left-hand side is equal to the first one in the right-handside. This leaves us with Z W q i j ¶ u k , ± ¶ x j ¶y ¶ x i dx = ( l k , ± ) Z W u k , ± ( x ) y ( x ) dx (cid:18) Z Y ∗ r ( y ) c ( y ) dy (cid:19) . (3.46)Choosing c as test function in (2.13) reveals that Z Y ∗ r ( y ) c ( y ) dy = a ( c , c ) = n . pectral Asymptotics in Porous Media 19Hence N (cid:229) i , j = Z W q i j ¶ u k , ± ¶ x j ¶y ¶ x i dx = ( l k , ± ) n Z W u k , ± ( x ) y ( x ) dx , and − N (cid:229) i , j = ¶¶ x i q i j ¶ u k , ± ¶ x j ( x ) ! = ( l k , ± ) n u k , ± ( x ) in W . Thus the convergence (3.33) holds for the whole sequence E . As regards (3.38), we noticethat for fixed k ≥ f ∈ D ( W ) one has (keep (2.9) in mind)lim E ′ ∋ e → e Z W ( P e u k , ± e ) f ( x ) r ( x e ) c G ( x e ) dx = ZZ W × Y ∗ u k , ± ( x , y ) f ( x ) r ( y ) dxdy . Hence, as E ′ ∋ e → e ( P e u k , ± e ) r e c e G ⇀ Z Y ∗ u k , ± ( · , y ) r ( y ) dy in L ( W ) − weak . Using once again the decomposition W e = Q e ∪ ( W e \ Q e ) and the equality Q e = W ∩ e G , weget as E ′ ∋ e → e Z W e ( P e u k , ± e )( P e u l , ± e ) r ( x e ) dx → ZZ W × Y ∗ u k , ± ( x , y ) u l , ± ( x ) r ( y ) dxdy , for fixed l ≥
1. This together with (3.30) and (3.44) yields l k , ± n Z W u l , ± u k , ± dx − N (cid:229) j = a ( c j , c ) Z W ¶ u k , ± ¶ x j u l , ± dx = ± d k , l , k , l = , , · · · (3.47)If k = l , then by Green’s formula the sum in the left-hand side vanishes and (3.47) reducesto the desired result. This concludes the proof. Remark . • Permuting k and l in (3.47) and adding side by side the resulting equalityto (3.47) we realize that the eigenfunctions { u k , ± } ¥ k = are orthonormalized by Z W u l , ± ( x ) u k , ± ( x ) dx = ± d k , l n ( l k , ± + l l , ± ) k , l = , , · · ·• If l k , ± is simple (this is the case for l , ± ), then by Theorem 3.5, l k , ± e is also simple,for small e , and we can choose the eigenfunctions u k , ± e such that the convergence re-sults (3.34)-(3.36) hold for the whole sequence E . In this case, the following correctortype result holds:lim E ∋ e → (cid:13)(cid:13)(cid:13) D x ( P e u k , ± e ( · )) − D x u k , ± ( · ) − D y u k , ± ( · , . e ) (cid:13)(cid:13)(cid:13) L ( W ) N = . Acknowledgments
The author is grateful to Dr. Jean Louis Woukeng for helpful discussions.0 Hermann Douanla
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