Homogenization of Steklov spectral problems with indefinite density function in perforated domains
aa r X i v : . [ m a t h . A P ] A ug H OMOGE NIZATION OF S T E KLOV S PE CT RAL P ROB LEMS WI T H I NDE FI NITE D E NSI TY F UNCT IONI N P E RFORATED D OMAI NS H ERMANN D OUANLA ∗ Department of Mathematical SciencesChalmers University of TechnologyGothenburg, SE-41296, Sweden
Abstract
The asymptotic behavior of second order self-adjoint elliptic Steklov eigenvalue prob-lems with periodic rapidly oscillating coefficients and with indefinite (sign-changing)density function is investigated in periodically perforated domains. We prove that thespectrum of this problem is discrete and consists of two sequences, one tending to − ¥ and another to + ¥ . The limiting behavior of positive and negative eigencouples de-pends crucially on whether the average of the weight over the surface of the referencehole is positive, negative or equal to zero. By means of the two-scale convergencemethod, we investigate all three cases. AMS Subject Classification:
Keywords : Homogenization, eigenvalue problems, perforated domains, indefinite weightfunction, two-scale convergence.
In 1902, with a motivation coming from Physics, Steklov[33] introduced the followingproblem D u = W¶ u ¶ n = rl u on ¶W , (1.1)where l is a scalar and r is a density function. The function u represents the steady statetemperature on W such that the flux on the boundary ¶W is proportional to the temperature.In two dimensions, assuming r =
1, problem (1.1) can also be interpreted as a membranewith whole mass concentrated on the boundary. This problem has been later referred toas Steklov eigenvalue problem (Steklov is often transliterated as ”Stekloff”). Moreover,eigenvalue problems also arise from many nonlinear problems after linearization (see e.g.,the work of Hess and Kato[15, 16] and that of de Figueiredo[13]). This paper deals with the ∗ E-mail address: [email protected]
Hermann Douanlalimiting behavior of a sequence of second order elliptic Steklov eigenvalue problems withindefinite(sign-changing) density function in perforated domains.Let W be a bounded domain in R Nx (the numerical space of variables x = ( x , ..., x N ) ),with C boundary ¶W and with integer N ≥
2. We define the perforated domain W e asfollows. Let T ⊂ Y = ( , ) N be a compact subset of Y with C boundary ¶ T ( ≡ S ) andnonempty interior. For e >
0, we define 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 ) Ni = denotes the outer unit normal vector to S with respect to Y ∗ .We are interested in the spectral asymptotics (as e →
0) of the following Steklov eigen-value problem − N (cid:229) i , j = ¶¶ x i (cid:18) a i j ( x e ) ¶ u e ¶ x j (cid:19) = W e N (cid:229) i , j = a i j ( x e ) ¶ u e ¶ x j n i ( x e ) = r ( x e ) l e u e on ¶ T e u e = ¶W , (1.2)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 | (1.3)for all x ∈ R N and for almost all y ∈ R Ny , where | x | = | x | + · · · + | x N | . The densityfunction r ∈ C per ( Y ) changes sign on S , that is, both the set { y ∈ S , r ( y ) < } and { y ∈ S , r ( y ) > } are of positive N − 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 = − ¥ . teklov Eigenvalue Problems with Sing-changing Density Function 3The asymptotic behavior of the eigencouples depends crucially on whether the average ofthe density r over S , M S ( r ) = R S r ( y ) d s ( y ) , is positive, negative or equal to zero. All threecases are carefully investigated in this paper.The homogenization of spectral problems has been widely explored. In a fixed do-main, homogenization of spectral problems with point-wise positive density function goesback to Kesavan [18, 19]. Spectral asymptotics in perforated domains was studied byVanninathan[35] and later in many other papers, including [11, 12, 17, 28, 29, 32] andthe references therein. Homogenization of elliptic operators with sing-changing densityfunction in a fixed domain with Dirichlet boundary conditions has been investigated byNazarov et al. [22, 23, 24] via a combination of formal asymptotic expansion with Tar-tar’s energy method. In porous media, spectral asymptotics of elliptic operator with signchanging density function is studied in [10] with the two scale convergence method.The asymptotics of Steklov eigenvalue problems in periodically perforated domainswas studied in [35] for the laplace operator and constant density ( r =
1) using asymptoticexpansion and Tartar’s test function method. The same problem for a second order periodicelliptic operator has been studied in [29] (with C ¥ coefficients) and in [11] (with L ¥ coef-ficient) but still with constant density ( r = L ¥ coefficientsand a sing-changing density function. We obtain accurate and concise homogenizationresults in all three cases: M S ( r ) > M S ( r ) = M S ( r ) < i) If M S ( r ) >
0, then the positive eigencouples behave like in the case of point-wise positivedensity function, i.e., for k ≥ l k , + e is of order e and e l k , + e converges as e → k th eigenvalue of the limit Dirichlet spectral problem, corresponding extendedeigenfunctions converge along subsequences.As regards the ”negative” eigencouples, l k , − e converges to − ¥ at the rate e and thecorresponding eigenfunctions oscillate rapidly. We use a factorization technique ([20,35]) to prove that l k , − e = e l − + x k , − e + o ( ) , k = , · · · where ( l − , q − ) is the first negative eigencouple to the following local Steklov spectralproblem − div ( a ( y ) D y q ) = Y ∗ a ( y ) D y q · n = lr ( y ) q on S q Y − periodic , (1.4)and { x k , ± e } ¥ k = are eigenvalues of a Steklov eigenvalue problem similar to (1.2). Wethen prove that { l k , − e e − l − e } converges to the k th eigenvalue of a limit Dirichlet spectral Hermann Douanlaproblem which is different from that obtained for positive eigenvalues. As regardseigenfunctions, extensions of { u k , − e ( q − ) e } e ∈ E - where ( q − ) e ( x ) = q − ( x e ) - converge alongsubsequences to the k th eigenfunctions of the limit problem. ii) If M S ( r ) =
0, then the limit spectral problem generates a quadratic operator pencil and l k , ± e converges to the ( k , ± ) th eigenvalue of the limit operator, extended eigenfunc-tions converge along subsequences as well. This case requires a new convergenceresult as regards the two-scale convergence theory, Lemma 2.9. iii) The case when M S ( r ) < M S ( r ) >
0, just replace r with − r .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 theLebesgue measure denoted by dx = dx ... dx N . The usual gradient operator will be denotedby D . For the sake of simple notations we hide trace operators. The rest of the paper isorganized as follows. Section 2 deals with some preliminary results while homogenizationprocesses are considered in Section 3. We first recall the definition and the main compactness theorems of the two-scale conver-gence method. Let W be a smooth open bounded set in R Nx (integer N ≥
2) and Y = ( , ) N ,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 [1, 25, 27] are cornerstones of the two-scale con-vergence 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 ) . teklov Eigenvalue Problems with Sing-changing Density Function 5 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 ) -weaku e → u in L ( W ) ¶ u e ¶ x j s −→ ¶ u ¶ x j + ¶ u ¶ y j in L ( W ) ( ≤ j ≤ N ) 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 (2.2) for y ∈ D ( W ) ⊗ ( L per ( Y ) / R ) .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 (2.3) u e → u in L ( W ) (2.4) ¶ u e ¶ x j s −→ ¶ u ¶ x j + ¶ u ¶ y j in L ( W ) ( ≤ j ≤ N ) (2.5) 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 (2.6) 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 (2.5) holds. Hermann DouanlaIn the sequel, S e stands for ¶ T e and the surface measures on S and S e are denotedby d s ( y ) ( y ∈ Y ), d s e ( x ) ( x ∈ W , e ∈ E ), respectively. The space of squared integrablefunctions, with respect to the previous measures on S and S e are denoted by L ( S ) and L ( S e ) respectively. Since the volume of S e grows proportionally to e as e →
0, we endow L ( S e ) with the scaled scalar product[3, 30, 31] ( u , v ) L ( S e ) = e Z S e u ( x ) v ( x ) d s e ( x ) (cid:0) u , v ∈ L ( S e ) (cid:1) . Definition 2.1 and theorem 2.2 then generalize as
Definition 2.6.
A sequence ( u e ) e ∈ E ⊂ L ( S e ) is said to two-scale converge to some u ∈ L ( W × S ) if as E ∋ e → e Z S e u e ( x ) f ( x , x e ) d s e ( x ) → ZZ W × S u ( x , y ) f ( x , y ) dxd s ( y ) for all f ∈ C ( W ; C per ( Y )) . Theorem 2.7.
Let ( u e ) e ∈ E be a sequence in L ( S e ) such that e Z S e | u e ( x ) | d s e ( x ) ≤ Cwhere C is a positive constant independent of e . There exists a subsequence E ′ of E suchthat ( u e ) e ∈ E ′ two-scale converges to some u ∈ L ( W ; L ( S )) in the sense of definition 2.6. In the case when ( u e ) e ∈ E is the sequence of traces on S e of functions in H ( W ) , one canlink its usual two-scale limit with its surface two-scale limits. The following propositionwhose proof can be found in [3] clarifies this. Proposition 2.8.
Let ( u e ) e ∈ E ⊂ H ( W ) be such that k u e k L ( W ) + e k Du e k L ( W ) N ≤ C , where C is a positive constant independent of e and D denotes the usual gradient. Thesequence of traces of ( u e ) e ∈ E on S e satisfies e Z S e | u e ( x ) | d s e ( x ) ≤ C ( e ∈ E ) and up to a subsequence E ′ of E, it two-scale converges in the sense of Definition 2.6 tosome u ∈ L ( W ; L ( S )) which is nothing but the trace on S of the usual two-scale limit, afunction in L ( W ; H per ( Y )) . More precisely, as E ′ ∋ e → e Z S e u e ( x ) f ( x , x e ) d s e ( x ) → ZZ W × S u ( x , y ) f ( x , y ) dxd s ( y ) , Z W u e ( x ) f ( x , x e ) dxdy → ZZ W × Y u ( x , y ) f ( x , y ) dxdy , for all f ∈ C ( W ; C per ( Y )) . teklov Eigenvalue Problems with Sing-changing Density Function 7In our homogenization process, more precisely in the case when M S ( r ) =
0, we willneed a generalization of (2.2) to periodic surfaces. Notice that (2.2) was proved for the firsttime in a deterministic setting by Nguetseng and Woukeng in [27] but to the best of ourknowledge its generalization to periodic surfaces is not yet available in the literature. Westate and prove it below.
Lemma 2.9.
Let ( u e ) e ∈ E ⊂ H ( W ) be such that as E ∋ e → u e s −→ u in L ( W ) (2.7) ¶ u e ¶ x j s −→ ¶ u ¶ x j + ¶ u ¶ y j in L ( W ) ( ≤ j ≤ N ) (2.8) for some u ∈ H ( W ) and u ∈ L ( W ; H per ( Y )) . Then lim e → Z S e u e ( x ) j ( x ) q ( x e ) d s e ( x ) = ZZ W × S u ( x , y ) j ( x ) q ( y ) dxd s ( y ) (2.9) for all j ∈ D ( W ) and q ∈ C per ( Y ) with R S q ( y ) d s ( y ) = .Proof. We first transform the above surface integral into a volume integral by adapting thetrick in [7, Section 3]. By the mean value zero condition over S for q we conclude that thereexists a unique solution J ∈ H per ( Y ∗ ) / R to ( − D y J = Y ∗ D y J ( y ) · n ( y ) = q ( y ) on S , (2.10)where n = ( n i ) Ni = stands for the outward unit normal to S with respect to Y ∗ . Put f = D y J .We get Z W e D x u e ( x ) j ( x ) · D y J ( x e ) dx = Z S e u e ( x ) j ( x ) D y J ( x e ) · n ( x e ) d s e ( x ) − Z W e u e ( x ) D x j ( x ) · D y J ( x e ) dx − e Z W e u e ( x ) j ( x ) D y J ( x e ) dx (2.11) = Z S e u e ( x ) j ( x ) q ( x e ) d s e ( x ) − Z W e u e ( x ) D x j ( x ) · f ( x e ) dx . Next, sending e to 0 yieldslim e → Z S e u e ( x ) j ( x ) q ( x e ) d s e ( x ) = ZZ W × Y ∗ [ D x u ( x ) + D y u ( x , y )] j ( x ) · f ( y ) dxdy + ZZ W × Y ∗ u ( x ) D x j ( x ) · f ( y ) dxdy = ZZ W × Y ∗ D y u ( x , y ) j ( x ) · f ( y ) dxdy . We finally have ZZ W × Y ∗ D y u ( x , y ) j ( x ) · f ( y ) dxdy = − ZZ W × Y ∗ u ( x , y ) j ( x ) D y J ( y ) dxdy + ZZ W × S u ( x , y ) j ( x ) f ( y ) · n ( y ) dxd s ( y )= ZZ W × S u ( x , y ) j ( x ) q ( y ) dxd s ( y ) . Hermann DouanlaThe proof is completed.We now gather some preliminary results. We introduce the characteristic function c G of G = R Ny \ Q with Q = [ k ∈ Z N ( k + T ) . It is clear 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 [8]. Proposition 2.10.
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 [8] 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 defines an open set in R N and W e \ Q e is the intersection of W with the collection of the holes crossing the boundary ¶W . The following result impliesthat the holes crossing the boundary ¶W are of no effects as regards the homogenizationprocess. Lemma 2.11. [26] Let K ⊂ W be a compact set independent of e . There is some e > such that W e \ Q e ⊂ W \ K for any < e ≤ e . We introduce the space F = H ( W ) × L (cid:0) W ; H , ∗ per ( Y ) (cid:1) and endow it with the following norm k v k F = k D x v + D y v k L ( W × Y ) ( v = ( v , v ) ∈ F ) , which makes it 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. 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 [11].teklov Eigenvalue Problems with Sing-changing Density Function 9 Lemma 2.12.
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. Put a ( u , v ) = N (cid:229) i , j = Z Y ∗ a i j ( y ) ¶ u ¶ y j ¶ v ¶ y i dy , l j ( v ) = N (cid:229) k = Z Y ∗ a k j ( y ) ¶ v ¶ y k dy , ( ≤ j ≤ N ) and l ( v ) = Z S r ( y ) v ( y ) d s ( y ) , 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 ) , (2.12) H per ( Y ∗ ) / R is a Hilbert space. Proposition 2.13.
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 (2.13) and u ∈ H per ( Y ∗ ) / R and a ( u , v ) = l ( v ) for all v ∈ H per ( Y ∗ ) / R (2.14) admit each a unique solution, assuming for (2.14) that M S ( 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 (2.15)0 Hermann Douanlawhere c j ( ≤ j ≤ N ) is the solution to (2.13). 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., [4]).We now visit the existence result for (1.2). The weak formulation of (1.2) 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 ) S e , v ∈ V e , (2.16)where ( r e u e , v ) S e = Z S e r e u e vd s e ( x ) . 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 [24].The bilinear form ( r e u , v ) S e defines a bounded linear operator K e : V e → V e such that ( r e u , v ) S 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 )) = W e a ( x e ) D x K e u · n ( x e ) = r e u on S e K e u ( x ) = ¶W , (2.17)we get a constant C e > k K e u k V e ≤ C e k u k L ( S e ) . But the trace operator V e → L ( S e ) is compact. The compactness of K e follows thereby. We can rewrite (2.16) asfollows K e u e = µ e u e , µ e = l e . We recall that (see e.g., [5]) in the case r ≥ S , the operator K e is positive and itsspectrum s ( K e ) lies in [ , k K e k ] and µ e = s e ( K e ) . Let L be a self-adjoint operator and let s ¥ p ( L ) and s c ( L ) be its set of eigenvalues of infinitemultiplicity and its continuous spectrum, respectively. We have s e ( L ) = s ¥ p ( L ) ∪ s c ( L ) bydefinition. The spectrum of K e is described by the following proposition whose proof issimilar to that of [24, Lemma 1]. Lemma 2.14.
Let r ∈ C per ( Y ) be such that the sets { y ∈ S : r ( y ) < } and { y ∈ S : r ( y ) > } are both of positive surface measure. Then for any e > , we have s ( K e ) ⊂ [ −k K e k , k K e k ] and µ = is the only element of the essential spectrum s e ( K e ) . Moreover, the discretespectrum of K e consists of two infinite sequencesµ , + e ≥ µ , + e ≥ · · · ≥ µ k , + e ≥ · · · → + , µ , − e ≤ µ , − e ≤ · · · ≤ µ k , − e ≤ · · · → − . teklov Eigenvalue Problems with Sing-changing Density Function 11 Corollary 2.15.
The hypotheses are those of Lemma 2.14. Problem (1.2) has a discrete setof eigenvalues consisting of two sequences < l , + e ≤ l , + e ≤ · · · ≤ l k , + e ≤ · · · → + ¥ , > l , + e ≥ l , − e ≥ · · · ≥ l k , − e ≥ · · · → − ¥ . We may now address the homogenization problem for (1.2).
In this section we state and prove homogenization results for both cases M S ( r ) > M S ( r ) =
0. The homogenization results in the case when M S ( r ) < M S ( r ) > r with − r . We start with the less technical case. M S ( 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 e Z S e r ( x e ) u k , + e u l , + e d s e ( x ) = d k , l k , l = , , · · · (3.1)and the homogenization results states as Theorem 3.1.
For each k ≥ and each e ∈ E, let ( l k , + e , u k , + e ) be the k th positive eigencoupleto (1.2) with M S ( r ) > and (3.1). Then, there exists a subsequence E ′ of E such that e 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) 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 S ( r ) q i j ¶ u ¶ x j (cid:19) = l u in W u = on ¶W Z W | u | dx = M S ( r ) , (3.6)2 Hermann Douanla and where u k ∈ L ( W ; H , ∗ per ( Y )) . 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 ) = − 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) (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 ( ≤ j ≤ N ) is the solution to the cell problem (2.13).Proof. We present only the outlines since this proof is similar but less technical to that ofthe case M S ( r ) = k ≥
1. By means of the minimax principle, as in [35], one easily proves the existenceof a constant C independent of e such that e 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 = (cid:18) e l k , + e (cid:19) e Z S e r ( x e ) u k , + e v d s e ( x ) (3.9)for any v ∈ V e . Bear in mind that e R S e r ( x e )( u k , + e ) dx = v = u k , + e in (3.9). Theboundedness of the sequence ( e l k , + e ) e ∈ E and the ellipticity assumption (1.3) imply at onceby means of Proposition 2.10 that the sequence ( P e u k , + e ) e ∈ E is bounded in H ( W ) . Theorem2.5 and Proposition 2.8 apply simultaneously and gives us u k = ( u k , u k ) ∈ F such that forsome l k ∈ R and some subsequence E ′ ⊂ E we have (3.2)-(3.5), where (3.4) is a directconsequence of (3.3) by the Rellich-Kondrachov theorem. For fixed e ∈ E ′ , let F e be as inLemma 2.12. Multiplying both sides of the first equality in (1.2) 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 = ( e l k , + e ) e Z S e ( P e u k , + e ) r ( x e ) F e d s e ( x ) . (3.10)Sending e ∈ E ′ to 0, keeping (3.2)-(3.5) and Lemma 2.12 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 × S u k y ( x ) r ( y ) dxd s ( y ) . 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 S ( r ) Z W u y dx for all F ∈ F . (3.11)teklov Eigenvalue Problems with Sing-changing Density Function 13which leads to the macroscopic and microscopic problems (3.6)-(3.7) without any majordifficulty. As regards the normalization condition in (3.6), we fix k , l ≥ j ∈ D ( W ) (Proposition 2.8)lim E ′ ∋ e → e Z S e ( P e u k , + e ) j ( x ) r ( x e ) d s e ( x ) = ZZ W × S u k ( x ) j ( x ) r ( y ) dxd s ( y ) . (3.12)But (3.12) still holds for any j ∈ H ( W ) . This being so, we write e Z S e ( P e u k , + e )( P e u l , + e ) r ( x e ) d s e ( x ) − M S ( r ) Z W u k u l dx = e Z S e ( P e u k , + e )( P e u l , + e − u l ) r ( x e ) d s e ( x ) + e Z S e ( P e u k , + e ) u l r ( x e ) d s e ( x ) (3.13) − M S ( r ) Z W u k u l dx According to (3.12) the sum of the last two terms on the right hand side of (3.13) goes tozero with e ∈ E ′ . As the remaining term on the right hand side of (3.13) is concerned, wemake use of the H ¨older inequality to get (cid:12)(cid:12)(cid:12)(cid:12) e Z S e ( P e u k , + e )( P e u l , + e − u l ) r ( x e ) d s e ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k r k ¥ (cid:18) e Z S e | P e u k , + e | d s e ( x ) (cid:19) (cid:18) e Z S e | P e u l , + e − u l | d s e ( x ) (cid:19) . Next the trace inequality (see e.g., [29]) yields e Z S e | P e u k , + e | d s e ( x ) ≤ c (cid:18) Z W e | P e u k , + e | dx + e Z W e | D ( P e u k , + e ) | dx (cid:19) (3.14) e Z S e | P e u l , + e − u l | d s e ( x ) ≤ c (cid:18) Z W e | P e u l , + e − u l | dx + e Z W e | D ( P e u l , + e − u l ) | dx (cid:19) , (3.15)for some positive constant c independent of e . But the right hand side of (3.14) is boundedfrom above whereas that of (3.15) converges to zero with e ∈ E ′ . This concludes the proof. Remark . • The eigenfunctions { u k } ¥ k = are in fact orthonormalized as follows Z W u k u l dx = d k , l M S ( 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 . • Replacing r with − r in (1.2), Theorem 3.1 also applies to the negative part of thespectrum in the case M S ( 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) = Y ∗ N (cid:229) i , j = a i j ( y ) ¶q¶ y i n j = lr ( y ) q ( y ) on S (3.16)and possesses a spectrum with similar properties to that of (1.2), two infinite (one positiveand another negative) sequences. We recall that since we have M S ( r ) >
0, problem (3.16)admits a unique nontrivial eigenvalue having an eigenfunction with definite sign, the firstnegative one (see e.g., [34]). In the sequel we will only make use of ( l − , q − ) , the firstnegative eigencouple to (3.16). After proper sign choice we assume that q − ( y ) > y ∈ Y ∗ . (3.17)We also recall that q − is d -H ¨older continuous(see e.g., [14]), hence can be extended to afunction living in C per ( Y ) still denoted by q − . Notice that we have Z S r ( y )( q − ( y )) d s ( y ) < , (3.18)as is easily seen from the variational equality (keep the ellipticity hypothesis (1.3) in mind) N (cid:229) i , j = Z Y ∗ a i j ( y ) ¶q − ¶ y j ¶q − ¶ y i dy = l − Z S r ( y )( q − ( y )) d s ( y ) . Bear in mind that problem (3.16) 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) = Q e N (cid:229) i , j = a i j ( x e ) ¶q e ¶ x i n j ( x e ) = e lr ( x e ) q ( x e ) on ¶ Q e , (3.19)where q e ( x ) = q ( x e ) . However, q e is not zero on ¶W . We now introduce the followingSteklov spectral 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) = W e N (cid:229) i , j = e a i j ( x e ) ¶ v e ¶ x i n j ( x e ) = x e e r ( x e ) v e on ¶ T e v e ( x ) = ¶W . (3.20)teklov Eigenvalue Problems with Sing-changing Density Function 15with new spectral parameters ( 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 ( y ) ∈ L ¥ per ( Y ) and e r ( y ) ∈ C per ( Y ) . As 0 < c − ≤ q − ( y ) ≤ c + < + ¥ ( c − , c + ∈ R ), the operator on the left hand side of (3.20) is uniformly elliptic andTheorem 3.1 applies to the negative part of the spectrum of (3.20) (see (3.18) and Remark3.2). The effective spectral problem for (3.20) reads − N (cid:229) i , j = ¶¶ x j (cid:18)e q i j ¶ v ¶ x i (cid:19) = x M S ( e r ) v in W v = ¶W Z W | v | dx = − M S ( e r ) . (3.21)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 , (3.22)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 . (3.23)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) . Notice that the spectrum of (3.21) is as follows0 > x > x ≥ x ≥ · · · ≥ x j ≥ · · · → − ¥ as j → ¥ . Making use of (3.19) when following the same line of reasoning as in [35, Lemma 6.1], weobtain that the negative spectral parameters of problems (1.2) and (3.20) verify: u k , − e = ( q − ) e v k , − e ( e ∈ E , k = , · · · ) (3.24)and l k , − e = e l − + x k , − e + o ( ) ( e ∈ E , k = , · · · ) . (3.25)The presence of the term o ( ) is due to integrals over W e \ Q e , which converge to zero with e ,remember that (3.19) holds in Q e but not W e . This trick, known as ”factorization principle”was introduced by Vaninathan[35] and has been used in many other works on averaging,see e.g., [2, 20, 23] just to cite a few. As will be seen below, the sequence ( x k , − e ) e ∈ E isbounded in R . In another words, l k , − e is of order 1 / e and tends to − ¥ as e goes to zero. It is6 Hermann Douanlanow clear why the limiting behavior of negative eigencouples is not straightforward as thatof positive ones and requires further investigations, which have just been made.Indeed, as the reader might be guessing now, the suitable orthonormalization conditionfor (3.20) is e Z S e e r ( x e ) v k , − e v l , − e d s e ( x ) = − d k , l k , l = , , · · · (3.26)which by means of ( . ) is equivalent to e Z S e r ( x e ) u k , − e u l , − e d s e ( x ) = − d k , l k , l = , , · · · (3.27)We may now state the homogenization theorem for the negative part of the spectrum of(1.2). Theorem 3.3.
For each k ≥ and each e ∈ E, let ( l k , − e , u k , − e ) be the k th negative eigencoupleto (1.2) with M S ( r ) > and (3.27). Then, there exists a subsequence E ′ of E such that l k , − e 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 ) (3.31) 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 S ( e r ) e q i j ¶ v ¶ x j (cid:19) = x v in W v = on ¶W Z W | v | dx = − M S ( e r ) , (3.32) and where v k ∈ L ( W ; H , ∗ per ( Y )) . 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 ; (3.33) (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 ∗ , (3.34) where e c j ( ≤ j ≤ N ) is the solution to the cell problem (3.23). teklov Eigenvalue Problems with Sing-changing Density Function 17 Remark . • The eigenfunctions { v k } ¥ k = are orthonormalized by Z W v k v l dx = − d k , l M S ( e r ) k , l = , , , · · ·• 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(3.29)-(3.31) hold for the whole sequence E . • Replacing r with − r in (1.2), Theorem 3.3 adapts to the positive part of the spectrumin the case M S ( r ) < M S ( r ) = We prove an homogenization result for both the positive part and the negative part of thespectrum simultaneously. We assume in this case that the eigenfunctions are orthonormal-ized as follows Z S e r ( x e ) u k , ± e u l , ± e d s e ( x ) = ± d k , l k , l = , , · · · (3.35)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 . (3.36)Indeed, the right hand side of (3.36) is positive. We recall that the following spectral prob-lem 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 , (3.37)has a spectrum consisting of two infinite sequences0 < l , + < l , + ≤ · · · ≤ l k , + ≤ . . . , lim n → + ¥ l k , + = + ¥ and 0 > l , − > l , − ≥ · · · ≥ l k , − ≥ . . . , lim n → + ¥ 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.
For each k ≥ and each e ∈ E, let ( l k , ± e , u k , ± e ) be the ( k , ± ) th eigencoupleto (1.2) with M S ( r ) = and (3.35). 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 ) (3.41)8 Hermann Douanla 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 , (3.42) and where u k , ± ∈ L ( W ; H , ∗ per ( Y )) . We have the following normalization condition Z W | u k , ± | dx = ± l k , ± n k = , , · · · (3.43) 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 S r ( y ) v ( y ) d s ( y ) − 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 ; (3.44) (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 ∗ , (3.45) where c j ( ≤ j ≤ N ) and c are the solutions to the cell problems (2.13) and (2.14),respectively.Proof. Fix k ≥
1, using the minimax principle, as in [35], we get a constant C independentof e such that | l 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 = l k , ± e Z S e r ( x e ) u k , ± e v d s e ( x ) (3.46)for any v ∈ V e . Bear in mind that R S e r ( x e )( u k , ± e ) d s e ( x ) = ± v = u k , ± e in (3.46).The boundedness of the sequence ( l k , ± e ) e ∈ E and the ellipticity assumption (1.3) imply atonce by means of Proposition 2.10 that the sequence ( P e u k , ± e ) e ∈ E is bounded in H ( W ) .Theorem 2.5 and Proposition 2.8 apply simultaneously and gives us u k , ± = ( u k , ± , u k , ± ) ∈ F such that for some l k , ± ∈ R and some subsequence E ′ ⊂ E we have (3.38)-(3.41), where(3.40) is a direct consequence of (3.39) by the Rellich-Kondrachov theorem. For fixed e ∈ E ′ , let F e be as in Lemma 2.12. Multiplying both sides of the first equality in (1.2) 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 S e ( P e u k , ± e ) r ( x e ) F e d s e ( x ) . teklov Eigenvalue Problems with Sing-changing Density Function 19Sending e ∈ E ′ to 0, keeping (3.38)-(3.41) and Lemma 2.12 in mind, we obtain a W ( u k , ± , F ) = l k , ± ZZ W × S (cid:16) u k , ± ( x , y ) y ( x ) r ( y ) + u k , ± y ( x , y ) r ( y ) (cid:17) dxd s ( y ) (3.47)The right-hand side follows as explained below. we have Z S e ( P e u k , ± e ) r ( x e ) F e d s e ( x ) = Z S e ( P e u k , ± e ) y ( x ) r ( x e ) d s e ( x )+ e Z S e ( P e u k , ± e ) y ( x , x e ) r ( x e ) d s e ( x ) . On the one hand we havelim E ′ ∋ e → e Z S e ( P e u k , ± e ) y ( x , x e ) r ( x e ) dx = ZZ W × S u k , ± y ( x , y ) r ( y ) dxd s ( y ) . On the other hand, owing to Lemma 2.9, the following holds:lim E ′ ∋ e → Z S e ( P e u k , ± e ) y ( x ) r ( x e ) d s e ( x ) = ZZ W × S u k , ± ( x , y ) y ( x ) r ( y ) dxd s ( y ) . We have just proved that ( l k , ± , u k , ± ) ∈ R × F solves the following global homogenizedspectral problem : Find ( l , u ) ∈ C × F such that a W ( u , F ) = l ZZ W × S ( u ( x , y ) y ( x ) + u ( x ) y ( x , y )) r ( y ) dxd s ( y ) for all F ∈ F . (3.48)To prove (i), choose F = ( y , y ) in (3.47) 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 S v ( y ) r ( y ) d s ( y ) (cid:21) dx Hence by the arbitrariness of 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 S v ( y ) r ( y ) d s ( y ) for any v in H per ( Y ∗ ) / R , which is nothing but (3.44).Fix x ∈ W , multiply both sides of (2.13) 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 (2.14) 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 (3.44).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 W × Y ∗ , (3.49)0 Hermann Douanlaby uniqueness of the solution to (3.44). Thus (3.45). But (3.49) still holds almost every-where in ( x , y ) ∈ W × S as S is of class C . Considering now F = ( y , y ) in (3.47) suchthat 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 × S u k , ± ( x , y ) r ( y ) y ( x ) dxd s ( y ) , which by means of (3.49) 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 (cid:18) Z Y ∗ a i j ( y ) ¶c ¶ y j ( y ) dy (cid:19) dx = − l k , ± N (cid:229) j = Z W ¶ u k , ± ¶ x j y ( x ) (cid:18) Z S r ( y ) c j ( y ) d s ( y ) (cid:19) dx (3.50) +( l k , ± ) Z W u k , ± ( x ) y ( x ) (cid:18) Z S r ( y ) c ( y ) d s ( y ) (cid:19) dx . Choosing c l ( ≤ l ≤ N ) as test function in (2.14) and c as test function in (2.13) weobserve that N (cid:229) j = Z Y ∗ a l j ( y ) ¶c ¶ y j ( y ) dy = Z S r ( y ) c l ( y ) d s ( y ) = a ( c l , c ) ( l = , · · · N ) . Thus, in (3.50), the second term in the left hand side is equal to the first one in the righthand side. 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 S r ( y ) c ( y ) d s ( y ) (cid:19) . (3.51)Choosing c as test function in (2.14) reveals that Z S r ( y ) c ( y ) d s ( y ) = a ( c , c ) = n . Hence 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 (3.38) holds for the whole sequence E . We now address (3.43).Fix k , l ≥ J ∈ H per ( Y ∗ ) / R be the solution to (2.10) where q is replaced with ourdensity function r . As in (2.11), we transform the surface integral into a volume integral Z S e ( P e u k , ± e )( P e u l , ± e ) r ( x e ) d s e ( x ) = Z W e ( P e u k , ± e ) D x ( P e u l , ± e ) · D y J ( x e ) dx + Z W e D x ( P e u k , ± e )( P e u l , ± e ) · D y J ( x e ) dx . (3.52)teklov Eigenvalue Problems with Sing-changing Density Function 21A limit passage in (3.52) as E ′ ∋ e → E ′ ∋ e → Z S e ( P e u k , ± e )( P e u l , ± e ) r ( x e ) d s e ( x )= ZZ W × Y ∗ u k , ± ( D x u l , ± + D y u l , ± ) · D y J dxdy + ZZ W × Y ∗ ( D x u k , ± + D y u k , ± ) u l , ± · D y J dxdy = Z W u k , ± (cid:18) Z Y ∗ D y u l , ± ( x , y ) · D y J ( y ) dy (cid:19) dx + Z W u l , ± (cid:18) Z Y ∗ D y u k , ± ( x , y ) · D y J ( y ) dy (cid:19) dx = ZZ W × S u k , ± ( x ) u l , ± ( x , y ) r ( y ) dxd s ( y ) + ZZ W × S u l , ± ( x ) u k , ± ( x , y ) r ( y ) dxd s ( y )= l l , ± n Z W u k , ± ( x ) u l , ± ( x ) dx + l k , ± n Z W u l , ± ( x ) u k , ± ( x ) dx = ( l k , ± + l l , ± ) n Z W u k , ± ( x ) u l , ± ( x ) dx . Where after the limit passage, we used the integration by part formula, then the weak for-mulation of (2.10) and finally (3.45) and integration by part. If k = l , the above limit passageand (3.35) lead to the desired result, (3.43), completing thereby the proof. Remark . • The eigenfunctions { u k , ± } ¥ k = are in fact orthonormalized as follows Z W u l , ± ( x ) u k , ± ( x ) dx = ± d k , l n ( l l , ± + l k , ± ) 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 convergenceresults (3.39)-(3.41) hold for the whole sequence E . Final Remark
After this paper was completed (see [9]) and submitted, we learned about an independentand similar work [6].
Acknowledgments
The author is grateful to Dr. Jean Louis Woukeng for helpful discussions.
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