Membranes with thin and heavy inclusions: asymptotics of spectra
MMEMBRANES WITH THIN AND HEAVY INCLUSIONS:ASYMPTOTICS OF SPECTRA
YURIY GOLOVATY
Abstract.
We study the asymptotic behaviour of eigenvalues and eigenfunc-tions of 2D vibrating systems with mass density perturbed in a vicinity ofclosed curves. The threshold case in which resonance frequencies of the mem-brane and thin inclusion coincide or closely situated is investigated. The per-turbed eigenvalue problem can be realized as a family of self-adjoint operatorsacting on varying Hilbert spaces. However the so-called limit operator whichis ultimately responsible for the asymptotics of eigenvalues and eigenfunctionsis non-self-adjoint and possesses the Jordan chains of length 2. Apart from thelack of self-adjointness, the operator has non-compact resolvent. As a conse-quence, its spectrum has a complicated structure, for instance, the spectrumcontains a countable set of eigenvalues with infinite multiplicity. Introduction and Statement of Problem
The mechanical systems with strongly inhomogeneous mass distributions havebecome the subject of intensive experimental and theoretical studies since the timeof Poisson and Bessel [1, Ch.2], and a lot of research has been devoted to theanalysis of vibrating systems with so-called added masses. Historically, the firstrelevant mathematical models in classical mechanics go back to the first half ofthe 20th century (see e.g. [2] and the references given there). Many authors haveinvestigated properties of strings and rods with the mass densities perturbed byfinite or infinite sums (cid:80) M k δ ( x − x k ), where δ is Dirac’s delta-function and M k is an added mass at the point x k . Recently, such models in dimensions two andthree with heavy inclusions of a different geometry are widely used not only inmechanics, but in various fields of science and technology such as physics of liquidcrystals, physical chemistry of polymers, micelles and microemulsions, moleculartheory, cell membrane theory [3–5]. For instance, cell membranes are known tocontain embedded proteins and various colloidal particles [6].In higher dimensions, the perturbation of mass densities by the δ -functions oftenleads to incorrect mathematical models, because the formal differential equationswhich appear have no mathematical meaning. As an example of such ill-posedproblem we can consider the eigenvalue problem for the Laplace operator − ∆ u = λ (1 + M δ ( x )) u in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R containing the origin. The equation has nonon-trivial solution, because any such solution u has a singularity at x = 0 andtherefore the product δ ( x ) u ( x ) = u (0) δ ( x ) is not defined. The new and at the Mathematics Subject Classification.
Key words and phrases.
Asymptotics of eigenvalues, eigenvalue of infinite multiplicity, quasi-mode, non-self-adjoint operator, concentrated mass, singular perturbation. a r X i v : . [ m a t h . SP ] J a n YURIY GOLOVATY same time obvious idea was instead to replace the δ -function with its regularization ε − q ( x/ε ), where q is a function of compact support, and study the asymptoticbehaviour of eigenvalues and eigenfunctions as ε →
0. The problem was firstinvestigated by E. S´anchez-Palencia [7,8], who proved the existence of the so-calledlocal eigenvibrations: the eigenfunctions are significant in a small neighborhood ofthe origin only.The model in which the density is perturbed by (cid:80) M k δ ( x − x k ) is not adequateeven in the one-dimensional case, when dealing with the large masses M k . Thevery heavy inclusions cause a strong local reaction of vibrating system, but thisphenomenon can not be described on the discrete set which is a support of the sumof Dirac’s functions. The geometry of small domains where the large masses areloaded should also have an effect on the form of eigenvibrations. In [9], asymptoticanalysis was applied to a spectral problem for the Sturm-Liouville operator withweight function of the form ρ ε ( x ) = ρ ( x ) + ε − m q ( x/ε ), where q is a function ofcompact support and m ∈ R . For the case m = 1 the perturbation is a δ -likesequence, but the most interesting cases of the limit behaviour of eigenfunctions as ε → m is greater than 1.These advanced models have attracted considerable attention in the mathema-tical literature over the last three decades (see e.g. [10] for a review). The spectralproperties of differential operators with weight functions having the form ρ ε ( x ) = ρ ( x ) + (cid:88) ε − m k q k,ε ( x ) , where q k,ε are compactly supported in vicinity of different sets, have been inves-tigated in numerous articles. We mention here [11] for the Laplace operator indimension 3, [12, 13] for ordinary differential operators of the fourth order and thebiharmonic operator, [14–16] for boundary value problems on junctions of a verycomplicated geometry, [17, 18] for the Sturm-Liouville operators on metric graphs.Since the 90s of the last century, a series of papers was published concerning 2D and3D elastic systems with many concentrated masses near the boundary [19–26]. Newasymptotic results for the spectral problems in domains surrounded by thin stiff andheavy bands, when the mass density and stiffness are simultaneously perturbed in aneighbourhood of the boundary, were obtained in [27–30]. The Neumann eigenvalueproblem for a membrane, almost the entire mass of which is concentrated aroundthe boundary, was studied in [31]. Another model in which the heavy inclusionswere regarded as rigid ones was treated in [32].In this paper we study the eigenvibration characteristics of a membrane withheavy and thin inclusion inside. Let Ω be a smooth bounded domain of R , andlet γ ⊂ Ω be a smooth closed curve. We will denote by ω ε the ε -neighborhood of γ (i.e., the union of all open balls of radius ε around a point on γ ). Consider ε > ω ε ⊂ Ω and the boundary ∂ω ε is smooth. Assume ρ isa smooth uniformly positive functions in Ω, and ρ ε ( x ) = (cid:40) ρ ( x ) , if x ∈ Ω \ ω ε ,ε − q ε ( x ) , if x ∈ ω ε . (1.1)To specify explicit dependence of q ε on ε we introduce the Fermi normal coordinatesin ω ε (see Fig. 1). Let α : [0 , | γ | ) → R be the unit-speed smooth parametrizationof γ with the natural parameter s , and | γ | is the length of γ . Then the vector ν = ( − ˙ α , ˙ α ) is a unit normal on γ . Set x = α ( s )+ rν ( s ) for ( s, r ) ∈ [0 , | γ | ) × ( − ε, ε ), EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 3 e r s e - gw eW Figure 1.
Membrane with heavy and thin inclusionwhere r is the signed distance from x to γ . Suppose that q ε ( x ) = q ( rε ) , where q is a smooth positive function in [ − , γ only. Let us consider the spectralproblem − ∆ u ε + au ε = λ ε ρ ε u ε in Ω , (cid:96)u ε = 0 on ∂ Ω , (1.2)where a ∈ C ∞ (Ω) and (cid:96)v = 0 denotes the Dirichlet, Neumann or Robin boun-dary conditions on ∂ Ω. Our goal is to describe the asymptotic behaviour of theeigenvalues λ ε and eigenfunctions u ε of (1.2) as ε → ε − m q ε of mass density has beentreated using the variational approach and the case m = 3 has been completelyinvestigated by asymptotic methods. Also, it has been shown that there existfive different limit behaviours for the spectrum and eigenspaces depending on m : m < m = 1, 1 < m < m = 2 and m >
2. These cases differ by the form ofeigenvibrations and place on the membrane where the main part of their “energy” isconcentrated in the limit. From the mathematical viewpoint, the difference betweenthe cases is that the spectral parameter in the limit eigenvalue problems appearsalternately both in a differential equation on Ω and in an equation on the strip γ × ( − , ω ε , and even in coupling conditions on γ . Thethreshold case m = 2 is the most difficult and interesting one to analyse, becausethe spectral parameter is included simultaneously in two differential equations, andthe limit spectral problem is associated with a non-self-adjoint operator. The maininsight of the present work is to exhibit the non-self-adjoint operator, the spectrumand generalized eigenspaces of which are ultimately responsible for the asymptoticsof eigenvalues and eigenfunctions of (1.2).To better understand what happens in dimension 2, we briefly discuss the similarmodel for a string. Recently, we have revised some results from [9] concerning thecase m = 2. In [35], we have studied the limiting behavior, as ε →
0, of eigenvaluesand eigenfunctions of the problem − y (cid:48)(cid:48) ε + Qy ε = λ ε ρ ε y ε , x ∈ ( a, b ) , (1.3) y ε ( a ) cos α + y (cid:48) ε ( a ) sin α = 0 , y ε ( b ) cos β + y (cid:48) ε ( b ) sin β = 0 (1.4) YURIY GOLOVATY with the weight function ρ ε given by (1.1), where Ω = ( a, b ) is an interval containingthe origin, and ω ε = ( − ε, ε ). For each real α and β the problem can be associatedwith a family of self-adjoint operators S ε in the weighted space L ( ρ ε , Ω). Theoperators S ε are defined by S ε φ = ρ − ε ( − φ (cid:48)(cid:48) + Qφ ) on functions φ ∈ W (Ω) obeyingboundary conditions (1.4). The spectra of S ε are real, discrete and simple. It isworth noting that the study of operator families acting on varying spaces entailssome mathematical difficulties. First of all, the question arises how to understandthe convergence of such families. Next, if these operators do converge in somesense, does this convergence implies the convergence of their spectra (see, e.g., [36,III.1], [37, 38]). By abandoning the self-adjointness, we have realized (1.3), (1.4)as a family of non-self-adjoint matrix operators A ε acting on the fixed Hilbertspace L ( ρ, ( a, b )) × L ( q, ( − , A ε and S ε coincide and the corresponding eigenspaces are isomorphic. It hasbeen proved that A ε converge in the norm resolvent sense (the resolvents of A ε converge in the uniform norm as ε →
0) to the matrix operator A associated withthe eigenvalue problem − v (cid:48)(cid:48) + Qv = λρv in ( a, ∪ (0 , b ) , − w (cid:48)(cid:48) = λqw in ( − , ,w (cid:48) ( −
1) = 0 , w (cid:48) (1) = 0 , v ( −
0) = w ( − , v (+0) = w (1) ,v ( a ) cos α + v (cid:48) ( a ) sin α = 0 , v ( b ) cos β + v (cid:48) ( b ) sin β = 0 . Surprisingly enough, A is not similar to a self-adjoint one, because it possessesmultiple eigenvalues with the Jordan chains of length 2. In view of [39, Ch.1],[40], the norm resolvent convergence A ε → A implies the “number-by-number”convergence of the corresponding eigenvalues and some results on the convergenceof eigenspaces.Problem (1.2) can also be associated with a family of matrix operators acting onthe same Hilbert space. However, in two dimensions, the principle difference is thatthe family does not converge in the norm resolvent topology. The limit operator P which can be obtained in weaker topology has non-compact resolvent. For thisreason, we do not introduce the matrix operators for the perturbed problem, butin the next section we construct the operator P , which is still non-self-adjoint.In Section 3 the spectrum of P and the structure of generalized eigenspaces arestudied. Because there is no uniform convergence of resolvents, in Sections 6 and 5we apply asymptotic methods and the method of quasimodes in order to describethe spectrum of (1.2) as perturbation of σ ( P ).2. Formal construction of the limit operator
It will be convenient to parameterize the curve γ by points of a circle. It willallow us not to indicate every time that functions on γ are periodic on s . Let S bethe circle of the length | γ | . Then ω ε is diffeomorphic to the cylinder S × ( − ε, ε ). Wealso set ω = S × ( − , ω ” after an equation,we mean that the equation is considered in the rectangle (0 , | γ | ) × ( − ,
1) and thecorresponding solution is a | γ | -periodic function on s .Let us denote by γ t the curve that is obtained from γ by flowing for “time” t along the normal vector field, i.e., γ t = { x ∈ R : x = α ( s ) + tν ( s ) , s ∈ S } . Thenthe boundary of ω ε consists of two curves γ − ε and γ ε . The domain Ω is divided by γ into subdomains Ω − and Ω + . Suppose that ∂ Ω − = ∂ Ω ∩ γ − and ∂ Ω + = γ + , where EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 5 γ − and γ + are two edges of the cut γ . In the sequel, the coordinate r increases inthe direction from Ω − to Ω + .For any ε >
0, problem (1.2) admits a self-adjoint operator realization in theweighted Lebesgue space L ( ρ ε , Ω). We introduce operator T ε defined by T ε φ = ρ − ε ( − ∆ φ + aφ )on functions φ ∈ W (Ω) obeying the boundary condition (cid:96)φ = 0 on ∂ Ω. Obviously,the spectrum of T ε is real and discrete.We look for the approximation to the eigenvalue λ ε and the corresponding eigen-function u ε of (1.2) in the form λ ε ∼ λ + · · · , u ε ( x ) ∼ (cid:40) v ( x ) + · · · if x ∈ Ω \ ω ε ,w ( s, rε ) + · · · if x ∈ ω ε . (2.1)To match the approximations for the eigenfunction, we hereafter assume that[ u ε ] − ε = 0 , [ u ε ] ε = 0 , [ ∂ r u ε ] − ε = 0 , [ ∂ r u ε ] ε = 0 , (2.2)where [ · ] t stands for the jump of a function across γ t . Since u ε solves (1.2) and thedomain ω ε shrinks to γ as ε →
0, the function v must be a solution of the equation − ∆ v + av = λρv in Ω \ γ that satisfies the boundary condition (cid:96)v = 0 on ∂ Ω. Ofcourse, v must also satisfy appropriate transmission conditions on γ . To find theseconditions, we must examine more closely the equation in (1.2) in a vicinity of γ .Returning to the local coordinates ( s, r ), we see that the vectors α = ( ˙ α , ˙ α ), ν = ( − ˙ α , ˙ α ) give the Frenet frame for γ . The Jacobian of transformation x = ˙ α ( s ) − r ˙ α ( s ) , x = ˙ α ( s ) + r ˙ α ( s )has the form J ( s, r ) = 1 − r κ ( s ), where κ = det( ˙ α, ¨ α ) is the signed curvature of γ . We see that J is positive for sufficiently small r , because the curvature κ isbounded on γ . In addition, the Laplace-Beltrami operator becomes∆ φ = J − (cid:0) ∂ s ( J − ∂ s φ ) + ∂ r ( J∂ r φ ) (cid:1) . In the local coordinates ( s, n ), where n = r/ε , the Laplacian can be written as∆ = (1 − εn κ ) − (cid:0) ε − ∂ n (1 − εn κ ) ∂ n + ∂ s (cid:0) (1 − εn κ ) − ∂ s (cid:1)(cid:1) . From this we readily deduce the representation∆ = ε − ∂ n − ε − κ ∂ n − n κ ∂ n + ∂ s − ε ( n κ ∂ n + 2 n κ ∂ s + n κ (cid:48) ∂ s ) + ε P ε , (2.3)where P ε is a PDE of the second order on s and the first one on n whose coefficientsare uniformly bounded in ω with respect to ε . Then using this representation and(2.1) we obtain the equation − ∂ n w = λqw in ω . Next, conditions (2.2) imply v ± = w ( · , ±
1) and ∂ n w ( · , ±
1) = 0, where v ± denote the one-side traces of v on γ ,i.e., v ± = v | γ ± . Combining the latter equalities, we can formally deduce that thepair ( v, w ) must be an eigenvector of the spectral problem − ∆ v + av = λρv in Ω \ γ, (cid:96)v = 0 on ∂ Ω , (2.4) − ∂ n w = λqw in ω, ∂ n w ( s, −
1) = 0 , ∂ n w ( s,
1) = 0 , (2.5) v − = w ( s, − , v + = w ( s, , s ∈ S (2.6)with the spectral parameter λ ; (2.4)–(2.6) will be regarded as the limit problem . YURIY GOLOVATY Properties of the limit operator
We use the following notation. The spectrum and resolvent set of a linear oper-ator T are denoted by σ ( T ) and (cid:37) ( T ), respectively, and the Hilbert space adjointoperator of T is T ∗ . For any z ∈ (cid:37) ( T ), R z ( T ) = ( T − z ) − is the resolvent operator.3.1. Spectrum of the limit operator.
We introduce the operators˚ A = ρ − ( − ∆ + a ) in L ( ρ, Ω) , dom ˚ A = { f ∈ W (Ω \ γ ) : (cid:96)f = 0 on ∂ Ω } ,B = − q − ∂ n in L ( q, ω ) , dom B = (cid:110) g ∈ W , ( ω ) : ∂ n g ( · , −
1) = ∂ n g ( · ,
1) = 0 (cid:111) , where W , ( ω ) is the anisotropic Sobolev space W , ( ω ) = (cid:8) g ∈ L ( ω ) : ∂ kn g ∈ L ( ω ) for k = 1 , (cid:9) . In the space L = L ( ρ, Ω) × L ( q, ω ) we consider the matrix operator P = (cid:18) ˚ A B (cid:19) , dom P = (cid:8) ( f, g ) ∈ dom ˚ A × dom B : f − = g ( · , − , f + = g ( · , (cid:9) . Now (2.4)–(2.6) can be written as P u = λu with the notation u = ( v, w ) (cid:62) . For sim-plicity of notation we often write ( v, w ) instead of ( v, w ) (cid:62) . Note that the operator P is non-self-adjoint. Direct computations show that P ∗ = (cid:18) A
00 ˚ B (cid:19) , dom P ∗ = (cid:8) ( f, g ) ∈ dom A × W , ( ω ) : ∂ r f − = − ∂ n g ( · , − , ∂ r f + = ∂ n g ( · , (cid:9) , where A is the restrictions of ˚ A to dom A = { f ∈ dom ˚ A : f = 0 on γ } and ˚ B isthe extension of B to the whole space W , ( ω ). Here ∂ r f ± stand for the one-sidetraces of the normal derivative of f on γ . The limit problem admits no self-adjointrealization, because it has generalized eigenvectors, as we will show below. Lemma 3.1.
The spectrum of B consists of a countable set of real eigenvalues ofinfinite multiplicity. Moreover, λ belongs to σ ( B ) if and only if λ is an eigenvalueof the Sturm-Liouville problem y (cid:48)(cid:48) + λq ( n ) y = 0 , n ∈ ( − , , y (cid:48) ( −
1) = 0 , y (cid:48) (1) = 0 . (3.1) Proof.
Obviously, the operator B is self-adjoint. For given λ ∈ C and g ∈ L ( q, ω ),the equation ( B − λ ) φ = g can be treated as the boundary value problem for theordinary differential equation − ( ∂ n + λq ( n )) φ = q ( n ) g ( s, n ) in ω, ∂ n φ ( s, −
1) = 0 , ∂ n φ ( s,
1) = 0with the parameter s ∈ S . If λ is not an eigenvalue of (3.1), then the problem has aunique solution φ ( s, · ) for each g ( s, · ) ∈ L ( − ,
1) and almost all s ∈ S . Moreover φ belongs to L ( q, ω ) in view of its integral representation via the Green function.Otherwise, if λ is an eigenvalue of (3.1) with eigenfunction y , the problem is ge-nerally unsolvable, because the corresponding homogeneous problem has infinitelymany linearly independent solutions of the form b ( s ) y ( n ), where b ∈ L ( γ ). There-fore the spectrum of B consists of all eigenvalues of (3.1), and the correspondingeigenspaces are infinite-dimensional. (cid:3) EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 7
Theorem 3.2.
The operator P has real discrete spectrum. Moreover σ ( P ) = σ ( A ) ∪ σ ( B ) . Proof.
Let us construct the resolvent of P in an explicit form. Given µ ∈ C , f ∈ L ( ρ, Ω) and g ∈ L ( q, ω ), we write ( ˚ A − µ ) v = f , ( B − µ ) w = g . The secondequation admits a unique solution w = R µ ( B ) g if µ ∈ (cid:37) ( B ), and then v is a solutionof the problem − ∆ v + av − µρv = ρf in Ω \ γ, (cid:96)v = 0 on ∂ Ω ,v − = w ( · , − , v + = w ( · , . (3.2)Suppose the operator T ( µ ) : W ( ω ) → L ( ρ, Ω) solves the problem − ∆ φ + aφ − µρφ = 0 in Ω \ γ, (cid:96)φ = 0 on ∂ Ω ,φ − = ψ ( · , − , φ + = ψ ( · ,
1) (3.3)for a given function ψ ∈ W ( ω ). If µ ∈ (cid:37) ( A ), then the problem has a uniquesolution φ = T ( µ ) ψ and hence T ( µ ) is bounded. Next, v can be represented as thesum of a solution of (3.2) subject to the homogeneous boundary conditions on γ ± and a solution of (3.3) with ψ = w . Therefore v = R µ ( A ) f + T ( µ ) w = R µ ( A ) f + T ( µ ) R µ ( B ) g, provided µ ∈ (cid:37) ( A ) ∩ (cid:37) ( B ). Then the resolvent of P can be written in the form R µ ( P ) = (cid:18) R µ ( A ) T ( µ ) R µ ( B )0 R µ ( B ) (cid:19) . The equality σ ( P ) = σ ( A ) ∪ σ ( B ) follows directly from this representation and thefact that T ( µ ) is bounded for µ ∈ (cid:37) ( A ). Clearly, σ ( P ) is real and discrete, because A and B are self-adjoint operators with discrete spectra. (cid:3) Corollary 3.3.
The operator P has non-compact resolvent.Proof. The entry R µ ( B ) of the matrix R µ ( P ) is a non-compact operator, which isclear from Lemma 3.1. The main reason for which R µ ( B ) is not compact is that thesecond derivative ∂ n is a non-elliptic PDE in the two-dimensional domain ω . (cid:3) Remark . The operator A can be represent as the direct sum A − ⊕ A + , where A − = ρ − ( − ∆ + a ) , dom A − = { f ∈ W (Ω − ) : (cid:96)f = 0 on ∂ Ω , f = 0 on γ } ,A + = ρ − ( − ∆ + a ) , dom A + = { f ∈ W (Ω + ) : f = 0 on γ } . Hence σ ( P ) = σ ( A − ) ∪ σ ( A + ) ∪ σ ( B ).3.2. Structure of generalized eigenspaces.
Let X µ be the generalized eigenspa-ce corresponding to µ ∈ σ ( P ), i.e., X µ = { h ∈ L : h ∈ ker( P − µ ) k for some k ∈ N } .If a vector h belongs to ker( P − µ ) j and ( P − µ ) j − h (cid:54) = 0, one says that h is ageneralized eigenvector of rank j . The eigenspace X µ = ker( P − µ ) is a subspaceof X µ ; eigenvectors are precisely the generalized eigenvectors of rank 1. YURIY GOLOVATY
Case λ ∈ σ ( A ) \ σ ( B ) . We look first for non-trivial solutions u = ( v, w ) of( P − λ ) u = 0. Since λ does not belong to σ ( B ), we have w = 0. This in turnimplies v | γ = 0, by (2.6). Therefore v must be a eigenvector of A corresponding to λ . All eigenvalues of A have finite multiplicities. Assume that λ is an eigenvalue ofmultiplicity K and V ,. . . , V K are the eigenfunctions of A such that (cid:90) Ω ρV i V j dx = δ i,j , i, j = 1 , . . . , K. (3.4)Here δ i,j is the Kronecker symbol. Then X λ is spanned by u = ( V , , u = ( V , , . . . , u K = ( V K , . (3.5)The generalized eigenvectors of rank 2 satisfy the equation ( P − λ ) u ∗ = u , where u is an eigenvector of P . There is no such vectors u ∗ = ( v ∗ , w ∗ ) in this case, becausethe second components of all the eigenvectors u k are zero. In fact, ( B − λ ) w ∗ = 0implies w ∗ = 0. Therefore ( A − λ ) v ∗ = v , where v is a linear combination of V j .The last equation is unsolvable, since A is self-adjoint.3.2.2. Case λ ∈ σ ( B ) \ σ ( A ) . Suppose that y is an eigenfunction of (3.1) correspond-ing to λ . Then B has a countable collection of linearly independent eigenfunctions (cid:8) b j ( s ) y ( n ) (cid:9) ∞ j =1 , where { b j } ∞ j =1 is a basis in L ( γ ) consisting of smooth functions.On the other hand, the problem − ∆ v + av = λρv in Ω \ γ, (cid:96)v = 0 on ∂ Ω , v − = y ( − b, v + = y (1) b (3.6)is uniquely solvable for any b ∈ C ∞ ( γ ), since λ does not belong to σ ( A ). Therefore P possesses the countable set of linearly independent eigenvectors u = ( v , b y ) , u = ( v , b y ) , . . . , u k = ( v j , b j y ) , . . . , (3.7)where v j is a solution of (3.6) with b j in place of b . Note that the values y ( ±
1) aredifferent from zero and hence all v j are non-trivial solutions. In this case, findinggeneralized eigenvectors of rank 2 leads to the equation ( B − λ ) w ∗ = by , which isunsolvable for b (cid:54) = 0. Hence X λ = X λ and dim X λ = ∞ .3.2.3. Case λ ∈ σ ( A ) ∩ σ ( B ) . Since λ belongs to σ ( B ), any eigenvector of P hasthe form ( v ( x ) , b ( s ) y ( n )), where v is a solution of (3.6). But now problem (3.6) isgenerally unsolvable, since λ is an point of σ ( A ). Proposition 3.5.
Let f ∈ L (Ω) and b ± ∈ W / ( γ ) . Assume that λ is an eigen-value of A of multiplicity K and the corresponding eigenfunctions V , . . . , V K satisfy (3.4) . Then the problem − ∆ v + av = λρv + f in Ω \ γ, (cid:96)v = 0 on ∂ Ω , v − = b − , v + = b + (3.8) admits a solution v ∈ W (Ω \ γ ) if and only if (cid:90) γ (cid:0) b + ∂ r V + k − b − ∂ r V − k (cid:1) dγ + (cid:90) Ω f V k dx = 0 (3.9) for all k = 1 , . . . , K .Proof. This statement is a simple consequence of the Fredholm alternative for theself-adjoint operator A with compact resolvent. Conditions (3.9) can be obtainedby multiplying the equation in (3.8) by V , . . . , V K in turn and then integrating byparts twice in view of the boundary conditions. (cid:3) EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 9
In view of Proposition 3.5, problem (3.6) is solvable if and only if ( b, Ψ k ) L ( γ ) = 0for all k = 1 , . . . , K , whereΨ k = y (1) ∂ r V + k − y ( − ∂ r V − k . (3.10)Let H λ be the subspace in L ( γ ) spanned by Ψ , . . . , Ψ K . Hence the solvability of(3.6) is equivalent to the orthogonality of b to H λ . Proposition 3.6.
Assume λ is an eigenvalue of A = A − ⊕ A + and k ± is themultiplicity of λ in the spectrum of A ± . Then dim H λ ≥ max( k + , k − ) .Proof. Since the operator A is a direct sum of A + and A − , we can choose a basisin the eigenspace of A such that V , . . . , V k + are identically equal to zero in Ω − and the rest of k − eigenfunctions V k + +1 , . . . , V K are identically equal to zero in Ω + .Note that k + + k − = K . Then H λ is the linear span of ∂ r V +1 , . . . , ∂ r V + k + , ∂ r V − k + +1 , . . . , ∂ r V − K . (3.11)In general, these normal derivatives are linearly dependent in L ( γ ), but the firstof k + derivatives are always linearly independent as well as the last of k − ones.Suppose, contrary to our claim, that the functions ∂ r V +1 , . . . , ∂ r V + k + are linearlydependent in L ( γ ). Then there exists an eigenfunction v of the Dirichlet typeproblem − ∆ v + av = λρv in Ω + , v = 0 on ∂ Ω + such that ∂ r v = 0 on the wholeboundary of Ω + , but this is impossible. The same conclusion can be drawn for thesecond part of the normal derivatives. (cid:3) Assume dim H λ = d and choose a basis { b j } ∞ j =1 in L ( γ ) such that b ,. . . , b d belong to H λ , while b k for j > d are elements of H ⊥ λ , and b j ∈ C ∞ ( γ ). Then( V , , . . . , ( V K , , ( v d +1 , b d +1 y ) , ( v d +2 , b d +2 y ) , . . . (3.12)is a countable set of linearly independent eigenvectors of P . Here v j is a solutionof (3.6) for b = b j which is orthogonal to the span of V , . . . , V K in L ( ρ, Ω). So X λ is infinite-dimensional.In this case, we can also find the generalized eigenvectors u ∗ = ( v ∗ , w ∗ ) of rank 2.They satisfy the equation ( P − λ ) u ∗ = u , where u = ( v, w ) is an eigenvector of P .Hence v ∗ and w ∗ solve the problem − ∆ v ∗ + av ∗ = λρv ∗ + ρv in Ω \ γ, (cid:96)v ∗ = 0 on ∂ Ω , (3.13) − ∂ n w ∗ = λqw ∗ + qw in ω, ∂ n w ∗ ( · , −
1) = 0 , ∂ n w ∗ ( · ,
1) = 0 , (3.14) v −∗ = w ∗ ( · , − , v + ∗ = w ∗ ( · , . (3.15)It is easily seen that (3.13)–(3.15) is unsolvable if w (cid:54) = 0. Therefore the generali-zed eigenvectors can be associated with some non-trivial linear combinations of theeigenvectors ( V , , . . . , ( V K ,
0) only. We set w = 0 and v = c V + · · · + c K V K .Then w ∗ ( s, n ) = b ∗ ( s ) y ( n ) and − ∆ v ∗ + av ∗ = λρv ∗ + ρ K (cid:88) j =1 c j V j in Ω \ γ, (cid:96)v ∗ = 0 on ∂ Ω , (3.16) v −∗ = y ( − b ∗ , v + ∗ = y (1) b ∗ . (3.17)By Proposition 3.5 and (3.4), the problem admits a solution if and only if c = − ( b ∗ , Ψ ) L ( γ ) , . . . , c K = − ( b ∗ , Ψ K ) L ( γ ) . Since v is non-zero, b ∗ must have a non-trivial projection onto the subspace H λ .But b ,. . . , b d have this property. Putting b ∗ = b j for j = 1 , . . . , d into (3.16), (3.17),we obtain the Jordan chains of length 2( V ( j ) , (cid:55)→ ( v ( j ) ∗ , b j y ) , j = 1 , . . . , d, where V ( j ) = c ( j )1 V + · · · + c ( j ) K V j with c ( j ) i = − ( b j , Ψ i ) L ( γ ) .There are no generalized eigenvectors of rank 3, because all the eigenvectorsof rank 2 have nonzero second components and the equation ( B − λ ) y ∗ = b j y isunsolvable for b j (cid:54) = 0. Hence the generalized eigenspace X λ of P has the basis u = ( V (1) , , u ∗ = ( v (1) ∗ , b y ) , . . . , u d = ( V ( d ) , , u ∗ d = ( v ( d ) ∗ , b d y ) ,u d +1 = ( v d +1 , b d +1 y ) , u d +2 = ( v d +2 , b d +2 y ) , . . . . (3.18)We summarize the information about the spectrum and generalized eigenspacesof the limit operator P that we have obtained. Theorem 3.7.
Let X λ be the generalized eigenspace corresponding to λ ∈ σ ( P ) and X λ be the eigenspace.(i) If λ ∈ σ ( A ) \ σ ( B ) , then X λ is finite dimensional, X λ = X λ , and the basisin X λ is given by (3.5) .(ii) The part σ ( B ) \ σ ( A ) of σ ( P ) consists of eigenvalues λ of infinite multiplicitywith eigenspaces X λ = X λ generated by vectors (3.7) .(iii) The part σ ( A ) ∩ σ ( B ) is also consists of eigenvalues λ of infinite multiplicity,but X λ (cid:54) = X λ . Apart from the eigenvectors, there exist the generalized eigenvectorsof rank , and the basis in X λ is given by (3.18) . The dimension of factor space X λ /X λ does not exceed the multiplicity of λ in the spectrum of A , namely max( k − , k + ) ≤ dim X λ /X λ ≤ k − + k + , where k − and k + are the multiplicity of λ in the spectrum of A − and A + respectively.Remark . The operator P has always the eigenvalue λ = 0 of infinite multiplicity,since 0 ∈ σ ( B ). The zero eigenvalue is the smallest infinite-fold one, because B isnon-negative. All negative eigenvalues, if they exist, have finite multiplicities.4. Asymptotics of eigenvalues in the case λ ∈ σ ( B ) \ σ ( A )We will first focus our attention on perturbations of infinite-fold eigenvalues. Inthis section, we construct the asymptotics of countable set of eigenvalues λ ε of (1.2)that converge to an infinite-fold eigenvalue λ of P when λ (cid:54)∈ σ ( A ). We look forthe asymptotics in the form λ ε ∼ λ + λ ε + · · · , u ε ( x ) ∼ (cid:40) v ( x ) + εv ( x ) + · · · if x ∈ Ω \ ω ε ,w ( s, rε ) + εw ( s, rε ) + · · · if x ∈ ω ε , (4.1)where ( v , w ) is an non-zero element of X λ . To match the expansions on γ ± ε , wewrite v k in the local coordinates ( s, n ). Then (2.2) becomes w ( s, ±
1) + εw ( s, ± − v ( s, ± ε ) − εv ( s, ± ε ) + · · · ∼ ,ε − ∂ n w ( s, ±
1) + ∂ n w ( s, ± − ∂ r v ( s, ± ε ) − ε∂ r v ( s, ± ε ) + · · · ∼ . After expanding v k ( s, ± ε ) and their derivatives into the Taylor series about n = ± s , we in particular derive w ( · , ±
1) = v ± ± ∂ r v ± and ∂ n w ( · , ±
1) = ∂ r v ± .Of course, w ( · , ±
1) = v ± and ∂ n w ( · , ±
1) = 0, since ( v , w ) ∈ dom P . EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 11
Substituting (4.1) into (1.2), and taking into account representation (2.3), weobtain that the pair ( v , w ) solves the problem − ∆ v + av = λ ρv + λ ρv in Ω \ γ, (cid:96)v = 0 on ∂ Ω , (4.2) − ∂ n w = λ qw − κ ∂ n w + λ qw in ω, (4.3) ∂ n w ( · , −
1) = ∂ r v − , ∂ n w ( · ,
1) = ∂ r v +0 , (4.4) v − = w ( · , −
1) + ∂ r v − , v +1 = w ( · , − ∂ r v +0 . (4.5)Problem (4.3), (4.4) has generally no solution, since λ ∈ σ ( B ). We can apply theFredholm alternative for the operator with compact resolvent associated with theSturm-Liouville problem (3.1). Then the compatibility condition for (4.3), (4.4)reads y ( − ∂ r v − − y (1) ∂ r v +0 + κ (cid:90) − ∂ n w y dn = λ (cid:90) − qw y dn on γ, (4.6)where y is an eigenfunction of (3.1) corresponding to λ . The relation can betreated as a spectral equation on λ for a certain pseudodifferential operator on γ .We introduce the Dirichlet-to-Neumann maps N ± λ in L ( γ ) as follows. Let z be thesolution of problem − ∆ z + az = λρz in Ω − , z = ϕ on γ, (cid:96)z = 0 on ∂ Ω (4.7)for given ϕ . Set N − λ ϕ = ∂ r z | γ . We follow [41] in assuming thatdom N − λ = (cid:8) ϕ ∈ L ( γ ) : z ∈ W (Ω − ) and ∂ r z | γ ∈ L ( γ ) (cid:9) . Likewise, we set N + λ ϕ = − ∂ r z | γ , where z is a solution of the problem − ∆ z + az = λρz in Ω + , z = ϕ on γ, (4.8)and dom N + λ = (cid:8) ϕ ∈ L ( γ ) : z ∈ W (Ω + ) and ∂ r z | γ ∈ L ( γ ) (cid:9) . The operators N ± λ transform the Dirichlet data on γ for solutions of the corresponding boundary valueproblems into the Neumann ones. Both operators are well-defined if λ (cid:54)∈ σ ( A ). Theminus sing in definition of N + λ indicates that the direction of axis r coincides withthe inward normal on γ = ∂ Ω + .It follows from (3.7) that w ( s, n ) = b ( s ) y ( n ) and v is a solution of (3.6)for b = b . Then ∂ r v − = y ( − N − λ b and ∂ r v +0 = − y (1) N + λ b . Consequently,condition (4.6) reads y ( − N − λ b + y (1) N + λ b + κ b (cid:90) − yy (cid:48) dn = λ b (cid:90) − qy dn. Suppose (cid:107) y (cid:107) L ( q, ( − , = 1 and write θ ± λ = y ( ± (cid:90) − yy (cid:48) dn = (cid:90) − ( y ) (cid:48) dn = (cid:0) θ + λ − θ − λ (cid:1) , we can finally rewrite (4.6) in the form N λ b = λ b , where N λ = θ − λ N − λ + θ + λ N + λ + ( θ + λ − θ − λ ) κ . Note that the values θ − λ and θ + λ do not depend on our choice of y , because alleigenvalues of the Sturm-Liouville problem are simple. Proposition 4.1.
The operator N λ is self-adjoint, bounded below and has compactresolvent for all λ (cid:54)∈ σ ( A ) . Proof.
The operators N − λ and N + λ are self-adjoint, bounded below and have com-pact resolvents [41, Th.3.1]. The linear combination θ − λ N − λ + θ + λ N + λ has the sameproperties, since θ − λ and θ + λ are positive. Finally, N λ is a perturbation of thislinear combination by the operator of multiplication by the bounded function ( θ + λ − θ − λ ) κ ( s ), which completes the proof. (cid:3) Denote by { λ ( k )1 } ∞ k =1 the eigenvalues of N λ . So we have calculated the countableset of correctors λ ( k )1 in asymptotics (4.1). To keep the mathematics rather simplewe suppose that the spectrum of N λ is simple. It means that the infinite-foldeigenvalue λ of P asymptotically splits into an infinite number of simple eigen-values of T ε under perturbation. Let { b ( k ) } ∞ k =1 be the collection of orthonormaleigenfunctions of N λ .Let us fix the corrector λ ( k )1 and set w = w ( k ) = b ( k ) y . Then v = v ( k ) is a uniquesolution of (3.6) with λ and b ( k ) in place of λ and b respectively. Such a choiceof ( v ( k ) , w ( k ) ) ensures that problem (4.3), (4.4) is solvable, since the compatibilitycondition N λ b ( k ) = λ ( k )1 b ( k ) holds. The problem admits the family of solutions w ( s, n ) = ˆ w ( k )1 ( s, n ) + b ( s ) y ( n ) , where b is an arbitrary L ( γ )-function and ˆ w ( k )1 is the partial solution subject tothe condition (cid:90) − qy ˆ w ( k )1 dn = 0 for all s ∈ S. (4.9)According to (4.2) and (4.5), with this choice of w the function v can be writtenas v = ˆ v ( k )1 + ˜ v , where ˆ v ( k )1 (resp. ˜ v ) is a solution of the problem − ∆ φ + aφ = λρφ + λ ( k )1 ρv ( k ) in Ω \ γ, (cid:96)φ = 0 on ∂ Ω , φ − = g − , φ + = g + (4.10)with the Dirichlet data g ± = ˆ w ( k )1 ( · , ± ∓ ∂ n v ( k ) | γ ± (resp. g ± = y ( ± b ). Theterm ˆ v ( k )1 is uniquely defined, while ˜ v along with b will be fixed below.Set a ( s ) = a ( s, v , w ) in the asymptotics of u ε solves the problem − ∆ v + av = λ ρv + λ ρv + λ ρv in Ω \ γ, (cid:96)v = 0 on ∂ Ω , (4.11) − ∂ n w = λ qw − ( κ ∂ n − λ q ) w − ( κ n∂ n − ∂ s + a − λ q ) w in ω, (4.12) ∂ n w ( · , −
1) = ∂ r v − − ∂ r v − , ∂ n w ( · ,
1) = ∂ r v +1 + ∂ r v +0 , (4.13) v − = w ( · , −
1) + ∂ r v − − ∂ r v − , v +2 = w ( · , − ∂ r v +1 − ∂ r v +0 . (4.14)By reasoning as above, we obtain that the solution w exists if and only if( N λ − λ ( k )1 ) b = λ b ( k ) + h ( k )1 , (4.15)where h ( k )1 = y ( − (cid:0) ∂ r ˆ v ( k )1 | γ − − ∂ r v ( k ) | γ − (cid:1) − y (1) (cid:0) ∂ r ˆ v ( k )1 | γ + + ∂ r v ( k ) | γ + (cid:1) + (cid:90) − (cid:0) κ nyy (cid:48) b ( k ) − y ( ∂ s b ( k ) − a b ( k ) ) (cid:1) dn − (cid:90) − ( κ ∂ n − λ ( k )1 q ) y ˆ w ( k )1 dn. (4.16)Since λ ( k )1 is a simple eigenvalue of N λ , the second corrector λ = λ ( k )2 can beuniquely calculated from the solvability condition for equation (4.15) λ ( k )2 = − ( h ( k )1 , b ( k ) ) L ( γ ) . EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 13
Moreover, there exists a unique solution b ( k )1 of (4.15) satisfying the condition( b ( k )1 , b ( k ) ) L ( γ ) = 0. Hence v ( k )1 is now uniquely defined by choosing ˜ v to be asolution of (4.10) with the Dirichlet data g ± = y ( ± b ( k )1 . The compatibility con-dition (4.15) allows us to solve problems (4.12), (4.13) and (4.11), (4.14) one afteranother and to find solutions w ( k )2 and v ( k )2 .The process used to find the leading terms of asymptotic expansions (4.1) can becontinued to systematically construct all other terms. For fixed λ ∈ σ ( B ) \ σ ( A )we consider the countable collection of formal approximations to eigenvalues andeigenfunctions of the perturbed problemΛ ( k ) ε = λ + λ ( k )1 ε + λ ( k )2 ε + λ ( k )3 ε , (4.17)ˆ u ( k ) ε ( x ) = (cid:40) v ( k )0 ( x ) + εv ( k )1 ( x ) + ε v ( k )2 ( x ) + ε v ( k )3 ( x ) if x ∈ Ω \ ω ε ,w ( k )0 ( s, rε ) + εw ( k )1 ( s, rε ) + ε w ( k )2 ( s, rε ) + ε w ( k )3 ( s, rε ) if x ∈ ω ε . (4.18)Let H be a Hilbert space with norm (cid:107) · (cid:107) and let T be a self-adjoint operator in H with a domain dom T . We say a pair ( µ, u ) ∈ R × dom T is a quasimode of T with the accuracy δ , if (cid:107) u (cid:107) = 1 and (cid:107) ( T − µ ) u (cid:107) ≤ δ . Of course, if δ = 0 , then µ isan eigenvalue of T with the eigenvector u . Proposition 4.2 ([42, p.139]) . Assume ( µ, u ) is a quasimode of T with accuracy δ > and the spectrum of T is discrete in the interval [ µ − δ, µ + δ ] . Then thereexists an eigenvalue µ ∗ of T such that | µ ∗ − µ | ≤ δ . In order to construct the quasimodes of T ε , we must modify the approximationsof eigenfunctions, because they do not belong to dom T ε . By construction, thefunctions v ( k ) j and w ( k ) j are smooth, since the coefficients a , ρ and q are smooth.But, in general, ˆ u ( k ) ε have jump discontinuities on ∂ω ε . Figure 2.
Plot of the function ζ .Let us define the function ζ plotted in Fig. 2. This function is smooth outsidethe origin, ζ ( r ) = 1 for r ∈ [0 , β/
2] and ζ ( r ) = 0 in the set R \ [0 , β ). We can choose β small enough such that the local coordinates ( s, r ) are well defined in ω β . Set η ( k ) ε = (cid:0) [ˆ u ( k ) ε ] ε + [ ∂ r ˆ u ( k ) ε ] ε ( r − ε ) (cid:1) ζ ( r − ε )+ (cid:0) [ˆ u ( k ) ε ] − ε + [ ∂ r ˆ u ( k ) ε ] − ε ( r + ε ) (cid:1) ζ ( − r − ε ) . (4.19)The function is different from zero in the set ω β + ε \ ω ε only. And it is easy tocheck that η ( k ) ε and ∂ r η ( k ) ε have the same jumps across the boundary of ω ε as ˆ u ( k ) ε and ∂ r ˆ u ( k ) ε respectively. Therefore the function U ( k ) ε = ˆ u ( k ) ε − η ( k ) ε belongs to the domain of T ε . Moreover we have not changed ˆ u ( k ) ε too much, sincesup x ∈ Ω \ ω ε (cid:0) | η ( k ) ε ( x ) | + | ∆ η ( k ) ε ( x ) | (cid:1) ≤ cε . (4.20)It follows from the explicit formula for η ( k ) ε and smallness of jumps of ˆ u ( k ) ε , ∂ r ˆ u ( k ) ε across ∂ω ε . All the jumps are of order O ( ε ), as ε →
0, by construction.We will henceforth write n ( k ) ε = (cid:107) U ( k ) ε (cid:107) − L ( ρ ε , Ω) . Lemma 4.3.
For each k ∈ N , the pair (Λ ( k ) ε , n ( k ) ε U ( k ) ε ) constructed above is aquasimode of the operator T ε with the accuracy O ( ε ) as ε → .Proof. For simplicity we shall drop the index k in the sequel and write (Λ ε , U ε ), v j , w j and λ j instead of (Λ ( k ) ε , U ( k ) ε ), v ( k ) j , w ( k ) j and λ ( k ) j . Set F ε = ( T ε − Λ ε ) U ε . Thenwe have F ε = ( − ∆ + a − Λ ε ρ ) U ε = (cid:88) j =0 ε j ( − ∆ v j + av j − ρ j (cid:88) i =0 λ i v j − i ) + ∆ η ε − aη ε + Λ ε ρη ε outside ω ε . From our choice of v j , we derive that the first sum in the right-handside vanishes. Therefore sup x ∈ Ω \ ω ε | F ε ( x ) | ≤ c ε , because of (4.20). Applyingrepresentation (2.3) of the Laplace operator in ω , we have∆ − a + ε − Λ ε q = (cid:88) j =0 ε j − p j + ε P ε − a ε , where p = ∂ n + λ q , p = − κ ∂ n + λ q , p = − n κ ∂ n + ∂ s − a ( s,
0) + λ q , p = n κ ∂ n + 2 n κ ∂ s + n κ (cid:48) ∂ s − n∂ n a ( s,
0) + λ q, and a ε ( s, n ) = a ( s, εn ) − a ( s, − εn∂ n a ( s, ω ε , we obtain F ε = ( − ∆ + a − ε − Λ ε q ) U ε = (cid:88) j =0 ε j − j − i (cid:88) i =0 p i w j − i − ε P ε U ε + a ε = − ε P ε U ε + a ε , since the functions w , . . . , w solve the equations (cid:80) j − ii =0 p i w j − i = 0, by construc-tion. Consequently sup x ∈ ω ε | F ε ( x ) | = ε sup x ∈ ω ε | P ε U ε ( x ) | +sup x ∈ ω ε | a ε ( x ) | ≤ c ε .Hence (cid:107) F ε (cid:107) L ( ρ ε , Ω) ≤ (cid:90) Ω \ ω ε ρ | F ε | dx + ε − (cid:90) ω ε q | F ε | dx ≤ c ε + c | ω ε | ε ≤ c ε . The main contribution to the L ( ρ ε , Ω)-norm of U ε is given by the integral ε − (cid:82) ω ε q | w | dx which is of order O ( ε − ), as ε →
0. Hence (cid:107) U ε (cid:107) L ( ρ ε , Ω) ≥ c ε − / for ε small enough, i.e., n ε ≤ c ε / . This bound along with (cid:107) F ε (cid:107) L ( ρ ε , Ω) ≤ c ε / yields (cid:107) ( T ε − Λ ε )( n ε U ε ) (cid:107) L ( ρ ε , Ω) = n ε (cid:107) F ε (cid:107) L ( ρ ε , Ω) ≤ c ε , (4.21)and this is precisely the assertion of the lemma. (cid:3) Theorem 4.4.
Suppose that λ is an eigenvalue of the limit operator P such that λ ∈ σ ( B ) \ σ ( A ) . Assume the spectrum { λ ( k )1 } k ∈ N of the operator N λ is simple.Then there exists a countable set of eigenvalues λ ε,k , k ∈ N , in the spectrum of T ε that converge to λ and admit the asymptotics λ ε,k = λ + ελ ( k )1 + O ( ε ) , as ε → . (4.22) EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 15 l l e , e l l e , l e ,k Figure 3.
Bifurcation of the eigenvalue λ ∈ σ ( B ) \ σ ( A ). Proof.
Fix I ∈ N . In view of Proposition 4.2 and Lemma 4.3, there exist eigenvalues λ ε, , . . . , λ ε,I of T ε and a constant c I such that | λ ε,k − λ − ελ ( k )1 | ≤ c I ε (4.23)for k = 1 , , . . . , I and ε small enough. Moreover these eigenvalues are pairwisedifferent. Suppose, contrary to our claim, that some eigenvalue λ ε of T ε simulta-neously satisfies two estimates (4.23), for example when k = 1 and k = 2. Then λ ε − λ − ελ (1)1 ≤ c I ε and λ + ελ (2)1 − λ ε ≤ c I ε . Adding these inequalitiesyields λ (2)1 − λ (1)1 ≤ c I ε for all ε small enough. But this is impossible, because thespectrum of N λ is simple and therefore λ (1)1 < λ (2)1 . Write δ εI = ε max k =1 ,...,I | λ ( k )1 | + c I ε . Hence the interval [ λ − δ εI , λ + δ εI ] contains at least I eigenvalues of T ε that possessthe asymptotics (4.22). (cid:3) Asymptotics of eigenvalues in the case λ ∈ σ ( A ) ∩ σ ( B )In view of Theorem 3.7, if the set σ ( A ) ∩ σ ( B ) is non-empty, the operator P possesses generalized eigenvectors of rank 2. This requires changing the structureof asymptotics λ ε ∼ λ + λ / ε / + λ ε + · · · , (5.1) u ε ( x ) ∼ (cid:40) v ( x ) + ε / v / ( x ) + εv ( x ) + · · · for x ∈ Ω \ ω ε ,w ( s, rε ) + ε / w / ( s, rε ) + εw ( s, rε ) + · · · for x ∈ ω ε . (5.2)Here ( v , w ) is an non-zero element of the eigenspace spanned by vectors (3.12). If y is a normalized eigenfunction of (3.1) corresponding to λ , then w ( s, n ) = b ( s ) y ( n )for some b ∈ L ( γ ). As above, substituting the series in (1.2) in particular yields − ∆ v / + av / = λ ρv / + λ / ρv in Ω \ γ, (cid:96)v / = 0 on ∂ Ω , (5.3) − ∂ s w / = λ qw / + λ / qw in ω, (5.4) ∂ n w / ( · , −
1) = 0 , ∂ n w / ( · ,
1) = 0 , (5.5) v − / = w / ( · , − , v +1 / = w / ( · , . (5.6) Since (5.4), (5.5) can be written as ( B − λ ) w / = λ / b y and B is self-adjoint, asolution w / exists if only if λ / b = 0 . (5.7)This condition is a branching point of our algorithm.5.1. Integer power asymptotics. If b is a non-zero function, then λ / = 0 bynecessity. Then problem (5.3)–(5.6) turns into the limit problem (2.4)–(2.6) and( v / , w / ) ∈ X λ . Without loss of generality we assume that ( v / , w / ) = 0,i.e., this vector is absorbed by the leading term of the asymptotics. Moreover, atrivial verification shows that all terms λ j/ , ( v j/ , w j/ ) with half-integer indexesin (5.1), (5.2) can be treated as equal to zero. In this case, we come back to theinteger power asymptotics (4.1), but the construction of quasimodes needs a slightmodification. The next terms λ , ( v , w ) solve problem (4.2)–(4.5), and thereforecondition (4.6) must hold. But now we cannot rewrite (4.6) in the form of thespectral equation for N λ , because this operator is not defined for λ ∈ σ ( A ).We will “extend” N λ to σ ( A ) by means of a restriction of the space in whichit acts. A slight change in the proof of Proposition 3.5 actually shows that bothproblems (4.7) and (4.8) are solvable for λ ∈ σ ( A ) if the function ϕ in the bound-ary condition on γ is orthogonal to the subspace H λ ⊂ L ( γ ) spanned by functions(3.10). Although solutions of the problems, in this case, are ambiguously deter-mined, we can subject them to some additional condition. Namely, there exists aunique solution z of (4.7) (or (4.8)) satisfying the condition ∂ r z | γ ⊥ H λ . So we candefine the Dirichlet-to-Neumann map M − λ ϕ = ∂ r z | γ on H ⊥ λ , where z is a solutionof (4.7) belonging to H ⊥ λ . Similarly, we define M + λ ϕ = − ∂ r z | γ , where z ∈ H ⊥ λ is asolution of (4.8). Both operators are well defined for λ ∈ C . In fact, we have M ± λ = ( I − P λ ) N ± λ ( I − P λ ) , where P λ is the orthogonal projector onto the subspace H λ . Moreover, M ± λ = N ± λ for λ ∈ C \ σ ( A ), since the subspace H λ is trivial in this case.Now solvability condition (4.6) becomes M λ b = λ b , where M λ = θ − λ M − λ + θ + λ M + λ + ( θ + λ − θ − λ ) κ . The pseudodifferential operator M λ has the same properties as N λ . Suppose thatthe spectrum of M λ is simple and λ ( k )1 is the k -th eigenvalue of M λ with thenormalized eigenfunction b ( k ) . We set w = b ( k ) y .Assume λ is an K -fold eigenvalue of A with the eigenspace V λ . In order toshorten notation, we introduce the vector V = ( V , . . . , V K ), where the eigenfunc-tions V k are subject to condition (3.4). Then the leading term v = v ( k )0 in (5.2)solves (3.6) for λ = λ and b = b ( k ) and has the form v ( k )0 ( x ) = ˆ v ( k )0 ( x ) + α · V ( x ) , where α is an arbitrary vector in R K and ˆ v ( k )0 is a partial solution of (3.6) suchthat ˆ v ( k )0 ⊥ V λ . The dot denotes the scalar product in R K . To determine v ( k )0 uniquely, we should calculate α . Next, we have w ( s, n ) = ˆ w ( k )1 ( s, n ) + b ( s ) y ( n ) , (5.8)where ˆ w ( k )1 is a partial solution of (4.3), (4.4) subject to condition (4.9), and b is anarbitrary L ( γ )-function. Assume that b = g + g , where g ∈ H λ and g ∈ H ⊥ λ . EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 17
Then problem (4.2), (4.5) for v admits solutions v = φ + ψ + α · V , α ∈ R K , (5.9)where φ and ψ solve the problems − ∆ φ + aφ = λ ρφ in Ω \ γ, (cid:96)φ = 0 on ∂ Ω , φ = y ( ± g on γ ± ; − ∆ ψ + aψ = λ ρψ + λ ( k )1 ρ (cid:16) ˆ v ( k )0 + α · V (cid:17) in Ω \ γ, (cid:96)ψ = 0 on ∂ Ω ,ψ ± = ˆ w ( k )1 ( · , ± ∓ ∂ r ˆ v ( k )0 + y ( ± g ∓ α · ∂ r V on γ respectively. Since g is orthogonal to H λ , the first problem is solvable and admitsa solution belonging to V ⊥ λ . As for the second one, its solvability conditions can bewritten in the form (cid:0) C λ − λ ( k )1 (cid:1) α = f , where C λ is a matrix with the entries c ij = (cid:90) γ + ∂ r V i ∂ r V j dγ + (cid:90) γ − ∂ r V i ∂ r V j dγ, i, j = 1 , . . . , K, and f is a vector with the components f i = λ ( k )1 (cid:90) Ω ρ ˆ v ( k )0 V i dx + (cid:90) γ + (cid:0) ˆ w ( k )1 − ∂ r ˆ v ( k )0 + y (1) g (cid:1) ∂ r V i dγ − (cid:90) γ − (cid:0) ˆ w ( k )1 + ∂ r ˆ v ( k )0 + y ( − g (cid:1) ∂ r V i dγ, i = 1 , . . . , K. If we suppose that λ ( k )1 is not an eigenvalue of C λ , then the solvability conditionsfor ψ can be fulfilled for any g ∈ H λ . It is enough to set α = (cid:0) C λ − λ ( k )1 (cid:1) − f .Then the problem has a solution φ ∈ V ⊥ λ .Note that the vector α has not yet been defined, because f depends on theunknown function g . Using representations (5.8) and (5.9) along with the factthat g ∈ H ⊥ λ , we can write the solvability condition for (4.12), (4.13) in the form( M λ − λ ( k )1 ) g = λ ( k )2 b ( k ) + λ ( k )1 g + h ( k )1 , (5.10)where h ( k )1 is given by (4.16). For the equation (5.10) to be meaningful, we needto ensure that the right hand side is orthogonal to H λ . Obviously, b ( k ) belongsto H ⊥ λ . Next, if λ ( k )1 is different from zero, there exists a unique vector g ∈ H λ such that λ ( k )1 g + h ( k )1 ∈ H ⊥ λ . With g in hand, we can uniquely defined f , α ,and finally the leading term ( v ( k )0 , w ( k )0 ). The solvability condition for (5.10) hasthe form λ ( k )2 = − ( h ( k )1 + λ ( k )1 g , b ( k ) ) L ( γ ) . Then there exists a unique solution g satisfying the condition ( g, b ( k ) ) L ( γ ) = 0. And finally, we can uniquely define w ( k )1 and v ( k )1 (up to the vector α ). This process can be continued to systematicallyconstruct all other terms λ ( k ) j , v ( k ) j and w ( k ) j in the approximation Λ ( k ) ε , ˆ u ( k ) ε of theform (4.17), (4.18). As in the previous section, ˆ u ( k ) ε can be improved to the element U ( k ) ε = ˆ u ( k ) ε − η ( k ) ε from the domain of operator T ε , where η ( k ) ε is given by (4.20).Summarizing results of the above calculations, we obtain the following statement. Lemma 5.1.
The pairs (Λ ( k ) ε , (cid:107) U ( k ) ε (cid:107) − L ( ρ ε , Ω) U ( k ) ε ) , k ∈ N , are quasimodes of T ε with the accuracy O ( ε ) as ε → that approximate the part of spectrum lying in avicinity of λ ∈ σ ( A ) ∩ σ ( B ) . The lemma can be proved similarly to Lemma 4.3.5.2.
Half-integer power asymptotics.
The set of quasimodes in Lemma 5.1does not approximate all eigenvalues of T ε that converge to λ . We assume that λ / in (5.2) is different from zero, and then b = 0, by (5.7). Recalling now (3.12),we have v = β · V , where β is an arbitrary vector in R K such that (cid:107) β (cid:107) = 1.In this case, we will use some finite-dimensional operator instead of M λ to splitthe limit multiple eigenvalue λ . Reasoning as above we deduce that the problem(5.3)-(5.6) has a solution of the form w / ( s, n ) = b / ( s ) y ( n ), v / = ˆ v / + β / · V where β / ∈ R K and ˆ v / is a partial solution of the problem − ∆ v / + av / = λ ρv / + λ / ρv in Ω \ γ, (cid:96)v / = 0 on ∂ Ω , (5.11) v − / = y ( − b / , v +1 / = y (1) b / (5.12)that is orthogonal to V λ in L ( ρ, Ω). By Proposition 3.5, the solvability conditionsfor (5.11), (5.12) can be written in the vector form (cid:90) γ b / Ψ dγ + λ / β = 0 , (5.13)where Ψ = (Ψ , . . . , Ψ K ) and Ψ k are given by (3.10). For the next terms we have − ∆ v + av = ρ ( λ v + λ / v / + λ v ) in Ω \ γ, (cid:96)v = 0 on ∂ Ω , (5.14) − ∂ n w = λ qw + λ / qw / in ω, (5.15) ∂ n w ( · , −
1) = ∂ r v − , ∂ n w ( · ,
1) = ∂ r v +0 , (5.16) v − = w ( · , −
1) + ∂ r v − , v +1 = w ( · , − ∂ r v +0 . (5.17)Problem (5.15), (5.16) is solvable if y (1) ∂ r v +0 − y ( − ∂ r v − + λ / b / = 0 . Since ∂ r v ± = β · ∂ r V ± , it can be written in the form β · (cid:0) y (1) ∂ r V + − y ( − ∂ r V − (cid:1) + λ / b / = 0 . Using notation (3.10), we have β · Ψ + λ / b / = 0 . (5.18)Multiplying this equality by Ψ (cid:62) , integrating over γ and recalling (5.13), we finallydiscover (cid:0) G λ − λ / (cid:1) β = 0, where G λ is the Gram matrix of Ψ ,. . . ,Ψ K . Thismatrix is semi-positive and its rank is equal to the dimension of H λ .Suppose ω is a positive simple eigenvalue of G λ with the eigenvector β . Sothere exist two different correctors λ / = ω and λ / = − ω in asymptotics (5.1)with the same leading term v = β · V in approximation (5.2). First assume that λ / = ω . From (5.18), we have b / = − ω − β · Ψ. Up to a function b , wecan find w ( s, n ) = ˆ w ( s, n ) + b ( s ) y ( n ), where ˆ w is a partial solution of (5.15),(5.16) subject to the condition ( ˆ w , y ) L ( q, ( − , = 0. Next, problem (5.14), (5.17)is solvable if and only if (cid:82) γ b Ψ dγ + ωβ / + λ β = h , where h = (cid:90) γ − (cid:0) ˆ w + ∂ r v (cid:1) ∂ r V dγ − (cid:90) γ + (cid:0) ˆ w − ∂ r v (cid:1) ∂ r V k dγ − ω (cid:90) Ω ρ ˆ v / V k dx. Also, a solution of the problem − ∂ n w / = q ( λ w / + ωw + λ w / ) in ω,∂ n w / ( · , −
1) = ∂ n v − / , ∂ n w / ( · ,
1) = ∂ n v +1 / . EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 19 exists if and only if y ( − ∂ r v − / − y (1) ∂ r v +1 / = ωb + λ b / . (5.19)Reasoning as in the previous step, we can rewrite this condition in the form (cid:0) G λ − ω (cid:1) β / = 2 ωλ β − ωh + f, where f = y ( − (cid:90) γ − ∂ r ˆ v / Ψ dγ − y (1) (cid:90) γ + ∂ r ˆ v / Ψ dγ. Since ω is an eigenvalue of G λ , the system admits a solution β / if and only if λ = ω β · ( ωh − f ) . (5.20)Although the unit vector β is defined up to the change of sign, λ is uniquelydefined by (5.20), because the transformation β (cid:55)→ − β implies that the vectors h and f also change their sings. We fix this solution such that β / · β = 0. Then b = ω (cid:16) y ( − ∂ r ˆ v − / − y (1) ∂ r ˆ v +1 / − β / · Ψ − λ b / (cid:17) , by (5.19). Assuming that λ +1 / = ω we have calculated λ +1 , w (+)1 / , w (+)1 and v (+)1 / in (5.1), (5.2). We can continue in this fashion obtaining the next terms of theasymptotics. Taking λ − / = − ω we can compute analogously the terms λ − k , w ( − ) k and v ( − ) k in the asymptotics of other eigenvalue and eigenfunction. A simple analysisof the foregoing formulas shows that w ( − )1 / = − w (+)1 / and v ( − )1 / = − v (+)1 / . Moreover,by construction, we have w ( ± )1 / ( s, n ) = ∓ ω − y ( n ) β · Ψ( s ) , v ( ± )1 / ( x ) = ± v ∗ ( x ) , where v ∗ is a solution of the problem − ∆ v + av = λ ρv + ωρ β · V in Ω \ γ, (cid:96)v = 0 on ∂ Ω , v = − ω − y ( ± β · Ψ on γ ± that is orthogonal to V λ .Let us summarize the above considerations in the following lemma. Lemma 5.2.
Let d be the dimension of H λ , where λ ∈ σ ( A ) ∩ σ ( B ) . Suppose thatall non-zero eigenvalues ω , . . . , ω d of the matrix G λ are simple and β , , . . . , β ,d are the corresponding normalized eigenvectors. Then the operator T ε possesses d pairs of the quasimodes (ˆ µ ± ε,j , n ± ε,j V ( ± ) ε,j ) , j = 1 , . . . , d , with the accuracy O ( ε / ) as ε → , where n ± ε,j is a normalizing factor and ˆ µ ± ε,j = λ ± ε / ω j + (cid:88) k =2 ε k/ λ ± k/ ,j ,V ( ± ) ε,j = β ,j · V ± ε / v ∗ ,j + (cid:88) k =2 ε k/ v ( ± ) k/ ,j − η ( ± ) ε,j in Ω \ ω ε ,V ( ± ) ε,j = ∓ ε / ω − j y ( rε ) β ,j · Ψ( s ) + (cid:88) k =2 ε k/ w ( ± ) k/ ,j ( s, rε ) in ω ε . The small correctors η ( ± ) ε,j are defined as in (4.19) with V ( ± ) ε,j in place of ˆ u ( k ) ε . Proof.
The proof differs from the proof of Lemma 4.3 only by estimate (4.21).The approximations to eigenvalues and eigenfunctions have been constructed up toorder O ( ε ) and therefore the remainder term F ε (with the notation of Lemma 4.3)can be also estimated by (cid:107) F ε (cid:107) L ( ρ ε , Ω) ≤ c ε / . But the leading term w in thisasymptotics is equal to zero and hence the L ( ρ ε , Ω)-norm of V ( ± ) ε,j is boundeduniformly with respect to ε . Then the normalizing factor n ± ε,j tends to some positivenumber as ε → (cid:107) ( T ε − ˆ µ ± ε,j )( n ± ε,j V ( ± ) ε,j ) (cid:107) L ( ρ ε , Ω) = n ± ε,j (cid:107) F ε (cid:107) L ( ρ ε , Ω) ≤ c ε / , which is the desired conclusion. (cid:3) l l e , e l l e , l e ,k + m e , – m e , – m e , m e , Figure 4.
Bifurcation of the eigenvalue λ ∈ σ ( A ) ∩ σ ( B ).In view of Lemmas 5.1 and 5.2, we can prove the following result in the sameway as Theorem 4.4. Theorem 5.3.
Suppose that the set σ ( A ) ∩ σ ( B ) is non empty and λ is an eigen-value of the limit operator P belonging to this intersection.(i) Assume the spectrum { λ ( k )1 } k ∈ N of the operator M λ is simple. Then thereexists a countable set of eigenvalues λ ε,k , k ∈ N , in the spectrum of T ε thatadmit the asymptotics λ ε,k = λ + ελ ( k )1 + O ( ε ) , as ε → . (ii) Assume dim H λ = d and all positive eigenvalues ω j of the matrix G λ aresimple. Then operator T ε also possesses d eigenvalues with the asymptotics µ − ε,j = λ − ε / ω j + ελ − ,j + O ( ε / ) ,µ + ε,j = λ + ε / ω j + ελ +1 ,j + O ( ε / ) , j = 1 , . . . , d, as ε → , where λ ± ,j can be calculated from (5.20) . In the case under consideration, the generalized eigenspace X λ contains theJordan chains of length 2. Note that problem (5.3)–(5.6) for ( v / , w / ) coincidesup to the multiplier λ / in the right-hand side with problem (3.13)–(3.15) forgeneralized eigenvectors. Inspecting the structure of V ( ± ) ε,j more closely we see that V ( ± ) ε,j = U j ± ε / U ∗ j + O ( ε ) , as ε → , EMBRANES WITH THIN AND HEAVY INCLUSIONS: ASYMPTOTICS OF SPECTRA 21 where the vectors U j = ( β ,j · V ( x ) , , U ∗ j = (cid:0) v ∗ ,j ( x ) , − ω − j y ( n ) β ,j · Ψ( s ) (cid:1) form a Jordan chain of P corresponding to λ . This observation has the followinggeometric interpretation. The operator T ε possesses pairs of eigenvalues with theasymptotics µ ± ε,j = λ ± ω j ε / + O ( ε ) for which the corresponding normalizedeigenfunctions u ( − ) ε,j and u (+) ε,j converge in L (Ω) to the same function v ,j = β ,j · V ,as ε →
0. Although these eigenfunctions remain orthogonal for all ε in the weightedspace L ( ρ ε , Ω), they make an infinitely small angle between them in L (Ω) with thestandard norm, and stick together at the limit. In particular, it leads to the loss ofcompleteness in L (Ω) for the limit eigenfunction collection. Interestingly enough,however, the plane π ε,j that is the span of u ( − ) ε,j and u (+) ε,j } has regular behaviouras ε →
0. The limit position of π ε,j is the 2-dimensional space π j spanned by thefunctions β ,j · V and v ∗ ,j .We actually have an example of singular perturbations in which the complete-ness property of perturbed eigenfunction collection passes in some sense into thecompleteness of generalized eigenfunctions of the limit non-self-adjoint operator.The non-self-adjoint operator P contains all the information about the asymptoticbehaviour of spectrum and eigenspaces of the family of self-adjoint operators T ε acting on varying Hilbert spaces L ( ρ ε , Ω). In particular, the Jordan chains of P are involved in the asymptotics of some eigenfunctions of T ε .As we mentioned above, the Dirichlet type eigenvalue problem − ∆ u ε = λ ε ρ ε u ε in Ω , u ε = 0 on ∂ Ωwith the weight function ρ ε ( x ) = ρ ( x ) + ε − m q ( rε ) has been studied in [34]. It hasbeen proved that for any m ≥ λ ε,k = ε m − ( ν k + o (1)) as ε →
0. Theorems 4.4 and 5.3 also containthis result for m = 2. Set q = (cid:82) − q ( t ) dt . Corollary 5.4.
Problem (1.2) has a series of small eigenvalues that admit theasymptotics λ ε,k = εν k + O ( ε ) as ε → , where ν k are eigenvalues of the problem − ∆ v + av = 0 in Ω \ γ, (cid:96)v = 0 on ∂ Ω , [ v ] γ = 0 , [ ∂ r v ] γ + νq v = 0 on γ with the spectral parameter ν in the coupling conditions. In addition, if is a pointof σ ( A ) , then (1.2) has also a finite number of small eigenvalues with asymptotics µ ± ε,j = ± ω j √ ε + O ( ε ) as ε → , j = 1 , . . . , d. Proof.
The existence of such eigenvalues follows from Theorems 4.4 and 5.3 alongwith the observation that λ = 0 belongs to σ ( P ) (see Remark 3.8). It remains toderive the eigenvalue problem for ν k . Let us look at (2.4)–(2.6) and put λ = 0: − ∆ v + av = 0 in Ω \ γ, (cid:96)v = 0 on ∂ Ω , v − = w ( · , − , v + = w ( · , , on γ,∂ n w = 0 in ω, ∂ n w ( · , −
1) = 0 , ∂ n w ( · ,
1) = 0 , We see that w = b ( s ) and hence v − = v + , i.e., v is continuous on γ . Since y = 1in this case, condition (4.6) for v = v reads as ∂ r v + − ∂ r v − + λ q b = 0 on γ . Tocomplete the proof we replace λ with ν and recall that b = v | γ . (cid:3) Asymptotics of eigenvalues in the case λ ∈ σ ( A ) \ σ ( B )For the sake of completeness, we briefly discuss the perturbation of eigenvalue λ of P which does not belong to σ ( B ). In view of part (i) of Theorem 3.7, λ is an eigenvalue of finite multiplicity. Also, we have v = α · V and w = 0 inasymptotics (4.1), where α ∈ R K \ { } . Then (4.3) and (4.4) imply − ∂ n w = λ qw in ω, ∂ n w ( · , −
1) = α · ∂ r V − , ∂ n w ( · ,
1) = α · ∂ r V + . Since λ (cid:54)∈ σ ( B ), there exists a unique solution of the problem. Suppose thefunctions W k , k = 1 , . . . , K , solve the problems − ∂ n w = λ qw in ω, ∂ n w ( · , −
1) = ∂ r V − k , ∂ n w ( · ,
1) = ∂ r V + k , and W = ( W , . . . , W K ). Thus we have w = α · W . Next, we can rewrite (4.2)and (4.5) in the form − ∆ v + av = λ ρv + λ ρ α · V in Ω \ γ, (cid:96)v = 0 on ∂ Ω , (6.1) v − = α · ( W ( · , −
1) + ∂ r V − ) , v +1 = α · ( W ( · , − ∂ r V + ) . (6.2)In general, the problem is unsolvable, because λ belongs to the spectrum of A which is the direct sum of A − and A + . To achieve the solvability one needs tochoose proper vectors α along with the parameter λ . Multiplying the equation in(6.1) by V , . . . , V K in turn and integrating by parts twice in view of the boundaryconditions (6.2) yield the spectral matrix equation Rα = λ α . The matrix R hasthe entries r ij = (cid:90) γ (cid:0) ( W i ( · , − ∂ r V + i ) ∂ r V + j + ( W i ( · , −
1) + ∂ r V − i ) ∂ r V − j (cid:1) dγ. Assume that R has K simple eigenvalues λ (1)1 , . . . , λ ( K )1 with the correspondingeigenvalues α (1)0 , . . . , α ( K )0 . For any pair ( λ ( k )1 , α ( k )0 ), we can solve (6.1), (6.2) andfind v ( k )1 = ˆ v ( k )1 + α ( k )1 · V up to the vector α ( k )1 ∈ R K . We continue in this fashionobtaining K different quasimodes with high enough accuracy of the operator T ε that approximate the part of spectrum lying in a vicinity of λ . Theorem 6.1.
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