On vibrating thin membranes with mass concentrated near the boundary: an asymptotic analysis
Abstract: We consider the spectral problem
in a smooth bounded domain of . The factor which appears in the first equation plays the role of a mass density and it is equal to a constant of order in an neighborhood of the boundary and to a constant of order in the rest of . We study the asymptotic behavior of the eigenvalues and the eigenfunctions as tends to zero. We obtain explicit formulas for the first and second terms of the corresponding asymptotic expansions by exploiting the solutions of certain auxiliary boundary value problems.
Keywords: Steklov boundary conditions, eigenvalues, mass concentration, asymptotic analysis, spectral analysis.
2010 Mathematics Subject Classification: Primary 35B25; Secondary 35C20, 35P05, 70Jxx, 74K15.
1 Introduction
We fix once for all a real number and a bounded connected open set in of class . Then, for small, we consider the Neumann eigenvalue problem
(1.1) 
in the unknowns (the eigenvalue) and (the eigenfunction). The factor is defined by
where
is the strip of width near the boundary of (see Figure 1). Here and in the sequel denotes the outer unit normal to .
It is wellknown that the eigenvalues of (1.1) have finite multiplicity and form an increasing sequence
In addition and the eigenfunctions corresponding to are the constant functions on . We will agree to repeat the eigenvalues according to their multiplicity.
Problem (1.1) arises in the study of the transverse vibrations of a thin elastic membrane which occupies at rest the planar domain (see e.g., [cohil]). The mass of the membrane is distributed accordingly to the density . Thus the total mass is given by
and it is constant for all . In particular, most of the mass is concentrated in a neighborhood of the boundary , while the remaining is distributed in the rest of with a density proportional to . The eigenvalues are the squares of the natural frequencies of vibration when the boundary of the membrane is left free. The corresponding eigenfunctions represent the profiles of vibration.
Then, we introduce the classical Steklov eigenvalue problem
(1.2) 
in the unknowns (the eigenvalue) and (the eigenfunction). The spectrum of (1.2) consists of an increasing sequence of nonnegative eigenvalues of finite multiplicity, which we denote by
One easily verifies that and that the corresponding eigenfunctions are the constants functions on . In addition, one can prove that for all we have
(see, e.g., Arrieta et al. [arrieta], see also Buoso and Provenzano [buosoprovenzano] and Theorem 2.17 here below). Accordingly, one may think to the ’s as to the squares of the natural frequencies of vibration of a free elastic membrane with total mass concentrated on the dimensional boundary with constant density . A classical reference for the study of problem (1.2) is the paper [steklov] by Steklov. We refer to Girouard and Polterovich [girouardpolterovich] for a recent survey paper and to the recent works of Lamberti and Provenzano [lambertiprovenzano1] and of Lamberti [lamberti1] for related problems. We also refer to Buoso and Provenzano [buosoprovenzano] for a detailed analysis of the analogous problem for the biharmonic operator.
The aim of the present paper is to study the asymptotic behavior of the eigenvalues of problem (1.1) and the corresponding eigenfunctions as goes to zero, i.e., when the thin strip shrinks to the boundary of . To do so, we show the validity of an asymptotic expansion for and as goes to zero. In addition, we provide explicit expressions for the first two coefficients in the expansions in terms of solutions of suitable auxiliary problems. In particular, we establish a closed formula for the derivative of at . We observe that such a derivative may be seen as the topological derivative of for the domain perturbation considered in this paper. We will confine ourselfs to the case when converges to a simple eigenvalue of (1.2). We observe that such a restriction is justified by the fact that Steklov eigenvalues are generically simple (see e.g., Albert [albert] and Uhlenbeck [uhl]).
As we have written here above, problems (1.1) and (1.2) concern the elastic behavior of a membrane with mass distributed in a very thin region near the boundary. In view of such a physical interpretation, one may wish to know whether the normal modes of vibration are decreasing or increasing when approaches .
To answer to this question one can compute the value of the derivative of at by exploiting the closed formula that we will obtain. When is a ball, we can find explicit expressions for the eigenvalues and for the corresponding eigenvectors (in this case every eigenvalue is double). In Appendix B we have verified that in such special case the eigenvalues are locally decreasing when approaches from above. Accordingly the Steklov eigenvalues of the ball are local minimizers of the . This result is in agreement with the value of the derivative of at that one may compute from our closed formula obtained for a general domain of class .
We observe here that asymptotics for vibrating systems (membranes or bodies) containing masses along curves or masses concentrated at certain points have been considered by several authors in the last decades (see, e.g., Golovaty et al. [gol1], Lobo and Pérez [lope1] and Tchatat [tchatatbook]). We also refer to Lobo and Pérez [loboperez1, loboperez2] where the authors consider the vibration of membranes and bodies carrying concentrated masses near the boundary, and to Golovaty et al. [gomezpereznazarov2, gomezpereznazarov1], where the authors consider spectral stiff problems in domains surrounded by thin bands. Let us recall that these problems have been addressed also for vibrating plates (see Golovaty et al. [golonape_plates1, golonape_plates2] and the references therein). We also mention the alternative approach based on potential theory and functional analysis proposed in Musolino and Dalla Riva [musolinodallariva] and Lanza de Cristoforis [lanza].
The paper is organized as follows. In Section 2 we introduce the notation and certain preliminary tools that are used through the paper. In Section 3 we state our main Theorems 3.1 and 3.5, which concern the asymptotic expansions of the eigenvalues and of the eigenfunctions of (1.1), respectively. In Theorem 3.1 we also provide the explicit formula for the topological derivative of the eigenvalues of (1.1). The proof of Theorems 3.1 and 3.5 is presented in Sections 4 and LABEL:sec:5. In Section 4 we justify the asymptotic expansions of Theorems 3.1 and 3.5 up to the zero order terms. Then in Section LABEL:sec:5 we justify the asymptotic expansions up to the first order terms and, as a byproduct, we prove the validity of the formula for the topological derivative. At the end of the paper we have included two Appendices. In the Appendix A we consider an auxiliary problem and prove its wellposedness. In the last Appendix B we consider the case when is the unit ball and prove that the Steklov eigenvalues are local minimizers of the Neumann eigenvalues for small enough.
2 Preliminaries
2.1 A convenient change of variables
Since is of class , it is wellknown that there exists such that the map is a diffeomorphism of class from to for all . We will exploit this fact to introduce curvilinear coordinates in the strip . To do so, we denote by the arc length parametrization of the boundary . Then, one verifies that the map defined by , for all , is a diffeomorphism and we can use the curvilinear coordinates in the strip . We denote by the signed curvature of , namely we set for all .
In order to study problem (1.1) it is also convenient to introduce a change of variables by setting . Accordingly, we denote by the function from to defined by for all . The variable is usually called ‘rapid variable’. We observe that in this new system of coordinates , the strip is transformed into a band of length and width (see Figures 2 and 3). Moreover, we note that if , then we have
(2.1) 
so that for all .
We will also need to write the gradient of a function on with respect to the coordinates . To do so we take
and we consider . Then we have
and therefore
(2.2) 
for all .
2.2 Some remarks about
We can write , where is the characteristic function of and
(2.3) 
Then we observe that for we have
(2.4) 
where is defined by
(2.5) 
By (2.1) it follows that . Then by (2.3) and (2.4) one verifies that there exists a real analytic map from to such that
(2.6) 
We are now legitimate to fix once for all a real number
such that  (2.7) 
2.3 Weak formulation of problem (1.1) and the resolvent operator
For all , we denote by the Hilbert space consisting of the functions in the standard Sobolev space endowed with the bilinear form
(2.8) 
The bilinear form (2.8) induces on a norm which is equivalent to the standard one. We denote such a norm by .
We note that the weak formulation of problem (1.1) can be stated as follows: a pair is a solution of (1.1) in the weak sense if and only if
Then, for all we introduce the linear operator from to itself which maps a function to the function such that
(2.9) 
We note that such a function exists by the Riesz representation theorem and it is unique because for all implies that .
In the sequel we will heavily exploit the following lemma. We refer to Oleĭnik et al. [oleinik, III.1] for its proof.
Lemma 2.10.
Let be a compact, selfadjoint and positive linear operator from a separable Hilbert space to itself. Let , with . Let be such that . Then, there exists an eigenvalue of the operator which satisfies the inequality . Moreover, for any there exists with , belonging to the space generated by all the eigenfunctions associated with an eigenvalue of the operator lying on the segment , and such that
We observe that the operator is a good candidate for the application of Lemma 2.10. Indeed, we have the following Proposition 2.11.
Proposition 2.11.
For all the map is a compact, selfadjoint and positive linear operator from to itself.
Proof.
The proof that is selfadjoint and positive can be effected by noting that for all . To prove that is compact we denote by the linear operator from to which takes a function to the unique element which satisfies the condition in (2.9). By the Riesz representation theorem one verifies that is well defined. In addition, we can prove that is bounded. Indeed, we have
and by a computation based on the Hölder inequality one verifies that
which implies that for all . Then we denote by the embedding map from to . Via the natural isomorphism from to , one deduces that is compact. Since , we conclude that also is compact. ∎
We conclude this subsection by observing that the norm of a function in is uniformly bounded by its norm for all . We will prove such a result in Proposition 2.15 below by exploiting the following Lemma 2.12.
Lemma 2.12.
There exists such that
for all and for all .
Proof.
We argue by contradiction and we assume that there exist a sequence and a sequence such that
(2.13) 
for all . Since is bounded there exist and a subsequence of , which we still denote by , such that as . Then we set
for all . We verify that and for all , and from (2.13), . Then is bounded in and we can extract a subsequence, which we still denote by , such that weakly in and strongly in , for some . Moreover, since one can verify that a.e. in , and thus is constant on . In addition,
(2.14) 
We now prove that (2.14) leads to a contradiction. Indeed, we can prove that . We consider separately the case when and the case when . For we verify that . Then , because . Since and is constant, it follows that . If instead , then, by an argument based on [phd, Lemmas 3.1.22, 3.1.28] we have . Since , it follows that . Since is constant on , we deduce that . ∎
We are now ready to prove Proposition 2.15.
Proposition 2.15.
Proof.
2.4 Known results on the limit behavior of
In the following Theorem 2.17 we recall some results on the limit behavior of the eigenelements of problem (1.1).
Theorem 2.17.
The following statements hold.

For all it holds

Let be a simple eigenvalue of problem (1.2) and let be such that . Then there exists such that is simple for all .
The proof of Theorem 2.17 can be carried out by using the notion of compact convergence for the resolvent operators, and can also be obtained as a consequence of the more general results proved in Arrieta et al. [arrieta] (see also Buoso and Provenzano [buosoprovenzano]).
From Theorem 2.17, it follows that the function which takes to can be extended with continuity at by setting for all .
3 Description of the main results
In this section we state our main Theorems 3.1 and 3.5 which will be proved in Sections 4 and LABEL:sec:5 below. We will use the following notation: if and is a simple eigenvalue of problem (1.2), then we take
with as in Theorem 2.17 and as in (2.7), so that is a simple eigenvalue of (1.1) for all . If is an invertible function, than denotes the inverse of , as opposed to and which denote the reciprocal of a real nonzero number or of a nonvanishing function.
In the following Theorem 3.1 we provide an asymptotic expansion of the eigenvalue up to a remainder of order .
Theorem 3.1.
In Theorem 3.5 here below we show an asymptotic expansion for the eigenfunction associated to .
Theorem 3.5.
Let and assume that is a simple eigenvalue of problem (1.2). Let be such that is a simple eigenvalue of problem (1.1) for all . Let be the unique eigenfunction of problem (1.2) associated with which satisfies the additional condition (3.4). For all , let be the unique eigenfunction of problem (1.1) corresponding to which satisfies the additional condition
(3.6) 
Then there exist and such that
(3.7) 
where the function is the extension by of to .
We shall present explicit formulas for in terms of and (see formula (4.4)) and we shall identify as the solution to a certain boundary value problem (see problem (LABEL:u1_problem_simple_true)). We also note that , so that the third term in (3.7) is in in (cf. Proposition 4.6).
The proof of Theorems 3.1 and 3.5 consists of two steps. In the first step (Section 4) we show that the quantity is of order as tends to zero. Moreover, we introduce the function and we show that is of order as tends to zero. In the second step (Section LABEL:sec:5) we complete the proof of Theorems 3.1 and 3.5 by proving the validity of (3.2) and (3.7) and we introduce the boundary value problem which identifies .
4 First step
We begin here the proof of Theorems 3.1 and 3.5. Accordingly, we fix and we take , , , , and as in the statements of Theorems 3.1 and 3.5. The aim of this section is to prove the following intermediate result.
Proposition 4.1.
We have
as  (4.2) 
and
in as .  (4.3) 
In other words, we wish to justify the expansions (3.2) and (3.7) up to a remainder of order . (We observe here that Theorem 2.17 states the convergence of to , but it does not provide any information on the rate of convergence.)
We introduce the following notation. We denote by the function from to defined by
(4.4) 
By a straightforward computation one verifies that solves the following problem
(4.5) 
Then for all we denote by the extensions by of to . We note that by construction . We also observe that the norm of is in as . Indeed, we have the following proposition.
Proposition 4.6.
There is a constant such that for all .
Proof.
Since is the extensions by of to , by the rule of change of variables in integrals we have
∎
We also observe that is uniformly bounded for . Namely, we have the following proposition.
Proposition 4.7.
There is a constant such that for all .
Proof.
We have
(4.8) 
Since on we have
Thus, by Proposition 4.6 and by (2.6) we deduce that
(4.9) 
for some . By (2.2) and by the rule of change of variables in integrals we have
From (4.4) we observe that
(4.10) 
and
(4.11) 
Since is assumed to be of class , a classical elliptic regularity argument shows that (see e.g., Agmon et al. [agmon1]). In addition, by the regularity of , we have that is of class from to . Thus, from (4.10) and (4.11) it follows that , . Then by condition (2.1) we verify that
(4.12) 
Now, by (4.8), (4.9), and (4.12) we deduce the validity of the proposition. ∎
We now consider the operator introduced in Section 2. We recall that is a compact selfadjoint operator from to itself. In addition, is an eigenvalue of (1.1) if and only if is an eigenvalue of and Theorem 2.17 implies that
Since is a simple eigenvalue of (1.1), we can prove that is also simple for small enough and we have the following Lemma 4.13.
Lemma 4.13.
There exist and such that, for all the only eigenvalue of in the interval
is .