Spectral gaps for the linear surface wave model in periodic channels

# Spectral gaps for the linear surface wave model in periodic channels

F.L. Bakharev , K. Ruotsalainen, J. Taskinen
###### Abstract

We consider the linear water-wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width . Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the continuous spectrum. We show that for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as : the endpoints are determined within corrections of order . The width of the first bands is shown to be . Finally, we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity.

Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178 Russia111The first named author was supported by the St. Petersburg State University grant 6.38.64.2012 as well as by the Chebyshev Laboratory - RF Government grant 11.G34.31.0026 and by JSC “Gazprom Neft”. The first and third named authors were also supported by the Academy of Finland project ”Functional analysis and applications”.

University of Oulu, Department of Electrical and Information Engineering, Mathematics Division, P.O. Box 4500, FI-90401 Oulu, Finland

University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68, FI-00014 Helsinki, Finland.

## 1 Introduction

### 1.1 Overview of the results

Research on wave propagation phenomena in periodic media has been very active during many decades. The topics and applications include for example photonic crystals, meta-materials, Bragg gratings of surface plasmon polariton waveguides, energy harvesting in piezoelectric materials as well as surface wave propagation in periodic channels, which is the subject of this paper. A standard mathematical approach consists of linearisation and posing a spectral problem for an elliptic, hopefully self-adjoint, equation or system.

Early on it was noticed that waves propagating in periodic media have spectra with allowed bands separated by forbidden frequency gaps. This phenomenon was first discussed by Lord Rayleigh [22]. It has also attracted some interest in coastal engineering because it provides a possible means of protection against wave damages [11, 14], for example by varying the bottom topography by periodic arrangements of sandbars. The existence of forbidden frequencies is conventionally related to Bragg reflection of water waves by periodic structures. Here, Bragg reflection is an enhanced reflection which occurs when the wavelength of an incident surface wave is approximately twice the wavelength of the periodic structure. This mechanism works, if the waves are relatively long so that the depth changes can effect them [14].

A similar phenomenon may also happen, when waves are propagating along a channel with periodically varying width. In [9], and later [13], the authors studied a channel, the wall of which had a periodic stepped structure. Using resonant interaction theory they were able to verify that significant wave reflection could occur. These results are based on the assumption of small wall irregularities.

Gaps in the continuous spectrum for equations or systems in unbounded waveguides have been studied in many papers, and we refer to [7] for an introduction to the topic. In [19] the authors studied the linear elasticity system and proved the existence of arbitrarily (though still finitely) many gaps, the number of them depending on a small geometric parameter; the approach is similar to Section 3.1, below, and the result is analogous to Corollary 3.2. In the setting of the linear water-wave problem, spectral gaps have been studied in [8], [12], [17] and [4], though the point of view is different from the present work.

In this paper we consider surface wave propagation using the linear water wave equation with spectral Steklov boundary condition on the free water surface, see the equations (1.8)–(1.10), which are called the original problem here. The water-filled domain forms an unbounded periodic channel consisting of infinitely many identical bounded containers connected by apertures of width , see Figure 1.1. The first results, Theorem 3.1 and Corollary 3.2 show that the essential spectrum of the original problem (which is expected to be non-empty due to the unboundedness of the domain) has gaps, and the number of them can be made arbitrarily large depending on the parameter . An explanation of this phenomenon can be outlined rather simply using the Floquet-Bloch theory, though a lot of technicalities will eventually be involved. Namely, if , the domain becomes a disjoint union of infinitely many bounded containers, and the water-wave problem reduces to a problem on a bounded domain (we call it the limit problem), hence it has a discrete spectrum consisting of an increasing sequence of eigenvalues . On the other hand, for , one can use the Gelfand transform to render the original problem into another bounded domain problem depending on the additional parameter . For each fixed this problem again has a sequence of eigenvalues . Moreover, by results of [15], [18], Theorem 3.4.6, and [16], Theorem 2.1, the essential spectrum of the problem (1.8)–(1.10) equals

 σ=∞⋃k=1Υϵk ,  Υϵk={Λϵk(η):η∈[0,2π)}, (1.1)

where the sets are subintervals of the positive real axis, or bands of the spectrum. (For the use of this so called Bloch spectrum in other problems, see for example [FG], or [2].) In general, those bands may overlap making connected, but in Theorem 3.1 we obtain asymptotic estimates for the lower and upper endpoints of : we show that for all and and for some constants . In view of (1.1) this implies the existence of a spectral gap between and for small and such that . However, since the estimates depend also on , we can only open a gap for finitely many , though the number of gaps tends to infinity as .

The asymptotic position (as ) of the gaps is determined more accurately in Theorems 3.5 and 3.6: those main results state that

 Υϵk=(Λ0k+Akϵ+O(ϵ3/2),Λ0k+Bkϵ+O(ϵ3/2))

where the numbers depend linearly on the three dimensional capacity of the set . This result also ensures that in case the bands do not degenerate into single points, which means that the spectrum of the original problem indeed has a genuine band-gap structure. Facts concerning the numbers , are discussed after Theorem 3.6.

As for the structure of this paper, we recall in Section 1.2 the exact formulation of the linear water-wave problem, its variational formulation as well as the parameter dependent problem arising from the Gelfand transform, and the limit problem. Section 2 contains the formal asymptotic analysis which relates the spectral properties of the original problem with the limit problem and which is rigorously justified in Secion 3. The main results, Theorems 3.1, 3.5 and 3.6 as well as Corollary 3.2 are also given in Section 3. The proofs are based on the max-min principle and construction of suitable test functions adjusted to the geometric characteristics of the domains under study.

Acknowledgement. The authors want to thank Prof. Sergey A. Nazarov for many discussions on the topic of this work.

### 1.2 Formulation of the problem, operator theoretic tools

Let us proceed with the exact formulation of the problem. We consider an infinite periodic channel (see (1.7)), consisting of water containers connected by small apertures of diameter . The coordinates of the points in the channel are denoted by , and stands for the projection of to the plane . We choose the coordinate system in such a way that the axis of the channel is in -direction and the free surface is in the plane .

###### Definition 1.1.

We describe the geometric assumptions on the periodicity cell in detail, as well as some related technical tools including the cut-off funcions. Let us denote by a domain with a Lipschitz boundary and compact closure such that its intersections with - and -planes are simply connected planar domains with positive area and contain the points and with , respectively; these points are fixed throughout the paper. Then the periodicity cell and its translates are defined by setting (see Figure 1.1)

 ϖ={x∈ϖ∙:x3<0,x1∈(0,1)},ϖj={x:(x1−j,x2,x3)∈ϖ}, j∈Z. (1.2)

Furthermore, we assume that the set is a bounded planar domain containing the origin and that the boundary is at least -smooth. We assume that is so small that the set is contained in and . We define the apertures between the container walls as the sets

 θϵj={x=(j,x′):ϵ−1(x′−(P2,P3))∈θ}, j∈Z. (1.3)

It is plain that for for all , by the choice of . We shall need at several places a cut-off function

 χθ∈C∞0(R3), (1.4)

which is equal to one in a neighbourhood of the set and vanishes outside another compact neighbourhood of . More precisely, we require that

 (supp(χθ)+(0,P2,P3))∩{x1=0}⊂∂ϖ, (supp(χθ)+(0,P2,P3)) ∩{x1>0}⊂ϖ (1.5)

(this is possible by the specifications made on ) and vanishes, if or . We also assume that , when . Furthermore, denoting , it follows from the above specifications that , if ; in particular vanishes on the free water surface . Finally, we shall need the scaled cut-off functions

 Xϵj=χθ(ϵ−1(x−Pj)). (1.6)

It is plain that also vanishes on for and that for , .

###### Definition 1.2.

The periodic water channel is defined by

 Πϵ=⋃j∈Z(ϖj∪θϵj), (1.7)

and it will be the main object of our investigation. The free surface of the channel is denoted by , and the wall and bottom part of the boundary is . The boundary of the isolated container , the periodicity cell, consists of the free surface and the wall and bottom with two apertures and .

###### Remark 1.3.

We shall use the following general notation. Given a domain , the symbol stands for the natural scalar product in , and , , for the standard Sobolev space of order on . The norm of a function belonging to a Banach function space is denoted by . For and , (respectively, ) stands for the Euclidean ball (resp. ball surface) with centre and radius . By (respectively, , , etc.) we mean positive constans (resp. constants depending on a parameter ) which do not depend on functions or variables appearing in the inequalities, but which may still vary from place to place. The gradient and Laplace operators and act in variable , unless otherwise indicated.

In the framework of the linear water-wave theory we consider the spectral Steklov problem in the channel ,

 −Δuϵ(x)=0 for all x∈Πϵ, (1.8) ∂nuϵ(x)=0 for a.e. x∈Σϵ, (1.9) ∂zuϵ(x)=λϵuϵ(x) for a.e. x∈Γϵ. (1.10)

Here is the velocity potential, is a spectral parameter related to the frequency of harmonic oscillations and the acceleration of gravity . By the geometric assumptions made above, the outward normal derivative is defined almost everywhere on . It coincides with on the free surface .

The rest of this section is devoted to presenting the operator theoretic tools which will be needed later to prove our results: Gelfand transform, variational formulation of the boundary value problems, and max-min-formulas for eigenvalues. The spectral problem (1.8)–(1.10) can be transformed into a family of spectral problems in the periodicity cell using the Gelfand transform. We briefly recall its definition:

 v(y,z)↦ V(y,z,η)=1√2π∑j∈Zexp(−iη(z+j))v(y,z+j), (1.11)

where on the left while and on the right. As is well known, the Gelfand transform establishes an isometric isomorphism between the Lebesgue spaces,

 L2(Πϵ)≃L2(0,2π;L2(ϖ)),

where is the Lebesgue space of functions with values in the Banach space endowed with the norm

 ∥V;L2(0,2π;B)∥=(∫2π0∥V(η);B∥2dη)1/2.

The Gelfand transform is also an isomorphism from the Sobolev space onto for . The space consists of Sobolev functions which satisfy the quasi-periodicity conditions

 u(0,x′)=e−iηu(1,x′), (0,x′)∈θϵ0, (1.12) ∂x1u(0,x′)=e−iη∂x1u(1,x′), (0,x′)∈θϵ0, (1.13)

whereas is the Sobolev space with the condition (1.12) only.

Applying the Gelfand transform to the differential equation (1.8) and to the boundary conditions (1.9)–(1.10), we obtain a family of model problems in the periodicity cell parametrized by the dual variable ,

 −ΔUϵ(x;η)=0, x∈ϖ, (1.14) ∂nUϵ(x;η)=0, x∈σϵ, (1.15) ∂zUϵ(x;η)=Λϵ(η)Uϵ(x;η), x∈γ, (1.16) Uϵ(0,x′;η)=e−iηUϵ(1,x′;η), x∈θϵ0, (1.17) ∂x1Uϵ(0,x′;η)=e−iη∂x1Uϵ(1,x′;η), x∈θϵ0. (1.18)

Here, is a new notation for the spectral parameter . More details on the use of the Gelfand-transform can be found e.g. in [19], Section 2.

The apertures disappear at so in that case the also quasi-periodicity conditions cease to exist. Hence, we can consider the problem (1.14)–(1.18) as a singular perturbation of the limit spectral problem

 −ΔU0(x)=0, x∈ϖ, (1.19) ∂nU0(x)=0, x∈σ, (1.20) ∂zU0(x)=Λ0U0(x), x∈γ (1.21)

with as a spectral parameter.

Our approach to the spectral properties of model and limit problems is similar to [20], Sections 1.2, 1.3.. We first write the variational form of the problem (1.14)–(1.18) for the unknown function as

 (∇Uϵ,∇V)ϖ=Λϵ(Uϵ,V)γ, V∈H1ϵ,η(ϖ), (1.22)

and the corresponding variational formulation of the limit problem for reads as

 (∇U,∇V)ϖ=Λ(U,V)γ, V∈H1(ϖ). (1.23)

We denote by the space endowed with the new scalar product

 (u,v)ϵ=(∇u,∇v)ϖ+(u,v)γ, (1.24)

and define a self-adjoint, positive and compact operator using

 (Bϵ(η)u,v)ϵ=(u,v)γ. (1.25)

The problem (1.22) is then equivalent to the standard spectral problem

 Bϵ(η)u=Mϵu (1.26)

with another spectral parameter

 Mϵ=(1+Λϵ)−1. (1.27)

Clearly, the spectrum of consist of 0 and a decreasing sequence of eigenvalues, which moreover can be calculated from the usual min-max formula

 Mϵk(η)=minEkmaxv∈Ek(Bϵ(η)v,v)ϵ(v,v)ϵ, (1.28)

where the minimum is taken over all subspaces of co-dimension . Using (1.24) and (1.25), we can write a max-min formula for the eigenvalues of the problem (1.22):

 Λϵk(η)=1Mϵk(η)−1=maxEkminv∈Ek(∇v,∇v)ϖ+(v,v)γ(v,v)γ−1 (1.29) = maxEkminv∈Ek∥∇v;L2(ϖ)∥2∥v;L2(γ)∥2

On the other hand, the connection (1.27) and the properties of the sequence mean that the eigenvalues (1.29) form an unbounded sequence

 0≤Λϵ1(η)≤Λϵ2(η)≤…≤Λϵk(η)≤…→+∞. (1.30)

The eigenfunctions can be assumed to form an orthonormal basis in the space . The functions are continuous and -periodic (see for example [5], Ch. 9). Hence the sets

 Υϵk={Λϵk(η):η∈[0,2π)} (1.31)

are closed connected segments, which may degenerate into single points; their relation to the original problem was already mentioned in (1.1).

The spectral concepts of the limit problem (1.19)–(1.21) can be treated in the same way as in (1.24)–(1.30). Since the quasi-periodicity conditions vanish for , the space is replaced by ; the norm induced by (1.24) is now equivalent to the original Sobolev norm of . We denote by the operator defined as in (1.24)–(1.25). The limit problem has an eigenvalue sequence like (1.30), however, neither the eigenvalues nor the operator depend on (cf. [19], Section 3). The first eigenvalue equals , and the first eigenfunction is the constant function. Analogously to (1.29) we can write

 Λ0k=maxFkminv∈Fk∥∇v;L2(ϖ)∥2∥v;L2(γ)∥2, (1.32)

where again is running over all subspaces of codimension . We denote by

 (U0k)∞k=1 (1.33)

an -orthonormal sequence of eigenfunctions corresponding to the eigenvalues (1.32).

###### Lemma 1.4.

For all there exists a constant such that

 |U0k(x)|≤Ck , |∇U0k(x)|≤Ck (1.34)

for all , (and hence for all , ).

###### Proof.

Let for example (the other case is treated similarly), and define the domains with boundary such that and still so small that

 G2∩{x1=0}⊂∂ϖ  and  ¯¯¯¯¯¯G2∩{x1>0}⊂ϖ. (1.35)

As a consequence, these domains are smooth enough so that we can use the local elliptic estimates [1], Theorem 15.2, to the solutions of the equation (1.19): this yields for every , a constant such that

 ∥U0k;Hl+1(Gn∩ϖ)∥≤Cl,k(∥U0k;Hl−1(Gn+1∩ϖ)∥+∥U0k;L2(Gn+1∩ϖ)∥)

for . Applying this first with and we get a bound for and then, with and , for . The standard embeddings and imply the result.

## 2 The formal asymptotic procedure

### 2.1 The case of a simple eigenvalue

To describe the asymptotic behaviour (as ) of the eigenvalues of the problem (1.14)-(1.18) we consider first the case is a simple eigenvalue of the problem (1.19)-(1.21) for some fixed . Let us make the following ansatz:

 Λϵk(η)=Λ0k+ϵΛ′k(η)+˜Λϵk(η), (2.1)

where is a correction term and a small remainder to be evaluated and estimated. In this section we derive the expression (2.13) for , cf. also (2.18) and (2.19), and the remainder will be treated in Section 3.2

The corresponding asymptotic ansatz for the eigenfunction reads as follows:

 Uϵk(x;η) = U0k(x) + χ0(x)wk0(ϵ−1(x−P0))+χ1(x)wk1(ϵ−1(x−P1)) + ϵU′k(x;η)+˜Uϵk(x;η),

where is as in (1.33). The functions and are of boundary layer type, and is given above (1.6).

The boundary layers depend on the “fast” variables (“stretched” coordinates)

 ξj=(ξj1,ξj2,ξj3)=ϵ−1(x−Pj),j=0,1.

They are needed to compensate the fact that the leading term in the expansion (2.1) does not satisfy the quasi-periodicity conditions (1.17)–(1.18). By Lemma 1.4 and the mean value theorem, the eigenfunction has the representation

 U0k(x)=U0k(Pj)+O(ϵ),x∈θϵj

near the points . We look for and as the solutions of the problems

 Δξ0wk0(ξ0)=0, ξ01>0, ∂ξ01wk0(ξ0)=0, ξ0∈{0}×(R2∖¯¯¯θ), wk0(ξ0)=ak0, ξ0∈{0}×θ,

and

 Δξ1wk1(ξ1)=0, ξ11<0, ∂ξ11wk1(ξ1)=0, ξ1∈{0}×(R2∖¯¯¯θ), wk1(ξ1)=ak1, ξ1∈{0}×θ

in the half spaces and , respectively; the meaning of the numbers will be explained below. Both of the functions can be extended to even harmonic functions in the exterior of the set :

 Δξjwkj(ξj)=0, ξj∈R3∖({0}×¯¯¯θ), (2.3) wkj(ξj)=akj, ξj∈∂({0}×¯¯¯θ).

Furthermore, the problem (2.3) admits a solution (see [21])

 wkj(ξj)=akjcap3θ|ξj|+˜wkj(ξj), (2.4) ˜wkj(ξj)=O(|ξj|−2) , ∇ξj˜wkj(ξj)=O(|ξj|−3), (2.5)

where is the 3-dimensional capacity of the set and (2.5) concerns large -behaviour. Moreover, the solution has a finite Dirichlet integral:

 (2.6)

for some constant .

We aim to choose the coefficients such that satisfies the quasi-periodicity conditions (1.17)–(1.18) . Clearly, for each

 Uϵk(P0;η) = e−iηUϵk(P1;η), ∂x1Uϵk(P0;η) = e−iη∂x1Uϵk(P1;η),

which together with the asymptotic expansion (2.1) yield the relations

 U0k(P0)+ak0=e−iη(U0k(P1)+ak1) and ak0=−e−iηak1

for the coefficients. Hence,

 ak1=−eiηak0,ak0=12(e−iηU0k(P1)−U0k(P0)). (2.7)

Now we can write a model problem for the main asymptotic correction term :

 −ΔU′k(x;η)=ΔWk(x) x∈ϖ, (2.8) (∂z−Λ0k)U′k(x;η)=Λ′k(η)U0k(x), x∈γ, (2.9) ∂nU′k(x;η)=0, x∈σ, (2.10)

where we denote

 Wk(x)=(1∑j=0χj(x)akjcap3(θ)|x−Pj|),x∈ϖ,. (2.11)

In addition to , the problem (2.8)–(2.10) will also determine the number in a unique way for every and . This will follow by requiring the solvability condition to hold in the Fredholm alternative, see Lemma 2.1 and its proof, below. Indeed, using the Green formula and the normalization in (1.33) we write ( is the surface measure):

 Λ′k(η)=Λ′k(η)∥U0k;L2(γ)∥2= (2.12) = ∫γ(∂zU′k(x;η)−Λ0kU′k(x;η))¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)ds(x)= = ∫∂ϖ(¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)∂nU′k(x;η)−U′k(x;η)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯∂nU0k(x))ds(x)= = ∫ϖ¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)ΔU′k(x;η)=−∫ϖ¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)ΔWk(x)dx.

Taking into account that the last integral converges absolutely and using the Green formula again yield

 Λ′k(η) = limr→01∑j=0∫Sr(Pj)∩ϖ¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(x)∂n(−akjcap3(θ)|x−Pj|)ds(x) = −2πcap3(θ)(ak0¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(P0)+ak1¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯U0k(P1));

see (2.11) and Remark 1.3 for notation. According to (2.7) we finally obtain

 Λ′k(η)=πcap3(θ)|U0k(P0)−e−iηU0k(P1)|2. (2.13)
###### Lemma 2.1.

Choosing as in (2.13), the problem (2.8)–(2.10) has a solution .

###### Proof.

The variational formulation of the problem (2.8)–(2.10) reads as

 (∇U′k,∇V)ϖ−Λ0k(U′k,V)γ=(∇Wk,∇V)ϖ−Λ′k(U0k,V)γ. (2.14)

We remark that the function is harmonic in , and since equals constant one in a neighbourhood of , the function vanishes there, hence, and are smooth as well as uniformly bounded everywhere in . Moreover, .

Using the definition of the operator (cf. (1.24), (1.25) and the remarks above (1.32)) we can rewrite (2.14) as follows:

 (U′k,V)0−(Λ0k+1)(BU′k,V)0=(Wk,V)0−Λ′k(BU0k,V)0−(BWk,V)0 (2.15)

which means that must be a solution of the equation

 (B−M0k)U′k=−M0k(Wk−BWk−Λ′kBU0k). (2.16)

Notice that is the solution of the homogeneous problem (2.16), so, by the Fredholm alternative, (2.16) is solvable, if and only if the right hand side of it is orthogonal to the function . This condition is satisfied by choosing as above, since

 (Wk−BWk−Λ′kBU0k,U0k)0=(∇Wk,∇U0k)ϖ−Λ′k∥U0k;L2(γ)∥2=0,

by (2.12) and . This last identity follows from the first Green formula, because the normal derivative of vanishes on due to the properties of the function , see below (1.5).

### 2.2 The case of a multiple eigenvalue

In this section we complete the asymptotic analysis by studying the behaviour of eigenvalues in the case some has multiplicity greater than one: we have

 Λ0k−1<Λ0k=…=Λ0k+m−1<Λ0k+m.

The ansatz (2.1) is used again. Furthermore, as in (1.33) we denote by an orthonormal system of eigenfunctions associated with the eigenvalue . Any eigenfunction corresponding to can be presented as a linear combination

 U0(x)=m−1∑j=0αjU0k+j(x).

Analogously to (2.1) we introduce the asymptotic ansatz

 Uϵ(x;η) = U0(x) + χ0(x)wk0(ϵ−1(x−P0))+χ1(x)wk1(ϵ−1(x−P1)) + ϵU′(x;η)+˜Uϵ(x;η).

Using the same argumentation as in the previous section we construct the boundary layers , , which satisfy the conditions

 wkj(ξj)=akjcap3θ|ξj|+O(|ξj|−2);

here the coefficients come from the equations (2.7), where is replaced by . The main asymptotic term is also treated in the same way as in Section 2.1. To use the Fredholm alternative for finding , , we write

 Λ′k+j(η)αj=Λ′k+j(η)(U0,U0k+j)γ,

and making use of the Green formula as above we get

 Λ′k+j(η)αj=m−1∑l=0βljαj,

where

 βlj=πcap3(θ)(U0k+l(P0)−e−iηU0k+l(P1))¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(U0k+j(P0)−e−iηU0k+j(P1)).

Hence, is an eigenvalue of the matrix . This matrix has rank one, because it can be represented in the form , where is a vector with components , . This means that

 Λ′k(η) = πcap3(θ)m−1∑l=0|U0k+l(P0)−e−iηU0k+l(P1)|2, (2.18) Λ′k+j(η) = 0,1≤j≤m−1. (2.19)

## 3 Existence and position of spectral gaps

### 3.1 Existence of gaps

The first estimate on the eigenvalues of the problem (1.22) can now be stated as follows.

###### Theorem 3.1.

For any there are numbers and such that for every and any dual variable , the eigenvalues of the problem (1.22) and the eigenvalues of the limit problem (1.23) are related as follows: