Open Access Research

The existence of eigenvalue problems for the waveguide theory

A Maher1* and EM Karachevskii2

Author affiliations

1 Department of Mathematics, Faculty of Science, Assiut University, Assiut, 71516, Egypt

2 Department of Applied Mathematics, Kazan State University, 18 Kremlyovskaya st., Kazan, 420008, Russia

For all author emails, please log on.

Citation and License

Boundary Value Problems 2012, 2012:133  doi:10.1186/1687-2770-2012-133


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/133


Received:5 April 2012
Accepted:1 November 2012
Published:13 November 2012

© 2012 Maher and Karachevskii; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, the existence of the eigenvalue problem for the waveguide theory is investigated. We used the Fourier transformation method for the solution of this problem. Also, we applied this problem to a dielectric waveguide. In this study, four theorems and two lemmas are obtained.

MSC: 35A22, 35P10.

Keywords:
partial differential equations; eigenvalue problems; Fourier transformation method

1 Basic preliminaries

A dielectric waveguide is a composite of its own index of refraction for each layer. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1">View MathML</a> is a layer, where the index of refraction is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M2">View MathML</a> and μ is a spectral parameter, then the waveguide process can be written in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M3">View MathML</a>

(1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M4">View MathML</a>

In order to obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M6">View MathML</a>, the process in all the waveguide for the common boundary of domains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M8">View MathML</a> is evaluated. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M6">View MathML</a> must be joined in the way that the obtained known functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M11">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M13">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M14">View MathML</a> will be the generalized solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M15">View MathML</a>

(2)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M16">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M18">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M14">View MathML</a>. If the boundary <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M20">View MathML</a> is sufficiently smooth, the condition of this junction may be put down in a natural form. Indeed, the contraction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M20">View MathML</a> is noninfinitely smooth in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M8">View MathML</a>, the functions which deteriorate their smoothness where the conditions themselves could be impossible to write. That is how the solution of this problem was progressing.

If the boundaries of domains are bad and there are several of them, it is not clear what the condition of the junction looks like. In this situation (connection), we need another approach to the solution of the set problem.

Since results of the junction must preserve the property of solution (being a generalized solution), we propose a new circuit system to solve the set problem. In general case, it is not solved.

The existence of eigenvalue is proved in [1] for the special case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M26">View MathML</a> - the circle. For more details, see [2-5] and [6].

Consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M27">View MathML</a>

(3)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M16">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M29">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M30">View MathML</a>

(4)

It is obvious that if we prove the existence of the eigenvalue (3), we obtain the following solution of the problem (1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M31">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M33">View MathML</a>, where they are found automatically joined by a required form.

2 Formulation of the problem

We consider the eigenvalue problem (3) where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M36">View MathML</a> are mutually exclusive (disjoint) measurable sets with a positive measure. If we introduce a new spectral parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M37">View MathML</a>, then the problem (1) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M38">View MathML</a>

(5)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M39">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M12">View MathML</a>.

The problem (5) is self-adjoint. This can be easily seen if we use the Fourier transformation. However, it does not influence the eigenvalue existence. Some examples of the problem (5) are known (with concrete <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M2">View MathML</a>, N and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M1">View MathML</a>) both with and without eigenvalues.

To use the Fourier transformation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M43">View MathML</a> of the distribution (generalized) function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M44">View MathML</a> of slow growth, we must be aware of the following well-known Parseval equality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M45">View MathML</a>

and Plancherel’s theorem: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M46">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M47">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M48">View MathML</a>

for all u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M47">View MathML</a>.

From now on, if it is not specifically indicated, the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M50">View MathML</a> is the norm in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M51">View MathML</a>.

3 The existence of negative eigenvalues for the general case

Let us consider the problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M52">View MathML</a>

(6)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M53">View MathML</a> is a measurable function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M54">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M56">View MathML</a> almost everywhere in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M39">View MathML</a> outside Ω, Ω is measurable and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M58">View MathML</a> is a linear pseudo-differential operator with constant coefficients. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M59">View MathML</a> argument quasi-polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M60">View MathML</a>, not depending on x and satisfying the following conditions for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M60">View MathML</a>:

(7)

(8)

(9)

We suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M65">View MathML</a>

(10)

for each sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M66">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M67">View MathML</a>.

Theorem 1The problem (6) has at least one negative eigenvalue if Ω is bounded.

It is necessary to introduce several lemmas before proving this theorem.

In each case, we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M68">View MathML</a>. By virtue of (8), there is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M69">View MathML</a> of the Fourier transformation which coincides with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M70">View MathML</a>. Considering (7), the real and even function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M71">View MathML</a> could be obtained.

Lemma 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M68">View MathML</a>. The problem (6) has a nonzero solution if and only if the nonzero solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M73">View MathML</a>has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M74">View MathML</a>

(11)

Proof Applying the Fourier transformation for (6) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M75">View MathML</a>

Hence, in particular, the integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M76">View MathML</a>

converges absolutely. From now on, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M77">View MathML</a>. It follows from latter relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M78">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M79">View MathML</a> means that the Fourier transformation has been determined under t. Hence, by virtue of Parseval’s equality, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M80">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M81">View MathML</a> outside Ω, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M44">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55">View MathML</a> is the solution of the problem (11). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M84">View MathML</a> where in Ω we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M85">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M86">View MathML</a>, by virtue of the latter equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M87">View MathML</a>. The necessity is proved.

Let us prove the sufficiency. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M73">View MathML</a> be the nonzero solution of the problem (11). Consider the new problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M89">View MathML</a>

(12)

in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M90">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M92">View MathML</a> outside Ω. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M93">View MathML</a>, applying the Fourier transformation for (12), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M94">View MathML</a>

From Parseval’s equality, the solution of the problem (12) exists and it is unique. In particular, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M96">View MathML</a>

Considering this inequality and (12), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M97">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M44">View MathML</a> is the solution of the problem (6). Thus, the lemma is proved. □

In the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M68">View MathML</a>, we consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100">View MathML</a> as an integral operator, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M101">View MathML</a>

We remember that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100">View MathML</a> is defined only when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M68">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M104">View MathML</a>, thus the Fourier transformation for the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M105">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M106">View MathML</a> coincides. That is why <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M107">View MathML</a>. If Ω is bounded, then the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M108">View MathML</a> of the integrated operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M110">View MathML</a>. It follows that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100">View MathML</a> is completely continuous. Its self-adjointness and positiveness are obvious. This enables us to write down the eigenvalues of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M100">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M113">View MathML</a>

(13)

It is well known that (see [7])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M114">View MathML</a>

(14)

where Sup is determined for all the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M115">View MathML</a>, for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M116">View MathML</a>.

From the known results for self-adjoint and quite continuous operators (see [7]), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M117">View MathML</a> continuously depends on μ, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M118">View MathML</a>

(15)

Lemma 2Let Ω be bounded when<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M91">View MathML</a>. Then

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M120">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M121">View MathML</a>,

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M122">View MathML</a>at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M67">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M125">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M126">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M127">View MathML</a>

Hence, the first statement follows from (9).

Let us prove the second statement. By virtue of (13), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M39">View MathML</a> outside Ω and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M129">View MathML</a>

which is applied to the last integral in Parseval’s inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M130">View MathML</a>

The following equations are correct:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M131">View MathML</a>

In a similar way, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M132">View MathML</a>

Thus, we have proved the following:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M133">View MathML</a>

The following estimate is obvious:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M134">View MathML</a>

(16)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M135">View MathML</a>

(17)

δ will be chosen in a way such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M136">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M139">View MathML</a>. Since Ω is bounded, we may always obtain the latter.

Considering (16) and (17), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M140">View MathML</a>

Hence, by virtue of (10), the lemma is proved. □

Proof of Theorem 1 At the first stage, we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M141">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55">View MathML</a>. By virtue of Lemmas 1 and 2, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M143">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M144">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M145">View MathML</a> is the eigenfunction corresponding to the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M146">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M147">View MathML</a>

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M148">View MathML</a>, we have the nonzero solution of the equation (11). It follows from Lemma 1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M149">View MathML</a> is the eigenvalue of the problem (6).

For the general case, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M150">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M152">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M153">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M55">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M155">View MathML</a>. The nonzero solutions of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M156">View MathML</a>

(18)

are chosen in such a way that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M157">View MathML</a>.

The integral operators defined by the right-hand sides of (11) and (18) are defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M159">View MathML</a> respectively. Since Ω is bounded, then both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M160">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M161">View MathML</a> uniformly converge by norm to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M163">View MathML</a> respectively. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M164">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M165">View MathML</a>

(19)

Considering the choice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M166">View MathML</a> and the property <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M167">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M168">View MathML</a>, we can easily prove the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M169">View MathML</a>. Noting that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M171">View MathML</a> for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M172">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M173">View MathML</a> is completely continuous. In this case, as we know, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M166">View MathML</a> contains the subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M173">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M177">View MathML</a> which converges by norm where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M178">View MathML</a>.

From (18) and (19) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M179">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M180">View MathML</a> by norm where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M181">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M182">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M183">View MathML</a> by norm and satisfies the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M184">View MathML</a>, i.e., when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M185">View MathML</a>, the equation (11) has a nonzero solution. Hence, the theorem is proved. □

4 Application to the problem of a dielectric waveguide

In the case of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M186">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M187">View MathML</a>

the condition (7) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M188">View MathML</a>

(20)

It is clear that in the case of n arbitrary, these requirements are not satisfied. However, it takes place in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M189">View MathML</a> important for the application. It can easily be proved when we use the spherical coordinates. Moreover, for the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M189">View MathML</a>, (9) also takes place. Let us make sure that (10) is satisfied when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M191">View MathML</a>.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M192">View MathML</a>

Consider the spherical coordinates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M193">View MathML</a>

(21)

The left-hand side of (20) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M194">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M195">View MathML</a>

It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M196">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M197">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M198">View MathML</a>

We can see that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M199">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M200">View MathML</a>

Taking into account that (10) is satisfied and denoting index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M201">View MathML</a> in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M2">View MathML</a> is the minimum, the problem (5) can be rewritten in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M203">View MathML</a>

(22)

when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M204">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M205">View MathML</a>; i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M39">View MathML</a>, outside <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M207">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M208">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M91">View MathML</a>.

The theorem may be applied to the problem (21). As a consequence of this theorem, we get the following:

Theorem 2If Ω is bounded, the problem (3) has an eigenvalueμfor which<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M210">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M211">View MathML</a> be the index at which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M2">View MathML</a> is maximum. Then the problem (3) may take the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M213">View MathML</a>

(23)

when

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M214">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M215">View MathML</a>

Now, we formulate the following theorem.

Theorem 3The problem (3) does not have an eigenvalueμfor which.

Proof Multiplying the equality (22) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M44">View MathML</a> and integrating it in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M86">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M219">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M221">View MathML</a>, then by virtue of the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M87">View MathML</a>, the latter is not impossible. □

By virtue of Theorems 2 and 3, we have

Theorem 4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M223">View MathML</a>be bounded. Then the problem (3) has an eigenvalueμwhich satisfies the condition.

Remark If the condition that the bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/133/mathml/M223">View MathML</a> is not valid, then the problem may not have eigenvalues.

5 Conclusions

This paper deals with the existence of eigenvalue problems for the waveguide theory. These problems are very important in the study of the mathematical analysis and mathematical physics. In this paper, we introduced four theorems and two lemmas.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The idea of this paper was introduced by the first author. The second author shared the first author in calculations.

Acknowledgements

We wish to thank the referees for their valuable comments which improved the original manuscript.

References

  1. Karchevskii, EM: The research (investigation) of the numerical method of solving the spectral problem for the theory of dielectric waveguides. Izv. Vysš. Učebn. Zaved., Mat.. 1, 10–17 (1999)

  2. Dautov, RZ, Karchevskii, EM: Existence and properties of solutions to the spectral problem of the dielectric waveguide theory. Comput. Math. Math. Phys.. 40, 1200–1213 (2000)

  3. Karchevskii, EM: The fundamental wave problem for cylindrical dielectric waveguides. Differ. Equ.. 36, 998–999 (2000). Publisher Full Text OpenURL

  4. Karchevskii, EM, Solov’ev, SI: Investigation of a spectral problem for Helmholtz operator on the plane. Differ. Equ.. 36, 563–565 (2000)

  5. Pinasco, JP: Asymptotic of eigenvalues and lattice points. Acta Math. Sin. Engl. Ser.. 22(6), 1645–1650 (2000)

  6. Snyder, A, Love, D: Optical Waveguide Theory, Chapman & Hall, New York (1987)

  7. Riss, F, Sekefalvi-Nad, B: Lectures on Functional Analysis, Mir, Moscow (1979)