Introduction

The aim of this book is to give a self-contained presentation of an asymptotic

theory for eigenvalue problems using layer potential techniques. This theory and

its application in the field of inverse problems have been developed over the last

number of years by the authors.

Since the early part of the twentieth century, the use of integral equations has

developed into a range of tools for the study of partial differential equations. This

includes the use of single- and double-layer potentials to treat classical boundary

value problems.

Our main objective in this book is to show how powerful the layer potential

techniques are for solving not only boundary value problems but also eigenvalue

problems if they are combined with the elegant theory of Gohberg and Sigal on

meromorphic operator-valued functions.

There are two prominent eigenvalue perturbation problems: one under variation

of domains or boundary conditions and the other due to the presence of small-

volume inclusions.

There have been several interesting works on the problem of eigenvalue changes

under variation of domains since the seminal formula of Hadamard [119]: the works

by Garabedian and Schiffer [113], Kato [149], Fujiwara and Ozawa [102], Sanchez

Hubert and Sanchez Palencia [221], Ward and Keller [248], Gadyl’shin and Il’in

[112], Gadyl’shin [110, 111], Daners [80], McGillivray [181], Noll [192], Planida

[212], Bruno and Reitich [61], Burenkov and Lamberti [62], and Kozlov [153].

For the second problem, Rauch and Taylor [216] have shown that the spec-

trum of a bounded domain does not change after imposing Dirichlet conditions on

compact subsets of capacity zero. Subsequently, many people have studied the as-

ymptotic expansions of the eigenvalues for the case of small holes with a Dirichlet or

a Neumann boundary condition. In particular, Ozawa provided in a series of papers

[202]–[207] leading-order terms (A0, A1, A2 in (0.3)) in eigenvalue expansions; see

also [248] and [177]. Besson [54] has proved the existence of a complete expansion

(0.3) of the eigenvalue perturbation in the two-dimensional case. Courtois [76] has

established a perturbation theory for the Dirichlet spectrum in a compactly per-

turbed domain in terms of the capacity of the compact perturbation. We also refer

to the book by Maz’ya, Nazarov, and Plamenevskii [178], where the method of

matched expansions [133, 134] has been used to construct asymptotic representa-

tions of eigenvalues of problems of conduction and elasticity theory for bodies with

small holes.

In this book, we shall consider both the first and the second eigenvalue prob-

lems. For the first problem, we consider an inclusion inside a bounded domain and

derive high-order asymptotic expansions of the perturbations of the eigenvalues that

are due to shape variations of the inclusion. We also study the effect of a change

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http://dx.doi.org/10.1090/surv/153/01