2 INTRODUCTION

in the boundary condition on a small part of the boundary on the eigenvalues. For

the second problem, we provide a complete asymptotic expansion of the eigenvalue

perturbation with respect to the size of the inclusion for a domain containing a

small inclusion. Our exposition is accompanied by many new applications of our

asymptotic theory, especially to imaging and optimal design. It is worth emphasiz-

ing that the asymptotic results derived in this book have numerous other important

applications in practice.

To be more precise, let Ω be a bounded domain in Rd, d ≥ 2, with a connected

Lipschitz boundary ∂Ω. Let ν denote the unit outward normal to ∂Ω. Suppose that

Ω contains a small inclusion D, of the form D = z + B, where B is a bounded Lip-

schitz domain in Rd containing the origin. We also assume that the “background”

is homogeneous with conductivity 1. Even though we will deal with other cases as

well, for the moment let us assume that the inclusion is grounded, meaning that

zero Dirichlet conditions are imposed on ∂D. Then the eigenvalue problem for the

domain with the inclusion is given by

(0.1)

⎧

⎪

⎪

⎨

⎪

⎪

⎩

∆u + µ u = 0 in Ω ,

∂u

∂ν

= 0 on ∂Ω,

u = 0 on ∂D,

where Ω := Ω \ D.

Let 0 = µ1 µ2 ≤ . . . be the eigenvalues of −∆ in Ω with Neumann conditions,

namely, those of the problem

(0.2)

⎧

⎨

⎩

∆u + µu = 0 in Ω,

∂u

∂ν

= 0 on ∂Ω.

The eigenvalues are arranged in an increasing sequence and counted according to

multiplicity. Fix j and suppose that the eigenvalue µj is simple. Note that this

assumption is not essential in what follows and, moreover, it is proved in [3, 4, 243]

that the eigenvalues are generically simple. By generic, we mean the existence of

arbitrary small deformations of ∂Ω such that in the deformed domain the eigenvalue

is simple. Throughout this book, the assumption of the simplicity is made for ease

of exposition. Then there exists a simple eigenvalue µj near µj associated to the

normalized eigenfunction uj ; that is, uj satisfies (0.1).

One of our goals in this book is to find complete asymptotic expansions for the

eigenvalues µj as tends to 0. In other words, we seek to find a series expansion of

the form

(0.3) µj = A0 + A1

n

+ A2

n+1

+ . . . .

Existence of such a series is a part of what we are going to prove.

The book consists of three parts. The first part is devoted to the theory devel-

oped by Gohberg and Sigal. In the second part, we provide rigorous derivations of

complete asymptotic expansions of eigenvalue perturbations such as (0.3). A key

feature of our work is the approach we develop: a general and unified boundary

integral approach with rigorous justification based on the Gohberg-Sigal theory ex-

plained in Part 1. By using layer potential techniques, we show that the square

roots of the eigenvalues are exactly the real characteristic values of meromorphic