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 u j that is, u j satisfies (0.1). One of our goals in this book is to find complete asymptotic expansions for the eigenvalues µ j as tends to 0. In other words, we seek to find a series expansion of the form (0.3) µ j = 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

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