INTRODUCTION 3 operator-valued functions that are of Fredholm type with index 0. We then pro- ceed from the generalized argument principle to construct their complete asymptotic expressions with respect to the perturbations. Our main idea is to reduce the eigen- value problem to the study of characteristic values of systems of certain integral operators. A similar approach is extended in Part 3 to investigate the band gap structure of the frequency spectrum for waves in a high contrast, two-component periodic medium. This provides a new tool for investigating photonic and phononic crystals and solving diﬃcult mathematical problems arising in these fields. Pho- tonic and phononic crystals have attracted enormous interest in the last decade because of their unique optical and acoustic properties. Such structures have been found to exhibit interesting spectral properties with respect to classical wave prop- agation, including the appearance of band gaps. An important example of these crystals consists of a background medium which is perforated by a periodic array of arbitrary-shaped holes with different material parameters. As we said, the method to derive the asymptotic expansion of the eigenvalues of problem (0.1) can be applied to other types of eigenvalue perturbation problems. As a first example, instead of being grounded, suppose that the inclusion may have a different conductivity, say 0 k = 1 +∞. Then the eigenvalue problem to be considered is (0.4) ⎧ ⎨∇ ⎩ · (1 + (k − 1)χ(D))∇u + µ u = 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where χ(D) denotes the indicator function of D which is of the form D = z + B, as before. Another example is the derivation of high-order terms in the asymptotic expansions of the eigenvalue perturbations resulting from small perturbations of the shape of the conductivity inclusion D. A third example is concerned with the effect of internal corrosion on eigenvalues. Suppose that ∂D contains a corroded part I of small Hausdorff measure |I| = and let a positive constant γ denote the surface impedance (the corrosion coeﬃcient) of I. The eigenvalue problem in the presence of corrosion consists of finding µ 0 such that there exists a nontrivial solution u to ⎧ ⎪∆u ⎪ ⎪ ⎪ + µ u = 0 in Ω \ D, − ∂u ∂ν + γχ(I)u = 0 on ∂D, u = 0 on ∂Ω, where χ(I) denotes the characteristic function on I. We will also derive an asymptotic expansion of the eigenvalues for the elasticity equations with Neumann boundary conditions in the presence of a small elastic inclusion. As will be shown in this book, these asymptotic expansions can be used for identifying the inclusions. We provide a general method for determining the lo- cations and/or shape of small inclusions by taking eigenvalue and eigenfunction measurements. It should be emphasized that in its most general form the inverse spectral problem is severely ill-posed and nonlinear. This has been the main ob- stacle to finding noniterative reconstruction algorithms with limited modal data.

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