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


· (1 + (k 1)χ(D))∇u + µ u = 0 in Ω,
= 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


+ µ u = 0 in \ D,

+ 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
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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