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.
The aim of this book is to give a self-contained presentation of an asymptotic theory
for eigenvalue problems using layer potential techniques with applications in the
fields of inverse problems, band gap structures, and optimal design, in particular the
optimal design of photonic and phononic crystals. Throughout this book, it is shown
how powerful the layer potentials 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. The general approach
in this book is developed in detail for eigenvalue problems for the Laplacian and
the Lamé system in the following two situations: one under variation of domains or
boundary conditions and the other due to the presence of inclusions.
The book will be of interest to researchers and graduate students working in the fields
of partial differential equations, integral equations, and inverse problems. Researchers
in engineering and physics may also find this book helpful.
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