**Mathematical Surveys and Monographs**

Volume: 153;
2009;
202 pp;
Hardcover

MSC: Primary 47; 31; 34; 35; 45; 30;

Print ISBN: 978-0-8218-4784-8

Product Code: SURV/153

List Price: $73.00

Individual Member Price: $58.40

**Electronic ISBN: 978-1-4704-1380-4
Product Code: SURV/153.E**

List Price: $73.00

Individual Member Price: $58.40

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#### Supplemental Materials

# Layer Potential Techniques in Spectral Analysis

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*Habib Ammari; Hyeonbae Kang; Hyundae Lee*

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.

#### Table of Contents

# Table of Contents

## Layer Potential Techniques in Spectral Analysis

- Contents v7 free
- Introduction 19 free
- Part 1: Gohberg and Sigal Theory 513 free
- Chapter 1. Generalized Argument Principle and Rouche's Theorem 715
- Part 2: Eigenvalue Perturbation Problems and Applications 1725
- Chapter 2. Layer Potentials 1927
- Chapter 3. Eigenvalue Perturbations of the Laplacian 3543
- Chapter 4. Vibration Testing for Detecting Internal Corrosion 7785
- Chapter 5. Perturbations of Scattering Frequencies of Resonators .... 9199
- Chapter 6. Eigenvalue Perturbations of the Lame System 103111
- Part 3. Photonic and Phononic Band Gaps and Optimal Design 119127
- Chapter 7. Floquet Transform, Spectra of Periodic Elliptic Operators, and Quasi-Periodic ... 121129
- Chapter 8. Photonic Band Gaps 133141
- Chapter 9. Phononic Band Gaps 153161
- Chapter 10. Optimal Design Problems 179187
- Bibliography 191199
- Index 201209 free

#### Readership

Graduate students and research mathematicians interested in PDE's, integral equations, and spectral analysis.