Hardcover ISBN:  9780821847848 
Product Code:  SURV/153 
202 pp 
List Price:  $78.00 
MAA Member Price:  $70.20 
AMS Member Price:  $62.40 
Electronic ISBN:  9781470413804 
Product Code:  SURV/153.E 
202 pp 
List Price:  $73.00 
MAA Member Price:  $65.70 
AMS Member Price:  $58.40 

Book DetailsMathematical Surveys and MonographsVolume: 153; 2009MSC: Primary 47; 31; 34; 35; 45; 30;
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 doublelayer potentials to treat classical boundary value problems.
The aim of this book is to give a selfcontained 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 operatorvalued 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.ReadershipGraduate students and research mathematicians interested in PDE's, integral equations, and spectral analysis.

Table of Contents

Chapters

Introduction

GohbergSigal theory

1. Generalized argument principle and Rouché’s theorem

Eigenvalue perturbation problems and applications

2. Layer potentials

3. Eigenvalue perturbations of the Laplacian

4. Vibration testing for detecting internal corrosion

5. Perturbations of scattering frequencies of resonators with narrow slits and slots

6. Eigenvalue perturbations of the Lamé system

Photonic and phononic band gaps and optimal design

7. Floquet transform, spectra of periodic elliptic operators, and quasiperiodic layer potentials

8. Photonic band gaps

9. Phononic band gaps

10. Optimal design problems


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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 doublelayer potentials to treat classical boundary value problems.
The aim of this book is to give a selfcontained 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 operatorvalued 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.
Graduate students and research mathematicians interested in PDE's, integral equations, and spectral analysis.

Chapters

Introduction

GohbergSigal theory

1. Generalized argument principle and Rouché’s theorem

Eigenvalue perturbation problems and applications

2. Layer potentials

3. Eigenvalue perturbations of the Laplacian

4. Vibration testing for detecting internal corrosion

5. Perturbations of scattering frequencies of resonators with narrow slits and slots

6. Eigenvalue perturbations of the Lamé system

Photonic and phononic band gaps and optimal design

7. Floquet transform, spectra of periodic elliptic operators, and quasiperiodic layer potentials

8. Photonic band gaps

9. Phononic band gaps

10. Optimal design problems