Layer Potential Techniques in Spectral Analysis
About this Title
Habib Ammari, Ecole Polytechnique, Palaiseau, France, Hyeonbae Kang, Inha University, Incheon, South Korea and Hyundae Lee, Inha University, Incheon, South Korea
Publication: Mathematical Surveys and Monographs
Publication Year 2009: Volume 153
ISBNs: 978-0-8218-4784-8 (print); 978-1-4704-1380-4 (online)
MathSciNet review: MR2488135
MSC: Primary 47F05; Secondary 35P20, 35R30, 47A10, 49Q12, 49R05, 74G99, 74H45
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.
Graduate students and research mathematicians interested in PDE's, integral equations, and spectral analysis.
Table of Contents
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 quasi-periodic layer potentials
- 8. Photonic band gaps
- 9. Phononic band gaps
- 10. Optimal design problems