##
**Layer potential techniques in spectral analysis.**
*(English)*
Zbl 1167.47001

Mathematical Surveys and Monographs 153. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4784-8/hbk). vi, 202 p. (2009).

The present book deals with using the layer potential technique to tackle the asymptotic theory for eigenvalue problems. The general approach in this book is developed in detail for the Laplacian and the Lamé system in two situations: one under variation of domains or boundary conditions, and the other due to the presence of small-volume inclusions. The book consists of three parts.

The first part is devoted to the theory developed by E. Gohberg and E. I. Sigal. In the second part, eigenvalue perturbation problems are studied. In Chapter 2, the layer potential technique associated with the Laplacian, the Helmholtz equation and the Lamé system is briefly reviewed. In Chapter 3, the authors derive complete expansions for the eigenvalues of the Neumann Laplacian in bounded singularly perturbed domains. Chapter 4 designs a simple method for detecting small internal corrosion by vibration analysis. Chapter 5 studies perturbations of scattering frequencies of resonators with narrow slits and slots. The book provides, on the one hand, results on the existence and localization of the scattering frequencies and, on the other hand, on the leading-order terms in their asymptotic expansions in terms of the characteristic width of the slits or the slots. In Chapter 6, the book develops the asymptotic theory for eigenvalue problems to the Lamé system with Neumann boundary condition. Complete asymptotic expansions of the perturbations due to the presence of an elastic inclusion are derived rigorously. Leading-order terms in these expansions are explicitly given. The inclusion may be hard or soft. A hard inclusion is characterized by the boundary condition \(u=0\) on its boundary, while a soft inclusion is characterized by the transmission condition on its boundary.

The third part of the book investigates the band gap structure of the frequency spectrum for waves in a high contrast, two-composed medium. This provides a new tool for investigating photonic and phononic crystals. Chapter 7 deals with the Floquet transform. Then the structure of spectra of periodic elliptic operators is dicussed. Also, quasi-periodic layer potentials for the Helmholtz equation and the Lamé system are investigated. A spectral and spatial representation of Green’s function in periodic domains is provided. In Chapter 8, the authors perform an analysis of the spectral properties of high contrast band gap materials, consisting of a background medium which is perforated by a periodic array of holes, with respect to the index ratio and small perturbations in the geometry of holes. Chapter 9 is devoted to phononic band gaps. The final Chapter 10 investigates shape optimization problems.

The first part is devoted to the theory developed by E. Gohberg and E. I. Sigal. In the second part, eigenvalue perturbation problems are studied. In Chapter 2, the layer potential technique associated with the Laplacian, the Helmholtz equation and the Lamé system is briefly reviewed. In Chapter 3, the authors derive complete expansions for the eigenvalues of the Neumann Laplacian in bounded singularly perturbed domains. Chapter 4 designs a simple method for detecting small internal corrosion by vibration analysis. Chapter 5 studies perturbations of scattering frequencies of resonators with narrow slits and slots. The book provides, on the one hand, results on the existence and localization of the scattering frequencies and, on the other hand, on the leading-order terms in their asymptotic expansions in terms of the characteristic width of the slits or the slots. In Chapter 6, the book develops the asymptotic theory for eigenvalue problems to the Lamé system with Neumann boundary condition. Complete asymptotic expansions of the perturbations due to the presence of an elastic inclusion are derived rigorously. Leading-order terms in these expansions are explicitly given. The inclusion may be hard or soft. A hard inclusion is characterized by the boundary condition \(u=0\) on its boundary, while a soft inclusion is characterized by the transmission condition on its boundary.

The third part of the book investigates the band gap structure of the frequency spectrum for waves in a high contrast, two-composed medium. This provides a new tool for investigating photonic and phononic crystals. Chapter 7 deals with the Floquet transform. Then the structure of spectra of periodic elliptic operators is dicussed. Also, quasi-periodic layer potentials for the Helmholtz equation and the Lamé system are investigated. A spectral and spatial representation of Green’s function in periodic domains is provided. In Chapter 8, the authors perform an analysis of the spectral properties of high contrast band gap materials, consisting of a background medium which is perforated by a periodic array of holes, with respect to the index ratio and small perturbations in the geometry of holes. Chapter 9 is devoted to phononic band gaps. The final Chapter 10 investigates shape optimization problems.

Reviewer: Dagmar Medková (Praha)

### MSC:

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

47A75 | Eigenvalue problems for linear operators |

47F05 | General theory of partial differential operators |

35P05 | General topics in linear spectral theory for PDEs |

35R30 | Inverse problems for PDEs |

74H45 | Vibrations in dynamical problems in solid mechanics |

47Nxx | Miscellaneous applications of operator theory |