Contents
Introduction 1
Part 1. Gohberg and Sigal Theory 5
Chapter 1. Generalized Argument Principle and Rouch´ e’s Theorem 7
1.1. Definitions and Preliminaries 8
1.2. Factorization of Operators 11
1.3. Main Results of the Gohberg and Sigal Theory 12
1.4. Concluding Remarks 16
Part 2. Eigenvalue Perturbation Problems and Applications 17
Chapter 2. Layer Potentials 19
2.1. Sobolev Spaces 19
2.2. Layer Potentials for the Laplace Equation 20
2.3. Layer Potentials for the Helmholtz Equation 22
2.4. Integral Representation of Solutions to the Lam´ e System 28
2.5. Concluding Remarks 33
Chapter 3. Eigenvalue Perturbations of the Laplacian 35
3.1. Eigenvalue Perturbations Due to Small Grounded Inclusions 35
3.2. Eigenvalue Perturbations Due to Conductivity Inclusions 47
3.3. Eigenvalue Perturbations Due to Shape Deformations 54
3.4. Splitting of Multiple Eigenvalues 60
3.5. Reconstruction of Inclusions 61
3.6. Reconstruction of Shape Deformations 68
3.7. Concluding Remarks 74
Chapter 4. Vibration Testing for Detecting Internal Corrosion 77
4.1. Introduction 77
4.2. Formal Derivations 77
4.3. Reconstruction Method 79
4.4. Numerical Results 81
4.5. Justification of the Asymptotic Expansion 84
4.6. Concluding Remarks 89
Chapter 5. Perturbations of Scattering Frequencies of Resonators with
Narrow Slits and Slots 91
5.1. Introduction 91
5.2. Perturbations of Scattering Frequencies Due to Narrow Slits 91
5.3. Perturbations of Scattering Frequencies of a Helmholtz Resonator 99
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