vi CONTENTS 5.4. Concluding Remarks 101 Chapter 6. Eigenvalue Perturbations of the Lam´ e System 103 6.1. Introduction 103 6.2. Hard Inclusion Case 104 6.3. Transmission Problem 108 6.4. Eigenvalue Perturbations Due to Shape Deformations 115 6.5. Reconstruction of Inclusions 115 6.6. Concluding Remarks 117 Part 3. Photonic and Phononic Band Gaps and Optimal Design 119 Chapter 7. Floquet Transform, Spectra of Periodic Elliptic Operators, and Quasi-Periodic Layer Potentials 121 7.1. Floquet Transform 121 7.2. Structure of Spectra of Periodic Elliptic Operators 122 7.3. Quasi-Periodic Layer Potentials for the Helmholtz Equation 123 7.4. Quasi-Periodic Layer Potentials for the Lam´ e System 126 7.5. Computations of Periodic Green’s Functions 128 7.6. Muller’s Method 131 7.7. Concluding Remarks 132 Chapter 8. Photonic Band Gaps 133 8.1. Introduction 133 8.2. Boundary Integral Formulation 134 8.3. Sensitivity Analysis with Respect to the Index Ratio 137 8.4. Photonic Band Gap Opening 146 8.5. Sensitivity Analysis with Respect to Small Perturbations in the Geometry of the Holes 146 8.6. Proof of the Representation Formula 147 8.7. Characterization of the Eigenvalues of 149 8.8. Concluding Remarks 150 Chapter 9. Phononic Band Gaps 153 9.1. Introduction 153 9.2. Asymptotic Behavior of Phononic Band Gaps 155 9.3. Criterion for Gap Opening 171 9.4. Gap Opening Criterion When Densities Are Different 175 9.5. Concluding Remarks 178 Chapter 10. Optimal Design Problems 179 10.1. Introduction 179 10.2. The Acoustic Drum Problem 179 10.3. An Optimal Control Approach in Shape Optimization 184 10.4. Maximizing Band Gaps in Photonic Crystals 186 10.5. Approximate Optimal Design Problems for Photonic Crystals 188 10.6. Concluding Remarks 189 Bibliography 191 Index 201
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