Layer potential techniques in spectral analysis

211 Pages·2009·1.769 MB·English

by  Habib Ammari, Hyeonbae Kang, and Hyundae Lee

Mathematical
Surveys
and
Monographs
Volume 153
Layer Potential Techniques
in Spectral Analysis
Habib Ammari
Hyeonbae Kang
Hyundae Lee
American Mathematical Society

Layer Potential Techniques
in Spectral Analysis

Mathematical
Surveys
and
Monographs
Volume 153
Layer Potential Techniques
in Spectral Analysis

Habib Ammari
Hyeonbae Kang
Hyundae Lee
American Mathematical Society
Providence, Rhode Island
EDITORIAL COMMITTEE
Jerry L. Bona Michael G. Eastwood
Ralph L. Cohen J. T. Stafford, Chair
Benjamin Sudakov
2000 Mathematics Subject Classification. Primary47A55, 47A75, 31A10, 34A55, 35R30,
35B34, 45Q05, 30E25.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-153
Library of Congress Cataloging-in-Publication Data
Ammari,Habib.
Layer potential techniques in spectral analysis / Habib Ammari, Hyeonbae Kang, Hyundae
Lee.
p.cm. —(Mathematicalsurveysandmonographs;v.153)
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4784-8(alk.paper)
1.Differentialequations,Elliptic. 2.Spectraltheory(Mathematics). 3.Compositematerials—
Spectra. 4.Eigenvalues. 5.Boundaryelementmethods. 6.Photoniccrystals. I.Kang,Hyeon-
bae. II.Lee,Hyundae,1976– III.Title.
QA377.A5668 2009
515(cid:1).3533—dc22 2008048317
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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
v
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
(cid:1)
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
Introduction
The aim of this book is to give a self-contained presentation of an asymptotic
theory for eigenvalue problems using layer potential techniques. This theory and
its application in the field of inverse problems have been developed over the last
number of years by the authors.
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.
Our main objective in this book is to show how powerful the layer potential
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.
Therearetwoprominenteigenvalueperturbationproblems: oneundervariation
of domains or boundary conditions and the other due to the presence of small-
volume inclusions.
Therehavebeenseveralinterestingworksontheproblemofeigenvaluechanges
undervariationofdomainssincetheseminalformulaofHadamard[119]: theworks
by Garabedian and Schiffer [113], Kato [149], Fujiwara and Ozawa [102], Sanchez
Hubert and Sanchez Palencia [221], Ward and Keller [248], Gadyl’shin and Il’in
[112], Gadyl’shin [110, 111], Daners [80], McGillivray [181], Noll [192], Planida
[212], Bruno and Reitich [61], Burenkov and Lamberti [62], and Kozlov [153].
For the second problem, Rauch and Taylor [216] have shown that the spec-
trum of a bounded domain does not change after imposing Dirichlet conditions on
compact subsets of capacity zero. Subsequently, many people have studied the as-
ymptoticexpansionsoftheeigenvaluesforthecaseofsmallholeswithaDirichletor
aNeumannboundarycondition. Inparticular,Ozawaprovidedinaseriesofpapers
[202]–[207] leading-order terms (A0,A1,A2 in (0.3)) in eigenvalue expansions; see
also[248]and[177]. Besson[54]hasprovedtheexistenceofacompleteexpansion
(0.3) of the eigenvalue perturbation in the two-dimensional case. Courtois [76] has
established a perturbation theory for the Dirichlet spectrum in a compactly per-
turbeddomain intermsof the capacity of thecompact perturbation. We also refer
to the book by Maz’ya, Nazarov, and Plamenevskii [178], where the method of
matched expansions [133, 134] has been used to construct asymptotic representa-
tions of eigenvalues of problems of conduction and elasticity theory for bodieswith
small holes.
In this book, we shall consider both the first and the second eigenvalue prob-
lems. For the first problem, we consider an inclusion inside a bounded domain and
derivehigh-orderasymptoticexpansionsoftheperturbationsoftheeigenvaluesthat
are due to shape variations of the inclusion. We also study the effect of a change
1
2 INTRODUCTION
in the boundary condition on a small part of the boundary on the eigenvalues. For
the second problem, we provide a complete asymptotic expansion of the eigenvalue
perturbation with respect to the size of the inclusion for a domain containing a
small inclusion. Our exposition is accompanied by many new applications of our
asymptotictheory, especiallytoimagingand optimaldesign. Itis worthemphasiz-
ingthattheasymptoticresultsderivedinthisbookhavenumerousotherimportant
applications in practice.
To be more precise, let Ω be a bounded domain inRd, d≥2, with a connected
Lipschitzboundary∂Ω. Letν denotetheunitoutwardnormalto∂Ω. Supposethat
ΩcontainsasmallinclusionD,oftheformD =z+(cid:3)B, whereB isaboundedLip-
schitz domain in Rd containing the origin. We also assume that the “background”
is homogeneous with conductivity 1. Even though we will deal with other cases as
well, for the moment let us assume that the inclusion is grounded, meaning that
zero Dirichlet conditions are imposed on ∂D. Then the eigenvalue problem for the
domain with the inclusion is given by
⎧
⎪⎪⎨ ∆u(cid:1)+µ(cid:1)u(cid:1) =0 in Ω(cid:1),
∂u(cid:1)
(0.1) =0 on ∂Ω,
⎪⎪⎩ ∂ν
u(cid:1) =0 on ∂D,
where Ω :=Ω\D.
(cid:1)
Let0=µ1 <µ2 ≤...betheeigenvaluesof−∆inΩwithNeumannconditions,
namely, those of the problem
⎧
⎨ ∆u+µu=0 in Ω,
(0.2) ∂u
⎩
=0 on ∂Ω.
∂ν
The eigenvalues are arranged in an increasing sequence and counted according to
multiplicity. Fix j and suppose that the eigenvalue µ is simple. Note that this
j
assumptionisnotessentialinwhatfollowsand,moreover,itisprovedin[3,4,243]
that the eigenvalues are generically simple. By generic, we mean the existence of
arbitrarysmalldeformationsof∂Ωsuchthatinthedeformeddomaintheeigenvalue
is simple. Throughout this book, the assumption of the simplicity is made for ease
of exposition. Then there exists a simple eigenvalue µ(cid:1) near µ associated to the
j j
normalized eigenfunction u(cid:1); that is, u(cid:1) satisfies (0.1).
j j
One of our goals in this book is to find complete asymptotic expansions for the
eigenvalues µ(cid:1) as (cid:3) tends to 0. In other words, we seek tofind a series expansion of
j
the form
(0.3) µ(cid:1)j =A0+A1(cid:3)n+A2(cid:3)n+1+....
Existence of such a series is a part of what we are going to prove.
The book consists of three parts. The first part is devoted to the theory devel-
oped by Gohberg and Sigal. In the second part, we provide rigorous derivations of
complete asymptotic expansions of eigenvalue perturbations such as (0.3). A key
feature of our work is the approach we develop: a general and unified boundary
integral approach with rigorous justification based on the Gohberg-Sigal theory ex-
plained in Part 1. By using layer potential techniques, we show that the square
roots of the eigenvalues are exactly the real characteristic values of meromorphic