Perturbation of the boundary in boundary-value problems of partial differential equations

216 Pages·2005·1.262 MB·English

by  Dan Henry, Jack Hale, Antônio Luiz Pereira

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PerturbationoftheBoundaryinBoundary-ValueProblems
ofPartialDifferentialEquations

LONDONMATHEMATICALSOCIETYLECTURENOTESERIES
ManagingEditor:ProfessorN.J.Hitchin,MathematicalInstitute,
UniversityofOxford,24-29StGiles,OxfordOX13LB,UnitedKingdom
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LondonMathematicalSocietyLectureNoteSeries.318
Perturbation of the Boundary in
Boundary-Value Problems of
Partial Differential Equations
DanHenry
witheditorialassistancefrom
JackHale
and
AntoˆnioLuizPereira
  
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
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Published in the United States of America by Cambridge University Press, New York
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Contents
Introduction page1
1 GeometricalPreliminaries 3
1.1 Somenotation 3
2 DifferentialCalculusofBoundaryPerturbations 18
3 ExamplesUsingtheImplicitFunctionTheorem 27
3.1 TorsionalRigidity 27
3.2 SimpleEigenvaluesoftheDirichletProblemfortheLaplacian 32
3.3 Capacity 36
3.4 Green’sFunction 39
3.5 SimpleEigenvaluesofRobin’sProblem 40
3.6 SimpleEigenvalueofaGeneralDirichletProblem 44
4 BifurcationProblems 46
4.1 MultipleEigenvaluesoftheDirichletProblemfor
theLaplacian 46
4.2 VariationofaTurningPoint 51
4.3 ABifurcationProblemwitTwo-DimensionalKernel 53
4.4 GenericSimplicityofEigenvaluesofaSelf-Adjoint2m-Order
DirichletProblem 55
5 TheTransversalityTheorem 60
6 GenericPerturbationoftheBoundary 79
6.1 GenericSimplicityofEigenvaluesoftheDirichlet
Problemfor(cid:1) 80
vii
viii Contents
6.2 SimplicityofEigenvalueswithaReflection-symmetry
Constraint 83
6.3 SimplicityofRealEigenvaluesofaGeneralSecond-Order
DirichletProblem 85
6.4 GenericSimplicityofEigenvaluesfortheNeumann
Problemfor(cid:1) 88
6.5 GenericSimplicityof(cid:1)u+ f(x,u,∇u)=0in(cid:2),u =0
on∂(cid:2) 89
6.6 GenericSimplicityofEigenvaluesofRobin’sProblem 94
6.7 GenericSimplicityofSolutionsof(cid:1)u+ f(u,x)=0in(cid:2),
∂u/∂N =g(k,u)on∂(cid:2) 100
6.8 GenericSimplicityofComplexEigenvaluesofaDirichlet
Problem 106
7 BoundaryOperatorsforSecond-OrderElliptic
Equations 114
7.1 WeaklySingularIntegralOperatorswithCr(α)Kernels 116
∗
7.2 IntegralEquationswithCr(α)Kernels 122
∗
7.3 FourierTransformandCompositionofWeakly
SingularKernels 126
7.4 LimitsofSingularIntegrals 135
7.5 FundamentalSolutions 139
7.6 CalculationofsomeboundaryOperators 141
8 TheMethodofRapidly-OscillatingSolutions 152
8.1 Introduction 152
8.2 FormalAsymptoticSolutions 154
8.3 ExactSolutions 158
8.4 Example6.8Revisited:GenericSimplicityof
ComplexEigenvalues 161
8.5 GenericSimplicityofSolutionsofaSystem 168
Appendix1 EigenvaluesoftheLaplacianinthePresence
ofSymmetry 183
Appendix2 OnMicheletti’sMetricSpace 188
References 199
Index 203