Table Of ContentLecture Notes in Mathematics 2077
Vladimir Maz'ya
Alexander Movchan
Michael Nieves
Green's Kernels
and Meso-Scale
Approximations
in Perforated
Domains
Lecture Notes in Mathematics 2077
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
Vladimir Maz’ya Alexander Movchan
(cid:2)
Michael Nieves
Green’s Kernels
and Meso-Scale
Approximations
in Perforated Domains
123
VladimirMaz’ya AlexanderMovchan
DepartmentofMathematics DepartmentofMathematicalSciences
Linko¨pingUniversity UniversityofLiverpool
Linko¨ping,Sweden Liverpool,UnitedKingdom
MichaelNieves
SchoolofEngineering
LiverpoolJohnMooresUniversity
Liverpool,UnitedKingdom
ISBN978-3-319-00356-6 ISBN978-3-319-00357-3(eBook)
DOI10.1007/978-3-319-00357-3
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013939619
MathematicsSubjectClassification(2010):4E10,35B40,35J08
(cid:2)c SpringerInternationalPublishingSwitzerland2013
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Preface
Thebookisbasedontheauthors’resultsonasymptoticapproximationsofGreen’s
kernelsforellipticboundaryvalueproblemsinperforateddomains.Anewfeature
is the uniformity of the asymptotics with respect to the independent variables.
Formal asymptotic approximations are supplied with estimates of the remainder
terms.Forthecasewhenthenumberofperforationsorinclusionsbecomeslarge,a
novelmethodofmeso-scaleasymptoticapproximationsisintroduced,anduniform
asymptotic approximations of Green’s kernels as well as solutions of boundary
valueproblemsinmultiplyperforateddomainsarepresented.Suchapproximations
donotrequireperiodicityorothertypicalconstraintsattributedtohomogenization
approximations.
Applicationsareconsideredforproblemsoflinearelasticityinplanarandthree-
dimensional domains containing multiple small holes or inclusions. Illustrative
computationalexamplesare included to compare asymptoticapproximationswith
accuratefiniteelementnumericalsimulations,whichdemonstratetheadvantagesof
theasymptoticmethod.
This book is addressed to mathematicians, physicists and engineers, as well as
researchstudents,interestedinasymptoticanalysisandnumericalcomputationsfor
solutionstopartialdifferentialequations.Therequiredbackgroundincludesabasic
theoryofpartialdifferentialequationsandelementsoffunctionalanalysis.
Acknowledgements TheauthorswouldliketothanktheUniversityofLiverpool forproviding
excellent academic facilitiesthroughout theduration oftheproject, whichhasledtothisbook.
The support of the UK Engineering and Physical Sciences Research Council via the grant
EP/F005563/1isgratefullyacknowledged.
Linko¨ping,Sweden V.Maz’ya
Liverpool,UK A.Movchan
Liverpool,UK M.Nieves
v
Contents
PartI Green’sFunctionsinSingularlyPerturbedDomains
1 UniformAsymptoticFormulaeforGreen’sFunctions
fortheLaplacianinDomainswithSmallPerforations................. 3
1.1 Green’sFunctionforaMulti-dimensionalDomain
withaSmallHole .................................................... 3
1.2 Green’sFunctionfortheDirichletProbleminaPlanar
DomainwithaSmallHole........................................... 9
1.2.1 AsymptoticApproximationoftheCapacitary
Potential..................................................... 12
1.2.2 UniformAsymptoticApproximation...................... 13
1.3 Corollaries............................................................ 17
2 MixedandNeumannBoundaryConditionsforDomains
withSmallHolesandInclusions:UniformAsymptotics
ofGreen’sKernels.......................................................... 21
2.1 MixedBoundaryValueProbleminaPlanarDomain
withaSmallHoleoraCrack........................................ 21
2.1.1 SpecialSolutionsofModelProblems ..................... 22
2.1.2 TheDipoleMatrixPPP ....................................... 24
2.1.3 Pointwise Estimate of a Solution
totheExteriorNeumannProblem......................... 25
2.1.4 AsymptoticPropertiesoftheRegularPart
oftheNeumannFunctioninR2nF....................... 27
2.1.5 MaximumModulusEstimateforSolutions
to the Mixed Problem in ˝ , with the
"
NeumannDataon@F ...................................... 30
"
2.1.6 ApproximationofGreen’sFunctionG.N/................. 31
"
2.1.7 SimplerAsymptoticFormulaeforGreen’s
FunctionG.N/ .............................................. 33
"
vii
viii Contents
2.2 MixedBoundaryValueProblemwiththeDirichlet
Conditionon@F ..................................................... 35
"
2.2.1 SpecialSolutionsofModelProblems ..................... 35
2.2.2 AsymptoticPropertyofthe RegularPart
ofGreen’sFunctioninR2nF ............................. 38
2.2.3 MaximumModulusEstimateforSolutions
to the Mixed Problem in ˝ , with the
"
DirichletDataon@F ....................................... 39
"
2.2.4 ApproximationofGreen’sFunctionG.D/................. 40
"
2.2.5 Simpler Asymptotic Representation
ofGreen’sFunctionG.D/ .................................. 43
"
2.3 The Neumann Function for a Planar Domain
withaSmallHoleorCrack.......................................... 44
2.3.1 SpecialSolutionsofModelProblems ..................... 45
2.3.2 MaximumModulusEstimateforSolutions
totheNeumannProblemin˝ ............................ 45
"
2.3.3 AsymptoticApproximationofN ......................... 48
"
2.3.4 Simpler Asymptotic Representation
ofNeumann’sFunctionN ................................. 49
"
2.4 Asymptotic Approximationsof Green’s Kernels
forMixedandNeumann’sProblemsinThreeDimensions........ 50
2.4.1 Special Solutions of Model Problems
inLimitDomains........................................... 51
2.4.2 ApproximationsofGreen’sKernels....................... 53
3 Green’sFunction for the Dirichlet Boundary Value
ProbleminaDomainwithSeveralInclusions........................... 59
3.1 DomainofDefinitionandtheGoverningEquations
fortheCaseofMultipleInclusions.................................. 59
3.2 Green’sFunctionfortheCaseofAnti-planeShear
foraDomainwithSeveralInclusions............................... 60
3.2.1 Estimatesforthe Functionsh.j/ and(cid:2).j/
intheUnboundedDomain ................................. 61
3.2.2 TheCapacitaryPotential................................... 61
3.2.3 A Uniform Asymptotic
Approximation of Green’s Function
for (cid:3)(cid:3) in a Two-DimensionalDomain
withSeveralSmallInclusions.............................. 65
3.3 SimplifiedAsymptoticFormulaeforGreen’sFunction
SubjecttoConstraintsontheIndependentVariables............... 70
Contents ix
4 NumericalSimulationsBasedontheAsymptoticApproximations... 75
4.1 AsymptoticFormulaeVersusNumericalSolution
fortheOperator(cid:3)(cid:3).................................................. 75
4.1.1 DomainandtheAsymptoticApproximation.............. 76
4.1.2 Example:A Configurationwith a Large
NumberofSmallInclusions ............................... 78
4.1.3 Example:AConfigurationwithInclusions
ofRelativelyLargeSize.................................... 79
5 Other Examples of Asymptotic Approximations
ofGreen’sFunctionsinSingularlyPerturbedDomains ............... 83
5.1 PerturbationofaSmoothExteriorBoundary....................... 83
5.2 Green’sFunctionfortheDirichlet–NeumannProblem
inaTruncatedCone.................................................. 84
5.3 TheDirichlet–NeumannProbleminaLongRod................... 87
5.3.1 CapacitaryPotential........................................ 88
5.3.2 AsymptoticApproximationofGreen’sFunction ......... 89
5.3.3 Green’s Function G Versus Green’s
M
FunctionsforUnboundedDomains........................ 92
5.3.4 TheDirichlet–NeumannProbleminaThinRod.......... 93
Part II Green’s Tensors for Vector Elasticity in Bodies
withSmallDefects
6 Green’sTensorfortheDirichletBoundaryValueProblem
inaDomainwithaSingleInclusion...................................... 97
6.1 Green’sRepresentationforVectorElasticity ....................... 97
6.1.1 GeometryandMatrixDifferentialOperators ............. 98
6.2 EstimatesfortheMaximumModulusofSolutions
ofElasticityProblemsinDomainswithSmallInclusions ......... 101
6.2.1 TheMaximumPrinciplein˝ ............................. 102
6.2.2 TheMaximumPrincipleinC!N............................ 102
6.2.3 TheOperatorNotations .................................... 105
6.3 Green’sTensorfora Three-DimensionalDomain
withaSmallInclusion ............................................... 109
6.3.1 Green’sMatricesforModelDomainsinThree
Dimensions ................................................. 109
6.3.2 TheElasticCapacitaryPotentialMatrix................... 110
6.3.3 AsymptoticEstimatesfortheRegularParth
ofGreen’sTensorinanUnboundedDomain ............. 118
6.3.4 AUniformAsymptoticFormulaforGreen’s
FunctionG inThreeDimensions......................... 119
"
6.4 Green’sTensorforaPlanarDomainwithaSmallInclusion....... 124
6.4.1 Green’s Kernels for Model Domains
inTwoDimensions......................................... 124
