Periodic Homogenization of Elliptic Problems

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1、IntroductiontoEllipticHomogenizationUniformInteriorEstimatesBoundaryEstimates-theDirichletConditionBoundaryRegularity:theNeumannConditionsPeriodicHomogenizationofEllipticProblemsCarlosKenigandZhongweiShenSummerSchoolonRecentAdvancesintheTheoryofHomogenizatio

2、nUniversityofChicago,June18-29,2012IntroductiontoEllipticHomogenizationUniformInteriorEstimatesBoundaryEstimates-theDirichletConditionBoundaryRegularity:theNeumannConditionsEllipticOperatorswithRapidlyOscillatingPeriodicCoefficientsConsiderafamilyofellipticop

3、erators@x@L"=aij;">0:@xi"@xjLetA=A(y)=(a(y));1i;jd;1;m:ijAssumethatAisrealanduniformlyelliptic.Aisperiodicw.r.t.Zd:A(y+z)=A(y)fory2Rdandz2Zd.MayalsoneedAsatisfiessomesmoothnessconditions(e.g.,AisHöldercontinuous).Aissymmetric,a=a.ijjiIntroducti

4、ontoEllipticHomogenizationUniformInteriorEstimatesBoundaryEstimates-theDirichletConditionBoundaryRegularity:theNeumannConditionsTheoryofHomogenizationGoal:Describethemacroscopicpropertiesofmicroscopicallyheterogeneousmedia.Considertheboundaryvalueproblem(L"(

5、u")=Fin;u"subjecttosomeboundaryconditions;whichdescribesastationaryprocessinastronglyinhomogeneousmediumwithaperiodicstructure.">0–theinhomogeneityscale"isverysmallrelativetootherlengthscalesintheproblem.IntroductiontoEllipticHomogenizationUniformInteriorEst

6、imatesBoundaryEstimates-theDirichletConditionBoundaryRegularity:theNeumannConditionsPeriodicHomogenizationAs"!0,u!ustronglyinL2()andweaklyinH1();"0whereu0isasolutionofanellipticsystemwithconstantcoefficients(thehomogenizedoreffectivesystem),(L0(u0)=Fin;u0subj

7、ecttothesamekindofboundarycondition;whereL0=div(Abr)andAb=(a^ij)maybecomputed“explicitly”,usingA(y).Thestronglyinhomogeneousmediumwithrapidlyoscillatingmicrostructure,suchascompositematerial,maybeapproximatelydescribedviaaneffectivehomogeneousmedium.Introdu

8、ctiontoEllipticHomogenizationUniformInteriorEstimatesBoundaryEstimates-theDirichletConditionBoundaryRegularity:theNeumannConditionsTopicsTobeCovered1.IntroductiontoEllipticHomogenization:weaksol