Homogenization

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1、PeriodicHomogenizationofEllipticProblems(Draft)ZhongweiShen1DepartmentofMathematicsUniversityofKentucky1SupportedinpartbyNSFgrantDMS-08552942Contents1EllipticOperatorswithPeriodicCoecients51.1Weaksolutions..................................61.2Asymptoticexpansionsandtheho

2、mogenizedoperator.............91.3Homogenizationofellipticsystems.......................131.4Convergenceresults................................191.5Notes........................................232InteriorEstimates252.1InteriorH•olderestimates.............................2

3、52.2InteriorLipschitzestimates............................302.3Areal-variablemethod..............................342.4InteriorW1;pestimates..............................412.5Asymptoticexpansionsoffundamentalsolutions................442.6Notes..............................

4、..........513RegularityfortheDirichletProblem533.1BoundaryH•olderestimates............................543.2CorrectorsfortheDirichletproblem.......................603.3BoundaryLipschitzEstimates..........................643.4TheDirichletprobleminC1;domains................

5、.....703.5Notes........................................744RegularityfortheNeumannProblems754.1BoundaryH•olderestimates............................754.2W1;pestimates..................................844.3MatrixofNeumannfunctions..........................884.4Correctorsfort

6、heNeumannproblems......................924.5BoundaryLipschitzEstimates..........................984.6LipschitzestimatesinC1;domains.......................1044.7Notes........................................10634CONTENTSChapter1SecondOrderEllipticOperatorswithPeriodicCoec

7、ientsInthismonographweshallbeconcernedwithafamilyofsecond-orderellipticoperatorsindivergenceformwithrapidlyoscillatingperiodiccoecients,
@x@1L"=divA(x=")r=aij;">0:(1.0.1)@xi"@xjWewillalwaysassumethatthecoecientmatrixA(y)=(aij(y))with1i;jdand1;misreal,bounde

8、dmeasurable,andsatisestheellipticitycondition212ddmjjaij(y)ijjjfory2Rand=(i)2R;(1.0.2)