Lihe Wang_Analysis of Elliptic Equations(Draft Version)

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1、AnalysisofEllipticEquationsLiheWang2DraftVersionContentsChapter0.Introduction30.0.1GeometricNotation...................90.0.2NotationforDerivatives.................90.0.3NotationforEstimates..................9Chapter1.FunctionSpaces111.1LpSpacesandHardyLittlewoodMaximalFunction.....121.2H•older

2、Spaces...........................181.3SobolevSpaces..........................24Chapter2.TheoryofWeakSolutions332.1LinearMethodsfortheExistenceofWeakSolutions.....332.2VariationalMethods.......................342.3EigenvaluesandEigenfunctions.................362.3.1NeumannBoundaryValueProblem...

3、.......372.4EquationswithLowerOrderTerms..............382.5Problems.............................46Chapter3.BasicPropertiesofHarmonicFunctions473.1BasicHarmonicFunctions....................473.2BasicPropertiesofHarmonicFunctions............493.2.1MeanValueIdentities..................493.2.2Harna

4、ckInequality....................513.2.3Green'sFormulaandIntegralRepresentations.....533.3EstimatesforHarmonicFunctions...............543.3.1MeanValueIdentityApproach.............543.3.2CacciopolliInequality..................563.3.3BernsteinMethod....................583.4Monotonicityformulas....

5、..................6134CONTENTS3.5Problems.............................62Chapter4.SchauderTheoryforPoissonEquation654.1TheMaximumPrincipleApproach...............674.2EnergyMethod..........................744.3CompactnessMethod......................784.4LocalSchauderEstimates....................8

6、54.4.1GeneralDiscussionofH•olderRegularity........884.5BoundaryEstimatesonFlatBoundaries............894.6Lipschitzdomainsasinvariantsets...............914.6.1BarrierFunctions.....................914.6.2HopfLemmaandSlopeEstimates...........924.6.3CEstimatesonLipschitzDomains..........934.7Maximu

7、mPrincipleApproachtoBoundaryRegularity....964.8BoundaryEstimatesonCurvedDomains...........984.9PatchingofInteriorEstimateswithBoundaryEstimates..1034.10SchauderEstimates........................1044.11Problems......................