[2011-Analytic Methods in Partial Differential.PDF

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1、AnalyticMethodsinPartialDierentialEquationsAcollectionofnotesandinsightsintotheworldofdierentialequations.GregoryT.vonNessiOctober31,2011Version::31102011ContentsiiContentsVersion::31102011ContentsiiWhereI'mComingFromviiiIClassicalApproaches1Chapter1:PhysicalMotiv

2、ations21.1Introduction...................................21.2TheHeatEquation...............................31.2.1ANoteonPhysicalAssumptions....................51.3ClassicalModelforTracFlow........................51.4TheTransportEquation............................6

3、1.4.1MethodofCharacteristics:AFirstLook................71.4.2Duhamel'sPrinciple...........................81.5MinimalSurfaces................................91.5.1DivergenceTheoremandIntegrationbyParts.............101.6TheWaveEquation...............................1

4、31.7ClassicationofPDE..............................131.8Exercises.....................................16Chapter2:Laplace'sEquation
u=0andPoisson'sEquation
u=f182.1Laplace'sEqn.inPolarCoordinates......................182.2TheFundamentalSolutionoftheLaplacian.......

5、...........212.3PropertiesofHarmonicFunctions.......................262.3.1WMPforLaplace'sEquation......................262.3.2PoissonIntegralFormula........................272.3.3StrongMaximumPrinciple.......................322.4EstimatesforDerivativesofHarmonicFun

6、ctions...............32GregoryT.vonNessiContentsiii2.4.1Mean-ValueandMaximumPrinciples.................342.4.2RegularityofSolutions.........................362.4.3EstimatesonHarmonicFunctions...................392.4.4AnalyticityofHarmonicFunctions...................

7、422.4.5Harnack'sInequality..........................442.5ExistenceofSolutions..............................472.5.1Green'sFunctions............................472.5.2Green'sFunctionsbyRe ectionMethods...............512.5.2.1Green'sFunctionfortheHalfspace.............

8、512.5.2.2Green'sFunctionforBalls..................552.5.3EnergyMethodsforPoisson'sEquation................572.5.4Perron'sMethodofSubharmon