(Lerner)Partial Differential Equations

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1、LectureNotesonPartialDifferentialEquationsUniversit´ePierreetMarieCurie(Paris6)NicolasLernerFebruary24,20112Contents1Introduction51.1Examples.................................51.2Comments.................................111.3Quotations.................................122VectorFields152.1OrdinaryDiffer

2、entialEquations.....................152.1.1TheCauchy-Lipschitzresult...................152.1.2MaximalandGlobalSolutions..................212.1.3Continuousdependence......................252.2VectorFields,Flow,FirstIntegrals...................292.2.1Definition,examples.......................292.2.2Lo

3、calStraighteningofanon-singularvectorfield.......342.2.32Dexamplesofsingularvectorfields..............382.3Transportequations...........................392.3.1Thelinearcase..........................392.3.2Thequasi-linearcase.......................422.3.3ClassicalsolutionsofBurgersequation............

4、.432.4One-dimensionalconservationlaws...................452.4.1Rankine-Hugoniotconditionandsingularsolutions......452.4.2TheRiemannproblemforBurgersequation..........463Fiveclassicalequations513.1TheLaplaceandCauchy-Riemannequations..............513.1.1Fundamentalsolutions......................513.

5、1.2Hypoellipticity..........................533.1.3Polarandsphericalcoordinates.................543.2Theheatequation............................553.3TheSchr¨odingerequation........................573.4TheWaveEquation............................593.4.1Presentation............................593.4.2T

6、hewaveequationinonespacedimension...........613.4.3Thewaveequationintwospacedimensions..........613.4.4Thewaveequationinthreespacedimensions..........6234CONTENTS4AnalyticPDE654.1TheCauchy-Kovalevskayatheorem...................655EllipticEquations715.1SomesimplefactsontheLaplaceoperator............

7、...715.1.1Themean-valuetheorem.....................715.1.2Themaximumprinciple.....................735.1.3Analyticityofharmonicfunctions................745.1.4Green’sfunction.........................776Hyper