harmonic analysis and partial differential equations - b. dahlberg, c. kenig

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1、HARMONICANALYSISANDPARTIALDIFFERENTIALEQUATIONSBJ?RNE.J.DAHLBERGCARLOSE.KENIGISSN0347-2809DEPARTMENTOFMATHEMATICSCHALMERSUNIVERSITYOFTECHNOLOGYANDTHEUNIVERSITYOFG?TEBORGG?TEBORG1985/1996FOREWORDTheselecturenotesarebasedonacourseIgaverstatUniversityofTexas,Austindu

2、ringtheacademicyear1983-1984andatUniversityofG?teborginthefallof1984.Mypurposeinthoselectureswastopresentsomeoftherequiredbackgroundinordertopresenttherecentresultsonthesolvabilityofboundaryvalueproblemsindomainswithbadboundaries.Thesenotesconcentrateontheboundar

3、yvalueproblemsfortheLaplaceoperator;foracompletesurveyofresults,werefertothesurveyarticlebyCarlosKenig;Iamverygratefulforthiskindpermissiontoincludeithere.ItisalsomypleasuretoacknowledgemygratitudetoPeterKumlinforexcellentworkinpreparingthesenotesforpublication.Jan

4、uary1985Bj?rnE.J.DahlbergiiiContents0Introduction11DirichletProblemforLipschitzDomain.TheSetup112ProofsofTheorem1.1andTheorem1.2193ProofofTheorem1.6254ProofofTheorem1.3375ProofofTheorem1.4476DirichletProblemforLipschitzdomains.Thenalargumentsforthe2L-theory517Exis

5、tenceofsolutionstoDirichletandNeumannproblemsforLipschitzpdomains.TheoptimalL-results57Index:::::::::::::::::::::::::::::::::::::::::64Appendix1C.E.Kenig:RecentProgressonBoundaryValueProblemsonLipschitzDomains:::::::::::::::::::::::::::::::67Appendix2B.E.Dahlberg/C

6、.E.Kenig:HardyspacesandtheNeumannpProbleminLforLaplace'sequationinLipschitzdomains:::::::::107iiiivChapter0IntroductionInthiscoursewewillstudyboundaryvalueproblems(BVP:s)forlinearellipticPDE:swithconstantcoecientsinLipschitz-domains,i.e.,domainswheretheboundary@lo

7、callyisgivenbythegraphofLipschitzfunction.Werecallthatafunction'isLipschitzifthereexistsaconstantM<1suchthatj'(x)?'(z)jMjx?zjforallxandz.y='(x)xTosolvetheBVP:swewillreformulatetheproblemsintermsofintegralequations.Itthereforebecomesnecessarytostudysingularintegral

8、operatorsofCalder?n-Zygmundpptype,whichweprovetobeL-boundedfor1