Elliptic Differential Equations of Divergence Form

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1、ELLIPTICDIFFERENTIALEQUATIONSOFDIVERGENCEFORMQINGHANContents1.GrowthofLocalIntegrals22.SchauderTheory63.DeGiorgiIterations144.Moser'sIterations22Inthisnote,wediscusstheregularityofweaksolutionsofellipticequationsofdiver-genceform.Let•beadomaininRn.Thefunctionu2H1

2、(•)isaweaksolutionifitsatis¯esZZaDuD'=f'forany'2H1(•);ijij0••whereweassume(i)theleadingcoe±cientsaij2L1(•)areuniformlyelliptic,i.e.,forsomepositiveconstant¸thereholdsa(x)»»¸¸j»j2foranyx2•and»2Rn;ijij2n(ii)thenonhomogeneoustermf2Ln+2(•).(BytheSobolevembedding,(ii)

3、istheleastassumptiononftohaveameaningfulequation.)Wewillprovevariousinteriorregularityresultsconcerningsolutionsu.Basically,therearetwoclassesofregularityresults,perturbationresultsandnon-perturbationresults.The¯rstisbasedonregularityassumptionsontheleadingcoe±ci

4、entsaij,whichareassumedtobeatleastcontinuous.Undersuchassumptions,wecomparesolutionsoftheunderlyingequationswithharmonicfunctions,orsolutionsofhomogeneousequationsofconstantcoe±cients.Then,regularitiesofsolutionsdependonhowclosesolutionsaretoharmonicfunctionsorho

5、wclosetheleadingcoe±cientsaijaretoconstantcoe±cients.Inthisdirection,wehaveSchauderestimatesandW2;p-estimates.Inthisnote,weonlydiscusstheSchauderestimates.Forthesecondclassofregularity,thereisnocontinuityassumptionontheleadingcoe±cientsaij.Hencetheresultisnotbase

6、donperturbations.TheiterationmethodsintroducedbyDeGiorgiandMoseraresuccessfulindealingwiththenon-perturbationsituation.Resultsprovedbythemarefundamental.ThisnoteispreparedforashortcourseintheSummerSchoolofGeometricAnalysisinUniversityofScienceandTechnologyofChina

7、,Hefei,China,inthesummerof2008.12HAN1.GrowthofLocalIntegralsLetBR(x0)betheballinRnofradiusRandcenterx0.Thewell-knownSobolevtheoremassertsthat,ifu2W1;p(BR(x0))withp>n,thenuisHÄoldercontinuouswithexponent®=1¡n=p.Inthe¯rstpartofthissection,weproveageneralresult,duet

8、oS.Campanato,whichcharacterizesHÄoldercontinuousfunctionsbythegrowthoftheirlocalintegrals.Thisresultisusefulforstudyingregularitiesofsolutionsofellipticdi®eren