Lipschitz区域上椭圆型方程组的内正则估计和Dirichlet问题

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1、中文摘要本文研究了Lipschitz区域内的形如(?u)a=一os(以；!；卢研“卢)+67卢Dj“卢+ca卢∥=0的变系数二阶椭圆型方程组．其中系数b和c属于Kato类．我们给出了该方程组的Green矩阵的逐点估计和弱解的内正则估计．随后应用由Shen发展起来的一种实方法把一类变系数椭圆型方程组LpDirichlet边值问题的指标从P=2提升至2

2、键词：椭圆型方程组；Lipschitz区域；Kato类；Dirichlet问题；正则性问题；Green矩阵AbstractWestudygeneralsecondordervariablecoefficientsellipticsystemsOnLipschitzdomainsin砜”withform(名’u)口=一Di(口嚣卢D1“卢)+67卢D∥芦+ca芦“J8：0．Thecoef-ficientsbandebelongtotheKatoclass．Thepoin饥'iseestimatesofGreenr

3、smatrixandinteriorestimatesfortheweaksolutionsareobtainedforsuchellipticsystems，Thenweapplyarealvariableargument，developedbyShen，toimprovetheindexofLesolvabilityofDirichletproblemsforaclassofvariableellipticsystemsfromP：2to2

4、perturbationpropertyfortheindexofL_psolvabilityofDirichletproblemsandLpregularityproblemsforaclassofyariableellipticsystemsisalsoestablished，whichisindependentoflayerpotentialtheoryorApweighttheory．KeyWords：Ellipticsystems；Lipschitzdomains；Katoclass；Dirichle

5、tproblems；Regularityproblems；Green’SmatrixCONTENTSCoNTENTSChapter1Introduction．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．1Chapter2Notationsandmainresults．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3Chapter3Green’Smatrixandinteriorestimate⋯⋯⋯⋯⋯⋯⋯8Chapter4Proofofmai

6、nresults⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯20Bibliography⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯29致谢⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯31个人简历⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯32Chapter1Introduction1IntroductionLetQbeaconnectedLipschitzdomainin瞅，丹≥3，andweconsidertheDirichletproblemforasecondorderellipticdifferentialoperator影，i．e

7、．(彭u)“=一Df(口妒Dju卢)+b尹Dju卢+c邮“pwhere1sf，J≤凡，1≤口，卢sm，and以0卢arereal，measurable，symmetric，i．e．口01B=∥jia，andsatisfyuniformlyellipticconditionpf毒f2≤口：卢o)铲眚fj≤t,-1f考f2(1．1)forsome肚>0．Moreover'thelowerordercoefficientsbandcbelongtotheKatoclasswhichhasbeenextensively

8、studied[2，171．Inrecentyear,manynewtechniquesinharmonicanalysisandpartialdifferentialequationshavebeenappliedtothestudyofboundaryvalueproblemsinLipschtizdo．mains．FortheDirichletproblemwithL9bound