Degenerate Elliptic Partial Differential Equations - Yihan Zhang.pdf

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1、DegenerateEllipticPartialDierentialEquationsYihanZhangDecember9,2015Contents1Introductionandpreliminaries12Basicconceptsandresults23Basicestimates43.1InteriorH1-estimatefor
u=f.........................43.2InteriorLp-estimatefor@(au)=f......................5iijj3.3Estimatefor@i(aijuj)

2、+biui+cu=0.....................63.3.1Uniformellipticity............................73.3.2Degenerateellipticity...........................74Diculties95Framework106Degeneratenon-characteristics116.1Estimateforu,u2.................................116.2Estimateforu1.......................

3、...........136.3Conclusion.....................................157Degenerateoncharacteristics177.1Energyestimate..................................177.1.1Sgn<0..................................197.1.2Sgn>0..................................207.2Higherregularity...........................

4、......218Hyperbolicequations248.1Stricthyperbolicity................................258.2Degeneratehyperbolicity.............................268.3Mixedtype....................................281IntroductionandpreliminariesWediscussregularityofsolutionofdegenerateellipticpartialdierent

5、ialequationsbyenergyestimates.Topicsincludinghyperbolicequations,mixedequations,maximumprincipleswillalsobeinvolved.Inthisnote,wemainlycareaboutHkandLpestimate.Wealwaysassumethataij=aji.Ellipticequations:1.Non-divergence:aijuij+biui+cu=f2.Divergence:@i(aijuj)+cu=fNotation:@aij1.=aij;k@

6、xk@c2.=c;k@xk3.Einsteinsummationconvention:repeatedindexmeanssummation.12BasicconceptsandresultsEllipticpartialdierentialequations:1.(Strict)elliptic:2naij(x)ijjj>0;8=(1;


;n)2R;8x2:Or,alleigenvaluesofaij(x).2.Degenerateelliptic:naij(x)ij0>0;82R;8x2:3.Uniformlyellipti

7、c:22njjaij(x)ijjj0<;82R;8x2:ForuniformlyellipticPDEs,wehavefollowingresults:1.Perturbationresults(assumeaij2C,andconsideraij'svalueonagivenpoint,thenequationhasconstantcoecients)(a)(Interior)SchauderTheory(1930s)If0,then2;aij;bi;c;f2C())u2C();and{}kukC2;(0)CkukL1()+