二階拋物型方程粘性解的若干問題.pdf

1、南昌航空大學(xué)碩士學(xué)位論文二階拋物型方程粘性解的若干問題姓名：郭林申請學(xué)位級別：碩士專業(yè)：偏微分方程及其應(yīng)用指導(dǎo)教師：易青20070601IIAbstractThispaperisdevotedtostudyofexistencecomparisontheemsofviscositysolutionsofDirichletproblemofsecondderparabolicequations.InChapterOnetheauthbri

2、eflyreviewshistyofviscositysolutionsthensetsfththeproblemsdiscussedinthispaper.InChapterTwothefollowingDirichletproblemisconsidered2()0()0(0)()tuFtxuDuDuutxuxx??=?=??=?()(0)()[0)TlTtxQTQtxQTQxQ∈=∈?=?∈(Ⅰ)UsingPerron’smeth

3、odthetheemisproved.Theem:Supposethatthereisasubsolutionuasupersolutionuof(Ⅰ)thatsatisfy()()0(0)(0)()utxutxuxuxx??????==???==?()lTtxQxQ∈?∈Then()sup():wtxutxuuu=≤≤uisasubsolutionof(Ⅰ)isasolutionof(Ⅰ).Remark:Fdefinitionsofu

4、?u?seeTheem2.3.1ofP14.Meoverexamplesofexistenceofviscositysolutionsarepresented.InChapterThreetheauthconsidersthefollowingDirichletproblem:2[()]()()(0)()()()(0)tTPTuTratxDuHtxuDutxQtQutxtxtxQQtQψ??∈=??=∈?=∪???(Ⅱ)Theauthp

5、rovesthecomparisontheemf(Ⅱ).Theem:AssumethatHsatisfiessomeproperconditions(SeeTheem3.2.1ofP21).Ifuisaboundeduscsubsolutionof(Ⅱ)ifvisaboundedlscsupersolutionof(Ⅱ)then()TuvtxQ≤∈MeoverifuvonTQaredefinedbysetting1()()()limsu

6、p()()()()()()TdsytxsyQusytxutxtxutxutxotherwiseψ→∈?∈∑≤?=???∩2()()()liminf()()()()()()TdsytxsyQvsytxvtxtxvtxvtxotherwiseψ→∈?∈∑≥?=???∩thenuvarestillanuscsubsolutionalscsupersolutionof(Ⅱ)respectively()TuvtxQ≤∈Meoverusingthe