Sobolev不等式中的最佳常數(shù)及其達到函數(shù).pdf

1、華中師范大學(xué)碩士學(xué)位論文Sobolev不等式中的最佳常數(shù)及其達到函數(shù)姓名：王丹申請學(xué)位級別：碩士專業(yè)：應(yīng)用數(shù)學(xué)指導(dǎo)教師：鄧引斌20090501⑧碩士學(xué)位論文MASTER’STHESISAbstractSobolevinequalitiesalsocalledSobolevimbeddingtheorems，aleverypopularamongwritersinpartialdifferentialequationsorinthecal

2、culusofvariationsInthispaper，weconsiderthefollowingSobolevimbeddinginequalities，HardySobolevinqualitiesandCagaxelliNirenbergSobolevinequalities：(／lul譬出)；≤s(磁Q)／I(△)鼉ui2如，tt∈日詈(Rn)，JRJR’where00fn，q2磊2n(厶群刪；≤c(厶1馴p如)；，讓∈艫啦

3、’，where1P佗，0≤sPandP≤g=元Fg二；8p(／lz|一如I訓(xùn)p出)；≤Q囊／Ixl一2nID讓12dz，牡∈曙(Rn)，，Rn，R“(1)(2)(3)where—ooD孚，口≤b≤凸1，P=i互害赫Remarks：Wewillpreciselydefine(一△)等insection2Usingtheknownresults，weshallsummarizetheexistenceandthepropertiesofth

4、eextremalfunctionsfortheaboveinequalitiesWeshallsummarizetheradialsymmetricoftheextremalfunctionsbythesphericallysymmetricrearrangement(alsocalledSchwarzsymmetricrearrangement)，andsummarizethepreciseexpressionsoftheextre

5、mal／unctionsandthebestconstantsbythepolarcoordinatetransformationorbytheFouriertran_sformation)MoreoverwesummerizethesolutionsofthecorrespondingpartialdifferentialequationsWhenwediscusstheexistenceofsolutionsforthepartia

6、ldifferentialequations，weusuallyrisetheMount&inPassTheoremIntheendofthispaper，wewillgivesomeexamplesfortheapplicationoftheextremalfunctionsThemainresultsinthispapermainlycomekomf11’【9j’[61，Is]，f15J，【18l，【51Keyword金Sobole