Sobolev Imbedding Theorems

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1、SobolevImbeddingTheoremsA.K.Pichler-Tennenberg¤UniversityofBristolJuly1997-March1998¤PartlysupportedbytheNu±eldFoundation.TheAuthorwishestothankV.A.Liskevichforhissupervisionandsupportthroughoutthewritingofthisproject.1Contents1Introduction31.1Preliminaries.......................

2、.......42WeakDerivativesandSobolevSpaces102.1LocalLebesguespaces.........................102.2Multi-indicesandweakderivatives...................102.3Thechainrule..............................122.4Sobolevspaces.............................163N-functionsandOrliczSpaces173.1N-functi

3、ons...............................173.2Orliczspaces..............................224RearrangementsandLorentzSpaces274.1Thedistributionfunctionandrearrangements............274.2Rearrangementinequalities.......................314.3LorentzSpaces.............................365Sobole

4、vImbeddingTheorems415.1Sobolev'stheorem............................415.2LimitingcaseofSobolev'stheorem..................516FurtherSobolevTypeImbeddings676.1RegularityestimatesforPDEsolutions................676.2ImbeddingsintoLorentzspaces....................726.3Furtherresults...

5、..........................7721IntroductionSobolev'simbeddingtheoremwas¯rstprovedbytheRussianmathematicianSergeiSobolev[27]in1938,andnumerousextensionsandimprovementshavebeenmadetoitsincethen-notablybyTrudinger[33],Moser[22]andAlvino[3].Ithasnumerouspracticalapplications-particula

6、rlyinprovidingregularityestimatesforsolutionsofpartialdi®erentialequations.Indeedthisisoneofthemainreasonsthatthisareaisstudied.InSobolev'stheorem,wearegivenameasurablefunction,uon•(aboundeddomain-anopenconnectedset-inRn)suchthatbothuandallofitspartialderivatives(takeninaweakerse

7、nse-describedlater-thantheclassicalone)oforderlessthanorequaltokareinLp(•).Wearealsogiventhatumaybeap-proximatedbycompactlysupportedk-timescontinuouslydi®erentiablefunctionson•.Fromthis,we¯ndcertainimprovedregularityestimatesforu.Thetypeofregularityestimateobtainedturnsouttodepen

8、donthevalueofkp:i)Ifkp