hilbert space methods for partial differential equations

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1、HilbertSpaceMethodsforPartialDierentialEquationsR.E.ShowalterElectronicJournalofDierentialEquationsMonograph01,1994.iPrefaceThisbookisanoutgrowthofacoursewhichwehavegivenalmostpe-riodicallyoverthelasteightyears.Itisaddressedtobeginninggraduatestudentsofmathematic

2、s,engineering,andthephysicalsciences.Thus,wehaveattemptedtopresentitwhilepresupposingaminimalbackground:thereaderisassumedtohavesomeprioracquaintancewiththeconceptsofin-ear"andcontinuous"andalsotobelieveL2iscomplete.AnundergraduatemathematicstrainingthroughLebesgu

3、eintegrationisanidealbackgroundbutwedarenotassumeitwithoutturningawaymanyofourbeststudents.Theformalprerequisiteconsistsofagoodadvancedcalculuscourseandamotivationtostudypartialdierentialequations.Aproblemiscalledwell-posedifforeachsetofdatathereexistsexactlyoneso

4、lutionandthisdependenceofthesolutiononthedataiscontinuous.Tomakethisprecisewemustindicatethespacefromwhichthesolutionisobtained,thespacefromwhichthedatamaycome,andthecorrespond-ingnotionofcontinuity.Ourgoalinthisbookistoshowthatvarioustypesofproblemsarewell-posed.T

5、heseincludeboundaryvalueproblemsfor(stationary)ellipticpartialdierentialequationsandinitial-boundaryvalueproblemsfor(time-dependent)equationsofparabolic,hyperbolic,andpseudo-parabolictypes.Also,weconsidersomenonlinearellipticboundaryvalueproblems,variationaloruni-

6、lateralproblems,andsomemethodsofnumericalapproximationofsolutions.Webrieflydescribethecontentsofthevariouschapters.ChapterIpresentsalltheelementaryHilbertspacetheorythatisneededforthebook.ThersthalfofChapterIispresentedinaratherbrieffashionandisin-tendedbothasarevi

7、ewforsomereadersandasastudyguideforothers.Non-standarditemstonoteherearethespacesCm(G),V,andV0.TherstconsistsofrestrictionstotheclosureofGoffunctionsonRnandthelasttwoconsistofconjugate-linearfunctionals.ChapterIIisanintroductiontodistributionsandSobolevspaces.Th

8、elatteraretheHilbertspacesinwhichweshallshowvariousproblemsarewell-posed.Weuseaprimitive(andnon-standard)notionofdistributionwhichisadequateforou