hilbert space methods for partial differential equations - r. showalter

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1、HilbertSpaceMethodsforPartialDierentialEquationsR.E.ShowalteriPrefaceThisbookisanoutgrowthofacoursewhichwehavegivenalmostpe-riodicallyoverthelasteightyears.Itisaddressedtobeginninggraduatestudentsofmathematics,engineering,andthephysicalsciences.Thus,wehav

2、eattemptedtopresentitwhilepresupposingaminimalbackground:thereaderisassumedtohavesomeprioracquaintancewiththeconceptsoflin-2ear"andcontinuous"andalsotobelieveLiscomplete.AnundergraduatemathematicstrainingthroughLebesgueintegrationisanidealbackgroundbutwe

3、darenotassumeitwithoutturningawaymanyofourbeststudents.Theformalprerequisiteconsistsofagoodadvancedcalculuscourseandamotivationtostudypartialdierentialequations.Aproblemiscalledwell-posedifforeachsetofdatathereexistsexactlyonesolutionandthisdependenceofth

4、esolutiononthedataiscontinuous.Tomakethisprecisewemustindicatethespacefromwhichthesolutionisobtained,thespacefromwhichthedatamaycome,andthecorrespond-ingnotionofcontinuity.Ourgoalinthisbookistoshowthatvarioustypesofproblemsarewell-posed.Theseincludeboundar

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

6、oblems,andsomemethodsofnumericalapproximationofsolutions.Webrie ydescribethecontentsofthevariouschapters.ChapterIpresentsalltheelementaryHilbertspacetheorythatisneededforthebook.ThersthalfofChapterIispresentedinaratherbrieffashionandisin-tendedbothasarevi

7、ewforsomereadersandasastudyguideforothers.m0Non-standarditemstonoteherearethespacesC(G),V,andV.ThenrstconsistsofrestrictionstotheclosureofGoffunctionsonRandthelasttwoconsistofconjugate-linearfunctionals.ChapterIIisanintroductiontodistributionsandSobolev

8、spaces.ThelatteraretheHilbertspacesinwhichweshallshowvariousproblemsarewell-posed.Weuseaprimitive(andnon-standard)notionofdistributionwhichisadequateforourpurposes.Ourdistributionsareconjugate-linearandhaveth