Treść książki
Przejdź do opcji czytnikaPrzejdź do nawigacjiPrzejdź do informacjiPrzejdź do stopki
8
Preface
canfindthedescriptionwhichcontainsthecharacterizationofthescaleof
Banachspacesandthescaleofdualspaces.
Theobjectofthelastpartistoshowtheapplicationsofthetheorygiven
beforeinspecificproblems.Weintroducetheclassofsectorialoperators,
whichwerewidelyconsideredi.a.byT.Kato,H.TanabeandD.Henry,
H.Amann,A.Lunardi.Theconnectionofthisclasswiththeoperatorsof
thetype(ωjM)isanalysedheretoo.Itisalsopossibletofindinthispart
somepropertiesofthescaleofBanachspaces,definedforsectorialoperators.
Finally,wedealwiththeinvestigationoftheexistenceandsmoothnessof
thesolutionsofsomeCauchy’sproblems,consideredondifferentspaceson
thescaleofBanachspaces.
Thestructureofthebookisasfollows:
Chapter1:introducesthedefinitionoffractionalpowersofoperatorand
describestheirbasicproperties.
Chapter2:firstdealswiththespacesDσ
panddescribestheirproperties
andshowsthattheycoincidewithsomeinterpolationspaces.Next,the
classicalapproachtothetheoryofinterpolationspacesisgivenanddiffer-
entmethodsofintroducingtheseinterpolationspaces,e.g.K-methodand
tracemethodofrealinterpolationandcomplexmethodofinterpolationare
presented.Inthischaptercanalsobefoundthedefinitionoftheoperator
ofthetype(ωjM(θ)).
Chapter3:thisshortchaptergivesthecharacterizationofthedomains
offractionalpowersofoperatorthroughinfinitesimalgeneratorsofbounded
semi-groupsorboundedanalyticsemi-groups.
Chapter4:thiscrucialchaptercontainstwosections.Thefirstone
considersinductivelimitsandprojectivelimitsofsequencesofBanachspaces
andtheirproperties.Thesecondoneshowsconstructionofthescaleof
Banachspacesforthelinearoperatorofthetype(ωjM(θ)),whichresolvent
setcontains0.NextthecharacterizationofthescaleofBanachspacesis
discussed.
Chapter5:theaimofthischapteristointroduceafewexamplesofthe
scalesoftheBanachspaces.Oneoftheexamplesleadstodefinitionofthe
fractionalSobolevspaces,whichareusefulspacesconsideredinthetheory
ofPartialDifferentialEquations.
Chapter6:presentssectorialoperators,describestheirpropertiesand
givessomebasicexamplesofsuchoperators.
Chapter7:isdevotedtosomeapplicationsofthetheoriesgivenbefore.
Atthebeginningweconsidertheoperatorsdefinedondifferentlevelsofthe
scale,thatistosayfortheoperatorA:D(A)⊆X→Xweconsiderits
