Degenerate diffusion [Elektronische Ressource] : behaviour at the boundary and kernel estimates / vorgelegt von Michal Chovanec

DEGENERATEDIFFUSIONBehaviour at the boundary andkernel estimatesMichal ChovanecApril 2010DEGENERATE DIFFUSIONBehaviour at the boundary and kernel estimatesDissertation zur Erlangung des Doktorgrades Dr. rer. nat. der Fakult at fur Math-ematik und Wirtschaftswissenschaften der Universit at Ulmvorgelegt von Michal Chovanec aus Bansk a BystricaGutachter: Prof. Dr. Wolfgang ArendtProf. Dr. Werner KratzProf. Tom ter Elst, PhD (Auckland)Dekan: Prof. Dr. Werner KratzTag der Promotion: 31. Mai 2010Abstract: We study evolution equations of the form:@u(t;x) =m(x)(4u)(t;x) t2R ; x2 ;+@tNwhere is a bounded domain in R and the function m : ! (0;1) is assumedto be measurable. Dirichlet boundary conditions are posed. We investigate underwhich conditions on m and @ the operator m4 generates a strongly continuoussemigroup on C ( ). In the second part of the thesis we obtain various estimates0pon the kernel of the semigroup generated by m4 on weighted L -spaces.ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Preliminaries 91.1 Sesquilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Associated operator, fundamentals of the semigroup theory . . . . . 111.3 Holomorphic semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Beurling-Deny conditions . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Irreducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .