Existence and representation of solutions of parabolic equations

Abstract

Let L L be a linear, second order parabolic operator in divergence form and let Q Q be a bounded cylindrical domain in E n + 1 {E^{n + 1}} . Let ∂ p Q {\partial _p}Q denote the parabolic boundary of Q Q . To each continuous function f f on ∂ p Q {\partial _p}Q there is a unique solution u u of the boundary value problem L u = 0 Lu = 0 in Q , u = f Q,u = f on ∂ p Q {\partial _p}Q . Moreover, for the given L L and Q Q , to each ( x , t ) ∈ Q (x,t) \in Q there is a unique nonnegative measure μ ( x , t ) {\mu _{(x,t)}} with support on ∂ p Q {\partial _p}Q such that the solution of the boundary value problem is given by u ( x , t ) = ∫ ∂ p Q f d μ ( x , t ) u(x,t) = \int _{{\partial _p}Q} {fd{\mu _{(x,t)}}} .