Existence and representation of solutions of parabolic equations

Abstract:

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