Two Results Relating an $L^p$ Regularity Condition and the $L^q$ Dirichlet Problem for Parabolic Equations

Abstract

We consider variations and generalizations of the initial Dirichlet problem for linear second order divergence form equations of parabolic type, with vanishing initial values and non-continuous lateral data, in the setting of Lipschitz cylinders. More precisely, lateral data in adequations of the Lebesgue classes $L^p$, and a family of Sobolev-type classes are considered. We also establish some basic connections between estimates related to solvability of each of these problems. This generalizes some of the well-known works for Laplace's equation, heat equation and some linear elliptic-type equations of second order.