Estimates of Caloric Measure and the Initial-Dirichlet Problem for the Heat Equation in Lipschitz Cylinders

Abstract

In this paper the authors prove unique solvability of the initial-Dirichlet problem for the heat equation in a cylindrical domain with Lipschitz base, lateral data in ${L^p},p \geqslant 2$, and zero initial values. A Poisson kernel for this problem is shown to exist with the property that its ${L^2}$-averages over parabolic rectangles are equivalent to ${L^1}$-averages over the same sets.