The parabolic Dirichlet problem with continuous and Hölder boundary data, and rough coefficients

Abstract: We provide very mild sufficient conditions for space-time domains (non-necessarily cylindrical) which ensure that the continuous Dirichlet problem and the H\"older Dirichlet problem are well-posed, for any parabolic operator in divergence form with merely bounded coefficients. Concretely, we show that the parabolic measure exists, even for unbounded domains, hence solving an open problem posed by Genschaw and Hofmann (2020). This problem has inherent difficulties because of its parabolic nature, as the behavior of solutions near the boundary may depend strongly on the values of the coefficients of the operator. One of our sufficient conditions, the time-backwards capacity density condition, is a quantitative version of the parabolic Wiener's criterion, and hence is adapted to the operator under consideration. The other condition, the time-backwards Hausdorff content condition, is (albeit slightly stronger) purely geometrical and independent of the operator, hence much easier to check in practice.