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

TL;DR

The paper addresses the existence of the parabolic measure and the solvability of both the continuous and Hölder Dirichlet problems for parabolic divergence-form operators with bounded coefficients on possibly non-cylindrical domains. It introduces the time-backwards capacity density condition (TBCDC) and the time-backwards Hausdorff content condition (TBHCC), proving that TBHCC implies TBCDC and yields parabolic measure existence and Hölder Dirichlet well-posedness, while also establishing a parabolic Wiener criterion. A Bourgain-type nondegeneracy estimate and Hölder decay up to the boundary are developed, leading to Hölder solvability; existence of the parabolic measure is then proved for general bounded-coefficient operators via approximation by smoother operators, including extensions to unbounded domains and boundary behavior at infinity. The results generalize prior work on parabolic measures and Dirichlet problems, providing operator-robust, geometry-driven criteria (TBHCC/TBCDC) and clarifying when the parabolic measure is a probability, with implications for non-cylindrical domains 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 Hölder 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.

Figures (6)

• Figure 2.1: In a rectangular/cylindrical domain, it only makes sense to prescribe boundary data on the red part of the boundary (initial values) and on the blue parts (lateral/boundary values). It does not make sense to prescribe boundary data on the black part because it has no influence on $\Omega$. Indeed, there is no part of $\Omega$ lying immediately to its right (future).
• Figure 2.2: A more general, and non-cylindrical, space-time domain: there are different kinds of vertical faces ($\mathcal{B}\Omega$, $\partial_s \Omega$ and $\partial_{ss}\Omega$), and even the point at infinity may belong to the boundary if $\Omega$ is unbounded.
• Figure 3.1: Because of the parabolic scaling, the big slab $\Phi_{ar}$ can be subdivided in smaller cubes ${\mathbf{Q}}_{ar}^i$ of the same size: one needs as many as to cover the front-most face $Q_r(0) \times \{0\}$.
• Figure 6.1: The boundary of $\Omega_R = \Omega \cap {\mathbf{Q}}_R(\mathbf{0})$ is divided into several lateral and vertical regions.
• Figure 6.2: The solutions $u_R$ grow (by the maximum principle) as $R$ grows.

Theorems & Definitions (75)

• Theorem 1.2: Bourgain's estimate and Hölder continuity up to the boundary
• Remark 1.5
• Remark 1.6
• Theorem 1.7: Well-posedness of the Hölder-Dirichlet problem
• Theorem 1.9: Existence of parabolic measure
• Remark 1.10
• Corollary 1.11
• Proposition 1.12: TBHCC$\implies$TBCDC for all $L$
• Corollary 1.13
• Remark 1.14