The $L^p$ regularity problem for parabolic operators

TL;DR

This paper establishes the solvability of the Regularity problem for certain parabolic PDEs with measurable coefficients under Carleson conditions, extending known results from elliptic to parabolic cases.

Contribution

It fully resolves the solvability of the parabolic Regularity problem under Carleson conditions, including cases with small norms, using a novel approach inspired by elliptic problem solutions.

Findings

Existence of a range (1, p_0) where the Regularity problem is solvable.

Solution applies to parabolic PDEs with elliptic, bounded, measurable coefficients.

Addresses challenges unique to the parabolic setting.

Abstract

In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + \mbox{div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a natural Carleson condition (a parabolic analog of the so-called DKP-condition). We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known even in the small Carleson case, that is, when the Carleson norm of coefficients is sufficiently small. In the elliptic case the analogous question was only fully resolved recently independently by two groups, with two very different methods: one involving two of the authors and S. Hofmann, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges.