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 the domain $\mathbb R^{n+1}_+\times\mathbb R$ 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 are independent of the spatial variable $x_{n+1}$ (which is transversal to the boundary). We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. An analogous result for the Dirichlet problem has been considered earlier by Auscher, Egert and Nyström, however the Regularity problem represents an additional step up in difficulty. In the elliptic case, the analog of the question considered here was resolved for both Dirichlet and Regularity problems by Hofmann, Kenig, Mayboroda and Pipher.
The main result of this paper complements a recent work of two of the authors with L. Li showing solvability of the parabolic Regularity problem for data in some $L^p$ spaces when the coefficients satisfy a natural Carleson condition (which is a parabolic analog of the so-called DKP-condition).
From: Martin Dindoš [view email]
[v1]
Mon, 8 Sep 2025 12:43:19 UTC (36 KB)
[v2]
Tue, 23 Jun 2026 15:53:45 UTC (39 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。