On the regularity problem for parabolic operators and the role of half-time derivative
Martin Dindoš
TL;DR
The paper addresses the parabolic $L^p$ Regularity problem for second-order equations on time-varying domains by showing that half-time regularity follows from spatial gradient control. The authors develop a reinforced weak solution framework on $\Omega = \mathcal{O} \times \mathbb{R}$ and establish a detailed parabolic non-tangential maximal function analysis, tying $D^{1/2}_t u$ and $D^{1/2}_t H_t u$ to $\tilde{N}(\nabla u)$ and the boundary data via $L^p$ estimates. A key contribution is extending well-posedness results from Lipschitz cylinders to the broader class of uniform time-varying domains by adapting corkscrew-region geometry and projecting arguments to time-slices, leveraging DPext-type techniques. The findings pave the way for solvability of the parabolic Regularity problem on large time-varying domains, enabling new applications in parabolic boundary value problems with minimal boundary regularity.
Abstract
In this paper we present the following result on regularity of solutions of the second order parabolic equation $\partial_t u - \mbox{div} (A \nabla u)+B\cdot \nabla u=0$ on cylindrical domains of the form $Ω=\mathcal O\times\mathbb R$ where $\mathcal O\subset\mathbb R^n$ is a is a uniform domain (it satisfies both interior corkscrew and Harnack chain conditions) and has a boundary that is $n-1$-Ahlfors regular. Let $u$ be a solution of such PDE in $Ω$ and the non-tangential maximal function of its gradient in spatial directions $\tilde{N}(\nabla u)$ belongs to $L^p(\partialΩ)$ for some $p>1$. Furthermore, assume that for $u|_{\partialΩ}=f$ we have that $D^{1/2}_tf\in L^p(\partialΩ)$. Then both $\tilde{N}(D^{1/2}_t u)$ and $\tilde{N}(D^{1/2}_tH_t u)$ also belong to $L^p(\partialΩ)$, where $D^{1/2}_t$ and $H_t$ are the half-derivative and the Hilbert transform in the time variable, respectively. We expect this result will spur new developments in the study of solvability of the $L^p$ parabolic Regularity problem as thanks to it it is now possible to formulate the parabolic Regularity problem on a large class of time-varying domains.