On the regularity problem for parabolic operators and the role of half-time derivative

TL;DR

This paper establishes regularity results for solutions of second order parabolic PDEs on complex domains, linking half-time derivatives of boundary data to interior solution regularity, thus advancing the solvability of the $L^p$ parabolic Regularity problem.

Contribution

It introduces a new regularity criterion involving half-time derivatives, enabling analysis of parabolic equations on more general time-varying domains.

Findings

Regularity of solutions is characterized via half-time derivatives.

Solutions' non-tangential maximal functions belong to $L^p$ under certain conditions.

The results facilitate solving the $L^p$ parabolic Regularity problem on complex domains.

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 $\Omega=\mathcal O\times\mathbb R$ 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 $\Omega$ and the non-tangential maximal function of its gradient in spatial directions $\tilde{N}(\nabla u)$ belongs to $L^p(\partial\Omega)$ for some $p>1$. Furthermore, assume that for $u|_{\partial\Omega}=f$ we have that $D^{1/2}_tf\in L^p(\partial\Omega)$. Then both $\tilde{N}(D^{1/2}_t u)$ and $\tilde{N}(D^{1/2}_tH_t u)$ also belong to $L^p(\partial\Omega)$, 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.