Solvability of the $L^p$ Dirichlet problem for the heat equation implies parabolic uniform rectifiability

TL;DR

This paper proves that the solvability of the $L^p$ Dirichlet problem for the heat equation implies that the boundary of the domain has a specific geometric regularity called parabolic uniform rectifiability, under minimal assumptions.

Contribution

It establishes that $L^p$ solvability of the heat equation's Dirichlet problem necessitates parabolic uniform rectifiability of the boundary, linking PDE solvability to geometric measure theory.

Findings

Solvability of the $L^p$ Dirichlet problem implies parabolic uniform rectifiability.

Weak-$A_ abla$ condition on caloric measure is sufficient for rectifiability.

Boundary regularity is characterized by geometric conditions in the parabolic setting.

Abstract

Let $\Omega \subset \mathbb{R}^{n+1}$ be an open set in space-time with boundary $\Sigma = \partial \Omega$. Under minimal and natural background assumptions - namely, that $\Sigma$ is time-symmetrically parabolic Ahlfors--David regular and that $\Omega$ satisfies an interior corkscrew condition - we treat a one-phase parabolic free boundary problem which establishes the necessity of parabolic uniform rectifiability for $L^p(d\sigma)$ solvability of the Dirichlet problem for the heat equation. More precisely, we prove that if the caloric measure associated with $\Omega$ satisfies a weak-$A_\infty$ condition with respect to the surface measure $\sigma = \mathcal{H}_{\mathrm{par}}^{n+1}\!\lfloor_{\Sigma}$, then $\Sigma$ is parabolically uniformly rectifiable, hence equivalently, that solvability of the Dirichlet problem for the heat (or adjoint heat) equation in $\Omega$ with boundary data in $L^p(d\sigma)$, for some $p \in (1,\infty)$, implies parabolic uniform rectifiability. Our main theorem thus identifies parabolic uniform rectifiability as the correct geometric framework for boundary regularity, and $L^p$ solvability, in the parabolic setting.