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

Abstract

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