Wiener's criterion and resolutivity for the $p$-parabolic equation

Abstract

We characterize boundary regularity for $u_t=Δ_p u$ in arbitrary bounded open subsets of $\mathbb{R}^{n+1}$ for $n\ge2$ and $p\in(2n/(n+1),2)\cup (2,\infty)$. We prove equivalent Wiener criteria in terms of capacitary potentials on prescribed cylinders and in terms of intrinsic capacities. Both require divergence at every positive obstacle height. Independently, we prove Perron resolutivity for continuous data on the full topological boundary of every nonempty bounded open set for all $n\geq 1$ and $p>1$. As an application of the Wiener criterion, we prove a Petrovski\uı-type criterion in the singular range, which gives regularity of the origin for $|x| 0$. For $2<p<n$, examples show that divergence at every sufficiently large height does not imply regularity.