PreprintCC0arXiv
Wiener's criterion and resolutivity for the p-parabolic equation
Abstract
We characterize boundary regularity for ut=Δp u in arbitrary bounded open subsets of ℝn+1 for n≥2 and p∈(2n/(n+1),2)∪ (2,∞). 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≥ 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|<C(−t)1/p for every C>0. For 2<p<n, examples show that divergence at every sufficiently large height does not imply regularity.
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BibTeX
@misc{avelin2026wieners,
title = {{Wiener's criterion and resolutivity for the $p$-parabolic equation}},
author = {Benny Avelin and Mikko Parviainen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26466},
eprint = {2610.26466},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26466},
note = {CC0},
}
RIS
TY - UNPB
TI - Wiener's criterion and resolutivity for the p-parabolic equation
AU - Benny Avelin
AU - Mikko Parviainen
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26466
UR - https://arxiv.org/abs/2610.26466
AB - We characterize boundary regularity for u_t=Δ_p u in arbitrary bounded open subsets of ℝ^n+1 for n≥2 and p∈(2n/(n+1),2)∪ (2,∞). 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≥ 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|<C(−t)^1/p for every C>0. For 2<p<n, examples show that divergence at every sufficiently large height does not imply regularity.
ER -