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Regularity of soda-can domains for the p-parabolic equation
Abstract
We classify the regularity of the origin for the p-parabolic equation in the soda-can domains Θl,θ={(x,t)∈ℝn×ℝ:0 < −t < θ|x|l < θ}, where l,θ>0. The classification covers every p>1 in dimensions n≥2. At the endpoint l=p, the origin is regular if and only if p≥2n/(n+1). We also prove that the origin admits a traditional barrier for every p>2 and l>0. When 2 < p < n and l < p, the origin is nevertheless irregular. This answers negatively the well-known open question whether one barrier suffices to characterize boundary regularity in the degenerate range p>2.
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BibTeX
@misc{avelin2026regularity,
title = {{Regularity of soda-can domains for the $p$-parabolic equation}},
author = {Benny Avelin and Mikko Parviainen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06497},
eprint = {2610.06497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06497},
}
RIS
TY - UNPB
TI - Regularity of soda-can domains for the p-parabolic equation
AU - Benny Avelin
AU - Mikko Parviainen
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06497
UR - https://arxiv.org/abs/2610.06497
AB - We classify the regularity of the origin for the p-parabolic equation in the soda-can domains Θ_l,θ={(x,t)∈ℝ^n×ℝ:0 < −t < θ|x|^l < θ}, where l,θ>0. The classification covers every p>1 in dimensions n≥2. At the endpoint l=p, the origin is regular if and only if p≥2n/(n+1). We also prove that the origin admits a traditional barrier for every p>2 and l>0. When 2 < p < n and l < p, the origin is nevertheless irregular. This answers negatively the well-known open question whether one barrier suffices to characterize boundary regularity in the degenerate range p>2.
ER -