Regularity of Weak Solutions near Singular Minimizing Cones for the One-Phase Free Boundary Problem
Abstract
Alt and Caffarelli proved an Allard-type ε-regularity theorem near the flat one-homogeneous minimizer of the one-phase free boundary problem. A natural follow-up question is whether an analogous regularity result holds near singular minimizing cones. In this paper, we prove such an ε-regularity theorem near a strongly integrable one-phase minimizing cone with an isolated singularity. More precisely, we show that, after a suitable translation and rotation, the free boundary of any non-degenerate weak solution sufficiently close to the cone is graphical over the free boundary of a leaf of the De Silva--Jerison--Shahgholian foliation. This result is analogous to recent ε-regularity theorems in minimal surface theory by Edelen and by Edelen--Spolaor; our proof adapts the strategy of Edelen to the one-phase setting.
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BibTeX
@misc{jeong2026regularity,
title = {{Regularity of Weak Solutions near Singular Minimizing Cones for the One-Phase Free Boundary Problem}},
author = {Minyoung Jeong},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08688},
eprint = {2611.08688},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08688},
}
RIS
TY - UNPB
TI - Regularity of Weak Solutions near Singular Minimizing Cones for the One-Phase Free Boundary Problem
AU - Minyoung Jeong
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08688
UR - https://arxiv.org/abs/2611.08688
AB - Alt and Caffarelli proved an Allard-type ε-regularity theorem near the flat one-homogeneous minimizer of the one-phase free boundary problem. A natural follow-up question is whether an analogous regularity result holds near singular minimizing cones. In this paper, we prove such an ε-regularity theorem near a strongly integrable one-phase minimizing cone with an isolated singularity. More precisely, we show that, after a suitable translation and rotation, the free boundary of any non-degenerate weak solution sufficiently close to the cone is graphical over the free boundary of a leaf of the De Silva--Jerison--Shahgholian foliation. This result is analogous to recent ε-regularity theorems in minimal surface theory by Edelen and by Edelen--Spolaor; our proof adapts the strategy of Edelen to the one-phase setting.
ER -