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arXiv · 2611.08688

Regularity of Weak Solutions near Singular Minimizing Cones for the One-Phase Free Boundary Problem

Minyoung Jeong
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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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Minyoung Jeong. 2026-11-05. Regularity of Weak Solutions near Singular Minimizing Cones for the One-Phase Free Boundary Problem. https://arxiv.org/abs/2611.08688

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