arXiv:2611.08688v1 Announce Type: new Abstract: Alt and Caffarelli proved an Allard-type $\epsilon$-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 $\epsilon$-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 $\epsilon$-regularity theorems in minimal surface theory by Edelen and by Edelen--Spolaor; our proof adapts the strategy of Edelen to the one-phase setting.
Continue to arXiv Mathematics →You are being forwarded in 6s — stay on this page