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arXiv · 2509.09409

New Homogeneous Solutions for the One-Phase Free Boundary Problem

Coleman Hines·James Kolesar·Peter McGrath
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Abstract

For each sufficiently large integer $k$, we construct a domain in the round $2$-sphere with $k$ boundary components which is the link of a cone in $\mathbb{R}^3$ admitting a homogeneous solution to the one-phase free boundary problem. This answers a question of Jerison-Kamburov, and also disproves a conjecture of Souam left open in earlier work. The method exploits a new connection with minimal surfaces, which we also use to construct an infinite family of homogeneous solutions in dimension four.

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Coleman Hines, James Kolesar, Peter McGrath. 2025-09-11. New Homogeneous Solutions for the One-Phase Free Boundary Problem. https://arxiv.org/abs/2509.09409

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