New Homogeneous Solutions for the One-Phase Free Boundary Problem
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915489947582464 |
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| author | Hines, Coleman Kolesar, James McGrath, Peter |
| author_facet | Hines, Coleman Kolesar, James McGrath, Peter |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09409 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New Homogeneous Solutions for the One-Phase Free Boundary Problem Hines, Coleman Kolesar, James McGrath, Peter Analysis of PDEs Differential Geometry 35R35 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. |
| title | New Homogeneous Solutions for the One-Phase Free Boundary Problem |
| topic | Analysis of PDEs Differential Geometry 35R35 |
| url | https://arxiv.org/abs/2509.09409 |