Regularity of Lipschitz free boundaries for weak solutions of Alt-Caffarelli type problems
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866912854806888448 |
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| author | Domingo-Pasarin, Joan Ros-Oton, Xavier |
| author_facet | Domingo-Pasarin, Joan Ros-Oton, Xavier |
| contents | Motivated by the Serrin problem, we study weak solutions of the generalised Alt-Caffarelli problem $-Δu = f$ in $Ω$, $u = 0$ on $\partialΩ$, $\partial_νu = Q$ on $\partialΩ$. Our main result establishes that if $Ω$ is Lipschitz, then it is actually $C^{\infty}$ (provided that $f$ and $Q$ are smooth). This was known before only for viscosity solutions. As a corollary, we obtain an alternative solution of Serrin's problem in the case of Lipschitz domains. We also discuss the characterisation of the regularity of Lipschitz domains in terms of their Poisson kernel. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_20493 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Regularity of Lipschitz free boundaries for weak solutions of Alt-Caffarelli type problems Domingo-Pasarin, Joan Ros-Oton, Xavier Analysis of PDEs 35R35, 35B65, 31B05 Motivated by the Serrin problem, we study weak solutions of the generalised Alt-Caffarelli problem $-Δu = f$ in $Ω$, $u = 0$ on $\partialΩ$, $\partial_νu = Q$ on $\partialΩ$. Our main result establishes that if $Ω$ is Lipschitz, then it is actually $C^{\infty}$ (provided that $f$ and $Q$ are smooth). This was known before only for viscosity solutions. As a corollary, we obtain an alternative solution of Serrin's problem in the case of Lipschitz domains. We also discuss the characterisation of the regularity of Lipschitz domains in terms of their Poisson kernel. |
| title | Regularity of Lipschitz free boundaries for weak solutions of Alt-Caffarelli type problems |
| topic | Analysis of PDEs 35R35, 35B65, 31B05 |
| url | https://arxiv.org/abs/2601.20493 |