Two-phase problems: Perron solutions and regularity of the Neumann problem in convex cones
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| Format: | Preprint |
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2024
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| _version_ | 1866929439722438656 |
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| author | Beck, Thomas De Silva, Daniela Savin, Ovidiu |
| author_facet | Beck, Thomas De Silva, Daniela Savin, Ovidiu |
| contents | We investigate a fully nonlinear two-phase free boundary problem with a Neumann boundary condition on the boundary of a general convex set $K \subset \mathbb{R}^n$ with corners. We show that the interior regularity theory developed by Caffarelli for the classical two-phase problem in his pioneer works \cite{C1,C2}, can be extended up to the boundary for the Neumann boundary condition under very mild regularity assumptions on the convex domain $K$. To start, we establish a general existence theorem for the Dirichlet two-phase problem driven by two different fully nonlinear operators, which is a result of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19538 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-phase problems: Perron solutions and regularity of the Neumann problem in convex cones Beck, Thomas De Silva, Daniela Savin, Ovidiu Analysis of PDEs We investigate a fully nonlinear two-phase free boundary problem with a Neumann boundary condition on the boundary of a general convex set $K \subset \mathbb{R}^n$ with corners. We show that the interior regularity theory developed by Caffarelli for the classical two-phase problem in his pioneer works \cite{C1,C2}, can be extended up to the boundary for the Neumann boundary condition under very mild regularity assumptions on the convex domain $K$. To start, we establish a general existence theorem for the Dirichlet two-phase problem driven by two different fully nonlinear operators, which is a result of independent interest. |
| title | Two-phase problems: Perron solutions and regularity of the Neumann problem in convex cones |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.19538 |