Optimal regularity for nonlocal elliptic equations and free boundary problems
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917611735875584 |
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| author | Ros-Oton, Xavier Weidner, Marvin |
| author_facet | Ros-Oton, Xavier Weidner, Marvin |
| contents | In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal regularity for nonlocal elliptic equations and free boundary problems Ros-Oton, Xavier Weidner, Marvin Analysis of PDEs 47G20, 35B65, 31B05, 60J75, 35K90 In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates. |
| title | Optimal regularity for nonlocal elliptic equations and free boundary problems |
| topic | Analysis of PDEs 47G20, 35B65, 31B05, 60J75, 35K90 |
| url | https://arxiv.org/abs/2403.07793 |