Higher regularity in nonlocal free boundary problems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918105545965568 |
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| author | Barrios, Begoña Ros-Oton, Xavier Weidner, Marvin |
| author_facet | Barrios, Begoña Ros-Oton, Xavier Weidner, Marvin |
| contents | We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,α}$ free boundaries are $C^\infty$. This is new even for the fractional Laplacian, as it was only known in case $s=\frac12$. We also establish a general result for overdetermined problems, showing that if the boundary condition is smooth, then so is $\partialΩ$. Our approach is very robust and works as well for the nonlocal obstacle problem, where it yields a new proof of the higher regularity of free boundaries, completely different from the one in [AbRo20]. In order to prove our results, we need to develop, among other tools, new integration by parts formulas and delicate boundary Hölder estimates for nonlocal equations with (local) Neumann boundary conditions that had not been studied before and are of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_20581 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher regularity in nonlocal free boundary problems Barrios, Begoña Ros-Oton, Xavier Weidner, Marvin Analysis of PDEs 35R35, 47G20, 35B65 We study the higher regularity in nonlocal free boundary problems posed for general integro-differential operators of order $2s$. Our main result is for the nonlocal one-phase (Bernoulli) problem, for which we establish that $C^{2,α}$ free boundaries are $C^\infty$. This is new even for the fractional Laplacian, as it was only known in case $s=\frac12$. We also establish a general result for overdetermined problems, showing that if the boundary condition is smooth, then so is $\partialΩ$. Our approach is very robust and works as well for the nonlocal obstacle problem, where it yields a new proof of the higher regularity of free boundaries, completely different from the one in [AbRo20]. In order to prove our results, we need to develop, among other tools, new integration by parts formulas and delicate boundary Hölder estimates for nonlocal equations with (local) Neumann boundary conditions that had not been studied before and are of independent interest. |
| title | Higher regularity in nonlocal free boundary problems |
| topic | Analysis of PDEs 35R35, 47G20, 35B65 |
| url | https://arxiv.org/abs/2507.20581 |