Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911594917658624 |
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| author | Gatti, Michele Scheuer, Julian Weth, Tobias |
| author_facet | Gatti, Michele Scheuer, Julian Weth, Tobias |
| contents | We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set $Ω\subseteq \mathbb{R}^n$. Specifically, we show that if $\overlineΩ$ has positive reach and the nonlocal normal derivative introduced in (Dipierro, Ros-Oton, Valdinoci, Rev. Mat. Iberoam. 33 (2017), no. 2, 377-416) is constant on an external surface parallel and sufficiently close to $\partial Ω$, then $Ω$ must be a ball. Remarkably, this conclusion remains valid under the sole assumption that $Ω$ is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on $Ω$. Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition Gatti, Michele Scheuer, Julian Weth, Tobias Analysis of PDEs 35N25, 35R11 (Primary) 35B06, 35B35 (Secondary) We investigate symmetry and quantitative approximate symmetry for an overdetermined problem related to the fractional torsion equation in a regular open, bounded set $Ω\subseteq \mathbb{R}^n$. Specifically, we show that if $\overlineΩ$ has positive reach and the nonlocal normal derivative introduced in (Dipierro, Ros-Oton, Valdinoci, Rev. Mat. Iberoam. 33 (2017), no. 2, 377-416) is constant on an external surface parallel and sufficiently close to $\partial Ω$, then $Ω$ must be a ball. Remarkably, this conclusion remains valid under the sole assumption that $Ω$ is convex. Moreover, we analyze the quantitative stability of this result under two distinct sets of assumptions on $Ω$. Finally, we extend our analysis to a broader class of overdetermined Dirichlet problems involving the fractional Laplacian. |
| title | Fractional Dirichlet problems with an overdetermined nonlocal Neumann condition |
| topic | Analysis of PDEs 35N25, 35R11 (Primary) 35B06, 35B35 (Secondary) |
| url | https://arxiv.org/abs/2507.13715 |