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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2602.07221 |
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| _version_ | 1866915781405573120 |
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| author | Djitte, Sidy M. Sueur, Franck |
| author_facet | Djitte, Sidy M. Sueur, Franck |
| contents | This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We consider the equation $(-Δ)^su=f(x,u)$ in $Ω$, $u=0$ in $Ω^c$ and establish a representation formula for partial derivatives of solutions in terms of the normal derivative $u/δ^s$. As a consequence, we prove that solutions to the overdetermined problem $(-Δ)^su=f(x,u)$ in $Ω$, $u=0$ in $Ω^c$, and $u/δ^s=0$ on $\partialΩ$ are globally Lipschitz continuous provided that $2s>1$. We also prove a Pohozaev-type identity for the Green function and, in particular, obtain a formula for the gradient of the Robin function, which extends to the fractional setting some results obtained by Brézis and Peletier in \cite{Bresiz} in the classical case of the Laplacian. Finally, an application to the nondegeneracy of critical points of the fractional Robin function in symmetric domains is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07221 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Brezis and Peletier type result for the fractional Robin function Djitte, Sidy M. Sueur, Franck Analysis of PDEs 35C15, 47G30 This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We consider the equation $(-Δ)^su=f(x,u)$ in $Ω$, $u=0$ in $Ω^c$ and establish a representation formula for partial derivatives of solutions in terms of the normal derivative $u/δ^s$. As a consequence, we prove that solutions to the overdetermined problem $(-Δ)^su=f(x,u)$ in $Ω$, $u=0$ in $Ω^c$, and $u/δ^s=0$ on $\partialΩ$ are globally Lipschitz continuous provided that $2s>1$. We also prove a Pohozaev-type identity for the Green function and, in particular, obtain a formula for the gradient of the Robin function, which extends to the fractional setting some results obtained by Brézis and Peletier in \cite{Bresiz} in the classical case of the Laplacian. Finally, an application to the nondegeneracy of critical points of the fractional Robin function in symmetric domains is discussed. |
| title | A Brezis and Peletier type result for the fractional Robin function |
| topic | Analysis of PDEs 35C15, 47G30 |
| url | https://arxiv.org/abs/2602.07221 |