A Brezis and Peletier type result for the fractional Robin function

Abstract

This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We consider the equation $(-\Delta)^su=f(x,u)$ in $\Omega$, $u=0$ in $\Omega^c$ and establish a representation formula for partial derivatives of solutions in terms of the normal derivative $u/\delta^s$. As a consequence, we prove that solutions to the overdetermined problem $(-\Delta)^su=f(x,u)$ in $\Omega$, $u=0$ in $\Omega^c$, and $u/\delta^s=0$ on $\partial\Omega$ 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 Br\'ezis 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.