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 $(-\Delta {)}^{s}u=f(x,u)$ 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 $(-\Delta {)}^{s}u=f(x,u)$ 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 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.

arXiv comment21 pages

References in corpus (4)

• Ten equivalent definitions of the fractional Laplace operator
• Some observations on the Green function for the ball in the fractional Laplace framework
• Integro-Differential Elliptic Equations
• Integral representation of solutions to higher-order fractional Dirichlet problems on balls