A Brezis and Peletier type result for the fractional Robin function

TL;DR

This paper extends classical results related to the Robin function and overdetermined problems to the fractional Laplacian setting, providing new representation formulas, identities, and regularity results for solutions.

Contribution

It introduces a representation formula for derivatives of solutions, proves Lipschitz regularity under certain conditions, and extends Brézis and Peletier's classical results to fractional operators.

Findings

Solutions are Lipschitz continuous if 2s>1.

A Pohozaev-type identity for the Green function is established.

A formula for the gradient of the fractional Robin function is derived.

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.