Shape derivative approach to fractional overdetermined problems

TL;DR

This paper proves that only spherical domains can admit solutions to a fractional overdetermined problem under certain conditions, extending classical symmetry results to a fractional setting using shape derivatives and symmetrization techniques.

Contribution

It introduces a novel approach combining shape derivatives and continuous Steiner symmetrization to analyze fractional overdetermined problems, covering cases outside previous frameworks.

Findings

Balls are the only convex domains with solutions for the fractional overdetermined problem.

Recovers classical symmetry results for p=1 and p=2 cases.

Extends methods to the fractional setting for p in (1,2).

Abstract

We use shape derivative approach to prove that balls are the only convex and $C^{1,1}$ regular domains in which the fractional overdetermined problem \begin{equation*} \left\{\begin{aligned} \Ds u&= \lambda_{s, p} u^{p-1}\quad\text{in}\quad\Om \\ u &= 0\quad \text{in}\quad\R^N\setminus \Om\\ u/d^s&=C_0\quad\text{on\;\; $\partial\O$} \end{aligned} \right. \end{equation*} admits a nontrivial solution for $p\in [1, 2]$ and where $\lambda_{s, p}= \lambda_{s, p}(\O)$ is the best constant in the family of Subcritical Sobolev inequalities. In the cases $p=1$ and $p=2$, we recover the classical symmetry results of Serrin, corresponding to the torsion problem and the first Dirichlet eigenvalue problem, respectively (see \cite{FS-15}). We note that for $p\in (1,2)$, the above problem lies outside the framework of \cite{FS-15}, and the methods developed therein do not apply. Our approach extends to the fractional setting a method initially developed by A. Henrot and T. Chatelain in \cite{CH-99}, and relies on the use of domain derivatives combined with the continuous Steiner symmetrization introduced by Brock in \cite{Brock-00}.