Symmetry and linear stability in Serrin's overdetermined problem via the stability of the parallel surface problem

Abstract

We consider the solution of the problem $$ -Δu=f(u) \ \mbox{ and } \ u>0 \ \ \mbox{ in } \ Ω, \ \ u=0 \ \mbox{ on } \ Γ, $$ where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with boundary $\Gamma$ of class $C^{2,\tau }$, $0<\tau <1$, and $f$ is a locally Lipschitz continuous non-linearity. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_{\nu }$ is constant on $\Gamma$, then $\Omega$ must be a ball. In [CMS2], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for a solution $u$ prove the estimate $$r_e-r_i\le C_{\delta }[u{]}_{{\Gamma }^{\delta }}$$ for some constant $C_{\delta }$ depending on $\delta >0$, where $r_e$ and $r_i$ are the radii of a spherical annulus containing $\Gamma$, ${\Gamma }^{\delta }$ is a surface parallel to $\Gamma$ at distance $\delta$ and sufficiently close to $\Gamma$, and $[u{]}_{{\Gamma }^{\delta }}$ is the Lipschitz semi-norm of $u$ on ${\Gamma }^{\delta }$; secondly, if in addition $u_{\nu }$ is constant on $\Gamma$, show that $$ [u]_{Γ^δ}=o(C_δ)\ \mbox{ as } \ δ\to 0^+. $$ In this paper, we prove that this strategy is successful. As a by-product of this method, for $C^{2,\tau }$-regular domains, we also obtain a linear stability estimate for Serrin's symmetry result. Our result is optimal and greatly improves the similar logarithmic-type estimate of [ABR] and the Hölder estimate of [CMV] that was restricted to convex domains.

References in corpus (1)

• Hölder stability for Serrin's overdetermined problem

Cited by in corpus (1)

• Optimal concavity of the torsion function