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

We consider the solution of the problem $$-\mathrm{\Delta }u=f(u)andu>0in\mathrm{\Omega },u=0on\mathrm{\Gamma },$$ where $\mathrm{\Omega }$ is a bounded domain in ${\mathbb{R}}^N$ with boundary $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$, then $\mathrm{\Omega }$ 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 $$r_e-r_i\le C_{\delta }[u{]}_{{\mathrm{\Gamma }}^{\delta }}$$ for some constant $C_{\delta }$ depending on $\delta >0$, where $r_e$ and $r_i$ are the radii of a spherical annulus containing $\mathrm{\Gamma }$, ${\mathrm{\Gamma }}^{\delta }$ is a surface parallel to $\mathrm{\Gamma }$ at distance $\delta$ and sufficiently close to $\mathrm{\Gamma }$, and $[u{]}_{{\mathrm{\Gamma }}^{\delta }}$ is the Lipschitz semi-norm of $u$ on ${\mathrm{\Gamma }}^{\delta }$; secondly, if in addition $u_{\nu }$ is constant on $\mathrm{\Gamma }$, show that $$[u{]}_{{\mathrm{\Gamma }}^{\delta }}=o(C_{\delta })as\delta \to 0^{+}\mathrm{.}$$ 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 H"{o}lder estimate of [CMV] that was restricted to convex domains.