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

We consider the solution of the problem $$ -Δu=f(u) \ \mbox{ and } \ u>0 \ \ \mbox{ in } \ Ω, \ \ u=0 \ \mbox{ on } \ Γ, $$ where $Ω$ is a bounded domain in $\mathbb{R}^N$ with boundary $Γ$ of class $C^{2,τ}$, $0<τ<1$, and $f$ is a locally Lipschitz continuous non-linearity. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_ν$ is constant on $Γ$, then $Ω$ 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 $$ r_e-r_i\le C_δ\,[u]_{Γ^δ} $$ for some constant $C_δ$ depending on $δ>0$, where $r_e$ and $r_i$ are the radii of a spherical annulus containing $Γ$, $Γ^δ$ is a surface parallel to $Γ$ at distance $δ$ and sufficiently close to $Γ$, and $[u]_{Γ^δ}$ is the Lipschitz semi-norm of $u$ on $Γ^δ$; secondly, if in addition $u_ν$ is constant on $Γ$, 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 $C^{2,τ}$-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.