A note on Serrin's overdetermined problem

Abstract

We consider the solution of the torsion problem $-Δu=1$ in $Ω$ and $u=0$ on $\partial Ω$. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_ν$ is constant on $\partial Ω$, then $Ω$ must be a ball. In a recent paper, it has been conjectured that Serrin's theorem may be obtained {\it by stability} in the following way: first, for the solution $u$ of the torsion problem prove the estimate $$ r_e-r_i\leq C_t\,\Bigl(\max_{Γ_t} u-\min_{Γ_t} u\Bigr) $$ for some constant $C_t$ depending on $t$, where $r_e$ and $r_i$ are the radii of an annulus containing $\partialΩ$ and $Γ_t$ is a surface parallel to $\partialΩ$ at distance $t$ and sufficiently close to $\partialΩ$; secondly, if in addition $u_ν$ is constant on $\partialΩ$, show that $$ \max_{Γ_t} u-\min_{Γ_t} u=o(C_t)\ \mbox{as} \ t\to 0^+. $$ In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains $Ω$ are ellipses.