A note on Serrin's overdetermined problem

Abstract

We consider the solution of the torsion problem $-\Delta u=1$ in $\Omega$ and $u=0$ on $\partial \Omega$. Serrin's celebrated symmetry theorem states that, if the normal derivative $u_{\nu }$ is constant on $\partial \Omega$, then $\Omega$ 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 $$r_e-r_i\le C_t(\underset{{\Gamma }_t}{\max }u-\underset{{\Gamma }_t}{\min }u)$$ for some constant $C_t$ depending on $t$, where $r_e$ and $r_i$ are the radii of an annulus containing $\partial \Omega$ and ${\Gamma }_t$ is a surface parallel to $\partial \Omega$ at distance $t$ and sufficiently close to $\partial \Omega$; secondly, if in addition $u_{\nu }$ is constant on $\partial \Omega$, 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 $\Omega$ are ellipses.