A note on Serrin's overdetermined problem

Abstract

We consider the solution of the torsion problem $$−Δu = N \quad\mathrm{in}\quad Ω,\quad u = 0\quad\mathrm{on}\quad ∂Ω,$$ where Ω is a bounded domain in R^N.

Serrin's celebrated symmetry theorem states that, if the normal derivative u_ν is constant on ∂Ω, then Ω must be a ball. In [6], it has been conjectured that Serrin's theorem may be obtained by stability in the following way: first, for the solution u of the torsion problem prove the estimate $$r_e − r_i ≤ C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$$ for some constant C_t depending on t, where r_e and r_i are the radii of an annulus containing ∂Ω and Γ_t is a surface parallel to ∂Ω at distance t and sufficiently close to ∂Ω secondly, if in addition u_ν is constant on ∂Ω, show that $$\max_{\Gamma_t} u-\min_{\Gamma_t} u = o(C_t) \quad\mathrm{as}\quad t → 0^+.$$

The estimate constructed in [6] is not sharp enough to achieve this goal. In this paper, we analyse a simple case study and show that the scheme is successful if the admissible domains Ω are ellipses.