A note on Serrin’s overdetermined problem

We consider the solution of the torsion problem
-Delta u = N in Omega, u = 0 on partial derivative Omega,
where Omega is a bounded domain in R-N.
Serrin's celebrated symmetry theorem states that, if the normal derivative u(v) is constant on partial derivative Omega, then Omega 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) <= Ct (max(Gamma i) u - min(Gamma i) u)
for some constant C-t depending on t, where r(e) and r(i) are the radii of an annulus containing partial derivative Omega and Gamma i is a surface parallel to partial derivative Omega at distance t and sufficiently close to partial derivative Omega; secondly, if in addition u(v) is constant on partial derivative Omega, show that
max(Gamma i) u - min(Gamma i) u = o(C-t) as 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 Omega are ellipses.