抄録
We consider the solution of the torsion problem
−Δu = N in Ω, u = 0 on ∂Ω,
where Ω is a bounded domain in RN.
Serrin's celebrated symmetry theorem states that, if the normal derivative uν 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
re − ri ≤ Ct(maxΓt u − minΓt u)
for some constant Ct depending on t, where re and ri are the radii of an annulus containing ∂Ω and Γt is a surface parallel to ∂Ω at distance t and sufficiently close to ∂Ω secondly, if in addition uν is constant on ∂Ω, show that
maxΓt u − minΓt u = o(Ct) 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 Ω are ellipses.
