Kodai Mathematical Journal

A note on Serrin's overdetermined problem

Giulio Ciraolo and Rolando Magnanini

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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

reri$C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$

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

$\max_{\Gamma_t} u-\min_{\Gamma_t} u$ = 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.

Article information

Kodai Math. J., Volume 37, Number 3 (2014), 728-736.

First available in Project Euclid: 30 October 2014

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Ciraolo, Giulio; Magnanini, Rolando. A note on Serrin's overdetermined problem. Kodai Math. J. 37 (2014), no. 3, 728--736. doi:10.2996/kmj/1414674618. https://projecteuclid.org/euclid.kmj/1414674618

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