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October 2014 A note on Serrin's overdetermined problem
Giulio Ciraolo, Rolando Magnanini
Kodai Math. J. 37(3): 728-736 (October 2014). DOI: 10.2996/kmj/1414674618

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 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 $$r_e − r_i ≤ C_t\Bigl(\max_{\Gamma_t} u-\min_{\Gamma_t} u\Bigr)$$ 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 $$\max_{\Gamma_t} u-\min_{\Gamma_t} u = o(C_t) \quad\mathrm{as}\quad t → 0^+.$$

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.

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Giulio Ciraolo. Rolando Magnanini. "A note on Serrin's overdetermined problem." Kodai Math. J. 37 (3) 728 - 736, October 2014. https://doi.org/10.2996/kmj/1414674618

Information

Published: October 2014
First available in Project Euclid: 30 October 2014

zbMATH: 1312.35010
MathSciNet: MR3273893
Digital Object Identifier: 10.2996/kmj/1414674618

Rights: Copyright © 2014 Tokyo Institute of Technology, Department of Mathematics

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Vol.37 • No. 3 • October 2014
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