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) &lt;= 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 -&gt; 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.

A note on Serrin’s overdetermined problem / G. Ciraolo, R. Magnanini. - In: KODAI MATHEMATICAL JOURNAL. - ISSN 0386-5991. - 37:3(2014), pp. 728-736. [10.2996/kmj/1414674618]

### A note on Serrin’s overdetermined problem

#### Abstract

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.
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Serrin’s problem; Parallel surfaces; overdetermined problems; method of moving planes; stability.
Settore MAT/05 - Analisi Matematica
2014
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/2434/675152`