Advances in Differential Equations

Approximate radial symmetry for overdetermined boundary value problems

Amandine Aftalion, Jérôme Busca, and Wolfgang Reichel

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Abstract

In this paper, we study the stability of Serrin's classical symmetry result for overdetermined boundary value problems [13]. We prove that if there exists a positive solution of $\Delta u +f(u) =0$ in $\Omega$ with $u=0$ on $\partial\Omega$ and if $\partial u / \partial \nu$ on $\partial\Omega$ is close to a constant, then the domain $\Omega$ is close to a ball. Additionally, we give an explicit estimate for the distance of the domain to a circumscribed and inscribed ball. The proof relies on the method of moving planes and new quantitative versions of the Hopf Lemma and Serrin's corner Lemma.

Article information

Source
Adv. Differential Equations Volume 4, Number 6 (1999), 907-932.

Dates
First available in Project Euclid: 15 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366030751

Mathematical Reviews number (MathSciNet)
MR1729395

Zentralblatt MATH identifier
0951.35046

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 35B50: Maximum principles

Citation

Aftalion, Amandine; Busca, Jérôme; Reichel, Wolfgang. Approximate radial symmetry for overdetermined boundary value problems. Adv. Differential Equations 4 (1999), no. 6, 907--932. https://projecteuclid.org/euclid.ade/1366030751.


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