Advances in Differential Equations

Approximate radial symmetry for overdetermined boundary value problems

Amandine Aftalion, Jérôme Busca, and Wolfgang Reichel

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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

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

First available in Project Euclid: 15 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Aftalion, Amandine; Busca, Jérôme; Reichel, Wolfgang. Approximate radial symmetry for overdetermined boundary value problems. Adv. Differential Equations 4 (1999), no. 6, 907--932.

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