### Approximate radial symmetry for overdetermined boundary value problems

#### 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

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