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1999 Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates
Nicola Garofalo, Elena Sartori
Adv. Differential Equations 4(2): 137-161 (1999). DOI: 10.57262/ade/1366291411

Abstract

Given a connected, bounded open set $\Omega_1 \subset \mathbb{R}^n$, we use a maximum principle, and compactness arguments to study the properties of the function $P(u,x)$ in (1.5) below associated to a weak solution of the exterior $p$-capacitary problem, \[ \hbox{\rm div\,}(|Du|^{p-2}Du) = 0\ \text{in} \ \Omega=\mathbb{R}^n \setminus \overline{\Omega_1}, \quad \quad 1<p<n, \] $u=1$ on $\partial{\Omega_1}$, $u(x)\to 0$ as $|x|\to \infty$. As a consequence of our results we prove spherical symmetry for the solution $u$ and for the condenser $\Omega_1$ when the overdetermined boundary condition $|Du|=c>0$ on $\partial\Omega_1$ is imposed. This provides a new proof of a recent result of Reichel \cite{31}.

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Nicola Garofalo. Elena Sartori. "Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates." Adv. Differential Equations 4 (2) 137 - 161, 1999. https://doi.org/10.57262/ade/1366291411

Information

Published: 1999
First available in Project Euclid: 18 April 2013

zbMATH: 0951.35045
MathSciNet: MR1674355
Digital Object Identifier: 10.57262/ade/1366291411

Subjects:
Primary: 35J65
Secondary: 31B20 , 35B05

Rights: Copyright © 1999 Khayyam Publishing, Inc.

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Vol.4 • No. 2 • 1999
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