Advances in Differential Equations

Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates

Nicola Garofalo and Elena Sartori

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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}.

Article information

Source
Adv. Differential Equations, Volume 4, Number 2 (1999), 137-161.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1366291411

Mathematical Reviews number (MathSciNet)
MR1674355

Zentralblatt MATH identifier
0951.35045

Subjects
Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 31B20: Boundary value and inverse problems 35B05: Oscillation, zeros of solutions, mean value theorems, etc.

Citation

Garofalo, Nicola; Sartori, Elena. Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates. Adv. Differential Equations 4 (1999), no. 2, 137--161. https://projecteuclid.org/euclid.ade/1366291411


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