## Advances in Differential Equations

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

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

Nicola Garofalo and Elena Sartori

#### 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