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# Mathematical Research Letters

## Volume 11 (2004)

### Number 1

### A theorem of Radò’s type for the solutions of a quasi-linear equation

Pages: 31 – 34

DOI: http://dx.doi.org/10.4310/MRL.2004.v11.n1.a4

#### Authors

#### Abstract

In this note, we prove a Radò type theorem for the solutions of the so-called $p$-harmonic equation $$\div(|\nabla u|^{p-2}\nabla u)=0$$ in any dimension. We show that a function $u\in C^1(\Omega)$ which is $p$-harmonic in $\Omega\setminus\{x\:u(x)=0\}$ is indeed $p$-harmonic in the whole domain $\Omega$. Our proof relies on a viscosity solution characterization of $p$-harmonic functions obtained in \cite{JLM}.