$p$-harmonic functions by way of intrinsic mean value properties

Abstract

Let $Ω\subset\mathbb{R}^n$ be a bounded domain satisfying the uniform exterior cone condition. We establish existence and uniqueness of continuous solutions of the Dirichlet Problem associated to certain intrinsic nonlinear mean value properties in $Ω$. Furthermore we show that, when properly normalized, such functions converge to the $p$-harmonic solution of the Dirichlet problem in $Ω$, for $p\in[2,\infty)$. The proof of existence is constructive and the methods are entirely analytic, a fundamental tool being the construction of explicit, $p$-independent barrier functions in $Ω$.