On the Dirichlet problem for solutions of a restricted nonlinear mean value property

Abstract

Let $\Omega \subset \mathbb R^d$ be a bounded domain and suppose that for each $x\in \Omega$ a radius $r = r(x)$ is given so that the ball $B_x = B(x,r)$ is contained in $\Omega$. For $0 \leq \alpha < 1 $, we consider the following operator in $\mathcal{C}(\overline{\Omega})$ $$ T_{\alpha}u(x) = \frac{\alpha}{2}\big ( \sup_{B_x} u + \inf_{B_x} u \big ) + (1-\alpha ) \int_{B_x} u, $$ and show that, under certain assumptions on $\Omega$ and the radius function $r(x)$, the Dirichlet problem with continuous boundary data has a unique solution $u\in \mathcal{C}(\overline{\Omega})$ satisfying $T_{\alpha}u = u$. The motivation comes from the study of so called $p$-harmonious functions and certain stochastic games.