A natural approach to the asymptotic mean value property for the $p$-Laplacian

Abstract

Let $1\le p\le\infty$. We show that a function $u\in C(\mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $\Delta_p^n u(x)=0$ if and only if the asymptotic formula $$ u(x)=\mu_p(\ve,u)(x)+o(\ve^2) $$ holds as $\ve\to 0$ in the viscosity sense. Here, $\mu_p(\ve,u)(x)$ is the $p$-mean value of $u$ on $B_\ve(x)$ characterized as a unique minimizer of $$ \inf_{\la\in\RR}\nr u-\la\nr_{L^p(B_\ve(x))}. $$ This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)'s obtained when $\mu_p(\ve,u)(x)$ is replaced by other kinds of mean values. The natural definition of $\mu_p(\ve,u)(x)$ makes sure that this is a monotonic and continuous (in the appropriate topology) functional of $u$. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic $p$-Laplace equation.