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.Comment: 19 pages, submitte