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

$$\begin{aligned} u(x)=\mu _p(\varepsilon ,u)(x)+o(\varepsilon ^2) \end{aligned}$$

holds as $\varepsilon \rightarrow 0$ in the viscosity sense. Here, $\mu _p(\varepsilon ,u)(x)$ is the p-mean value of u on $B_\varepsilon (x)$ characterized as a unique minimizer of

$$\begin{aligned} \Vert u-\lambda \Vert _{L^p(B_\varepsilon (x))} \end{aligned}$$

with respect to $\lambda \in {\mathbb {R}}$. This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)’s obtained when $\mu _p(\varepsilon ,u)(x)$ is replaced by other kinds of mean values. The natural definition of $\mu _p(\varepsilon ,u)(x)$ 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.