A natural approach to the asymptotic mean value property for the \(p\)-Laplacian (Q2406061)

The authors consider the normalized elliptic and parabolic \(p\)-Laplace equations, \(1 \leq p \leq \infty\). They consider the \(p\)-mean of a function, \(\mu_p\), and show that \[ u(x) = \mu_p(\varepsilon,u)(x) + o(\varepsilon^2) \] if and only if \(u\) is a viscosity solution of the normalized \(p\)-Laplace equation. They prove a similar characterization for the normalized \(p\)-heat equation, where they take the average over a heat ball equipped with an appropriate measure. The mean \(\mu_p\) is both monotone and continuous as a function of \(u\), unlike previous means, and also extends to the case \(p=1\).