Asymptotic statistical characterizations of \(p\)-harmonic functions of two variables (Q533526)

This paper is concerned with a characterization of \(p\)-harmonic functions in open sets \(\Omega\) of the plane. The main result states that a continuous function \(u\) is \(p\)-harmonic, \(1<p<\infty\), if and only if the following two equivalent conditions are satisfied: \[ u(x)=\frac{2}{p-1}\text{ median}^{}_{\partial B_\varepsilon(x)} u+\bigg(2-\frac{2}{p}\bigg)\int_{\partial B_\varepsilon(x)}u\,ds+o(\varepsilon^2)\tag{1} \] as \(\varepsilon\rightarrow 0\) holds for all \(x\in \Omega\) in the viscosity sense, or \[ u(x)=\frac{1}{p} \text{ median}^{}_{\partial B_\varepsilon(x)} u+\frac{p-1}{2p}\bigg[\max_{\overline B_\varepsilon(x)}u+\min_{\overline B_\varepsilon(x)}u\bigg]+o(\varepsilon^2)\tag{2} \] as \(\varepsilon\rightarrow 0\) holds for all \(x\in \Omega\) in the viscosity sense. Here the median of \(u\) over \(\partial B_r(x)\) is defined as the unique real number \(m\) such that the sets \(\{ s\in\partial B_r(x):u(s)\geq m\}\) and \(\{ s\in\partial B_r(x):u(s)\leq m\}\) have the same Hausdorff measure. The proof relies on various asymptotic analysis arguments applied in the viscosity sense.