Asymptotic mean value properties for the elliptic and parabolic double phase equations

We characterize an asymptotic mean value formula in the viscosity sense for the double phase elliptic equation -div(|∇u|p-2∇u+a(x)|∇u|q-2∇u)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -{\textrm{div}}(|\nabla u |^{p-2}\nabla u+ a(x)|\nabla u |^{q-2}\nabla u)=0 \end{aligned}$$\end{document}and the normalized double phase parabolic equation ut=|∇u|2-pdiv(|∇u|p-2∇u+a(x,t)|∇u|q-2∇u),1<p≤q<∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_t=|\nabla u |^{2-p}{\textrm{div}}(|\nabla u |^{p-2}\nabla u+ a(x,t)|\nabla u |^{q-2}\nabla u), \quad 1<p\le q<\infty . \end{aligned}$$\end{document}This is the first mean value result for such kind of nonuniformly elliptic and parabolic equations. In addition, the results obtained can also be applied to the p(x)-Laplace equations and the variable coefficient p-Laplace type equations.