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 $$-\mathrm{d}\mathrm{i}\mathrm{v}(\mid \mathrm{\nabla }u{\mid }^{p-2}\mathrm{\nabla }u+a(x)\mid \mathrm{\nabla }u{\mid }^{q-2}\mathrm{\nabla }u)=0$$ and the normalized double phase parabolic equation $$u_t=\mid \mathrm{\nabla }u{\mid }^{2-p}\mathrm{d}\mathrm{i}\mathrm{v}(\mid \mathrm{\nabla }u{\mid }^{p-2}\mathrm{\nabla }u+a(x,t)\mid \mathrm{\nabla }u{\mid }^{q-2}\mathrm{\nabla }u),1<p\le q<\mathrm{\infty }\mathrm{.}$$ 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.