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 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<∞. 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.