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

Abstract

We characterize an asymptotic mean value formula in the viscosity sense for the double phase elliptic equation $$ -{\rm div}(\lvert \nabla u \rvert^{p-2}\nabla u+ a(x)\lvert\nabla u \rvert^{q-2}\nabla u)=0 $$ and the normalized double phase parabolic equation $$ u_t=\lvert\nabla u \rvert ^{2-p}{\rm div}(\lvert \nabla u \rvert^{p-2}\nabla u+ a(x,t)\lvert\nabla u \rvert^{q-2}\nabla u), \quad 1<p\leq q<\infty. $$ 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.