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