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We characterize an asymptotic mean value formula in the viscosity sense forthe 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 thenormalized 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 valueresult for such kind of nonuniformly elliptic and parabolic equations. Inaddition, the results obtained can also be applied to the $p(x)$-Laplaceequations and the variable coefficient $p$-Laplace type equations.

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