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

TL;DR

This paper establishes an asymptotic mean value characterization for double phase elliptic and parabolic equations, extending the understanding of nonuniformly elliptic PDEs and their applications to variable exponent Laplace equations.

Contribution

It provides the first mean value formula for double phase elliptic and parabolic equations, broadening the theoretical framework for nonuniformly elliptic PDEs.

Findings

First mean value result for double phase equations

Applicable to p(x)-Laplace and variable coefficient p-Laplace equations

Enhances understanding of asymptotic properties of complex PDEs

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.