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Asymptotic mean value properties for the elliptic and parabolic double phase equations. (English) Zbl 1525.35014

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

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35D40 Viscosity solutions to PDEs
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35K92 Quasilinear parabolic equations with \(p\)-Laplacian

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