Second order Sobolev regularity for normalized parabolic $p(x)$-Laplace equations via the algebraic structure

Denote by Δ the Laplacian and by Δ_∞ the ∞-Laplacian. A fundamental inequality is proved for the algebraic structure of Δ vΔ_∞ v: for every v∈ C∞, | |D²vDv|²-Δ vΔ_∞ v-1/2[|D²v|²-(Δ v)²]|Dv|²| -2/2[|D²v|²|Dv|²-|D²vDv|²] Based on this, we prove the result: When n≥2 and p(x)∈(1,2)∪(2,3+2/n-2), the viscosity solutions to parabolic normalized $p(x)$-Laplace equation have the W2,2loc-regularity in the spatial variable and the W1,2loc-regularity in the time variable.