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

Abstract

Denote by $\Delta$ the Laplacian and by ${\Delta }_{\infty }$ the $\infty$-Laplacian. A fundamental inequality is proved for the algebraic structure of $\Delta v{\Delta }_{\infty }v$: for every $v\in C^{\infty }$, $$||D^{2}vDv{|}^2-\Delta v{\Delta }_{\infty }v-\frac{1}{2}[|D^{2}v{|}^2-(\Delta v{)}^2]|Dv{|}^2|\le \frac{n-2}{2}[|D^{2}v{|}^2|Dv{|}^2-|D^{2}vDv{|}^2]$$ Based on this, we prove the result: When $n\ge 2$ and $p(x)\in (1,2)\cup (2,3+\frac{2}{n-2})$, the viscosity solutions to parabolic normalized $p(x)$-Laplace equation have the $W_{loc}^{2,2}$-regularity in the spatial variable and the $W_{loc}^{1,2}$-regularity in the time variable.

arXiv admin note: text overlap with arXiv:1908.01547 by other authors