Second order regularity for elliptic and parabolic equations involving $p$-Laplacian via a fundamental inequality

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 following results: 1. For any $p$-harmonic functions $u$, $p\in (1,2)\cup (2,\infty )$, we have $$|Du{|}^{\frac{p-\gamma }{2}}Du\in W_{\mathrm{loc}}^{1,2},$$ with $\gamma <\min \{p+\frac{n-1}{n},3+\frac{p-1}{n-1}\}$. As a by-product, when $p\in (1,2)\cup (2,3+\frac{2}{n-2})$, we reprove the known $W_{\mathrm{loc}}^{2,q}$-regularity of $p$-harmonic functions for some $q>2$. 2. When $n\ge 2$ and $p\in (1,2)\cup (2,3+\frac{2}{n-2})$, the viscosity solutions to parabolic normalized $p$-Laplace equation have the $W_{\mathrm{loc}}^{2,q}$-regularity in the spatial variable and the $W_{\mathrm{loc}}^{1,q}$-regularity in the time variable for some $q>2$. Especially, when $n=2$ an open question in [17] is completely answered. 3. When $n\ge 1$ and $p\in (1,2)\cup (2,3)$, the weak/viscosity solutions to parabolic $p$-Laplace equation have the $W_{\mathrm{loc}}^{2,2}$-regularity in the spatial variable and the $W_{\mathrm{loc}}^{1,2}$-regularity in the time variable. The range of $p$ (including $p=2$ from the classical result) here is sharp for the $W_{\mathrm{loc}}^{2,2}$-regularity.

34 pages