preprint

Inserted: 12 jun 2024

Year: 2019

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 $v\in C^\infty$, $$\
{
D^2vDv
2} - {\Delta v \Delta_\infty v } -\frac12
Dv
^2\
\le \frac{n-2}2 . $$ Based on this, we prove the following results: 1. For any $p$-harmonic functions $u$, $p\in(1,2)\cup(2,\infty)$, we have $$
Du
^{\frac{p-\gamma}2}Du\in W^{1,2}_{\rm loc},$$ with $\gamma<\min\{p+\frac{n-1}{n},3+\frac{p-1}{n-1}\}$. As a by-product, when $p\in(1,2)\cup(2,3+\frac2{n-2})$, we reprove the known $W^{2,q}_{\rm loc}$-regularity of $p$-harmonic functions for some $q>2$. 2. When $n\ge 2$ and $p\in(1,2)\cup(2,3+\frac2{n-2})$, the viscosity solutions to parabolic normalized $p $-Laplace equation have the $W_{\rm loc}^{2,q}$-regularity in the spatial variable and the $W_{\rm loc}^{1,q}$-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 weakviscosity solutions to parabolic $p $-Laplace equation have the $W_{\rm loc}^{2,2}$-regularity in the spatial variable and the $W_{\rm loc}^{1,2}$-regularity in the time variable. The range of $p$ (including $p=2$ from the classical result) here is sharp for the $W_{\rm loc}^{2,2}$-regularity.