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

TL;DR

This paper establishes second order regularity results for solutions of elliptic and parabolic p-Laplacian equations using a new fundamental inequality involving the Laplacian and infinity-Laplacian, improving understanding of solution smoothness.

Contribution

The paper introduces a novel fundamental inequality for the algebraic structure of the Laplacian and infinity-Laplacian, leading to new regularity results for p-harmonic and p-Laplace equations.

Findings

Proves $W^{1,2}_{loc}$ regularity for certain p-harmonic functions.

Establishes $W^{2,q}_{loc}$ regularity for solutions to parabolic p-Laplace equations.

Answers an open question for the case n=2 in the regularity of parabolic p-Laplace solutions.

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 $v\in C^\infty$, $$\ | { |D^2vDv|^2} - {\Delta v \Delta_\infty v } -\frac12[|D^2v|^2-(\Delta v)^2]|Dv|^2\ | \le \frac{n-2}2 [|D^2v|^2{|Dv|^2}- |D^2vDv|^2 ]. $$ 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 weak/viscosity 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.