Note on an elementary inequality and its application to the regularity of $p$-harmonic functions

TL;DR

This paper establishes new Sobolev regularity results for p-harmonic functions using an elementary inequality, expanding understanding of their differentiability properties.

Contribution

It introduces a novel elementary inequality approach to prove Sobolev regularity of p-harmonic functions, a significant advance in regularity theory.

Findings

$|Du|^{rac{p-2+s}{2}}Du$ belongs to $W^{1,2}_{loc}$ for certain s

The regularity result applies to all p-harmonic functions

The proof relies on an elementary inequality

Abstract

We study the Sobolev regularity of $p$-harmonic functions. We show that $|Du|^{\frac{p-2+s}{2}}Du$ belongs to the Sobolev space $W^{1,2}_{loc}$, $s>-1-\frac{p-1}{n-1}$, for any $p$-harmonic function $u$. The proof is based on an elementary inequality.