Global second order Sobolev-regularity of $p$-harmonic functions

TL;DR

This paper establishes a global regularity result for p-harmonic functions, showing they belong to a second-order Sobolev space under specific conditions, using advanced matrix inequalities and analytical techniques.

Contribution

It extends classical local regularity results to a global setting for p-harmonic functions within a specific p-range, employing Cordes' inequalities and methods from Cianchi and Maz'ya.

Findings

p-harmonic functions are in W^{2,2}_{loc} globally for 1<p<3+2/(n-2)

The proof utilizes Cordes' matrix inequalities and techniques from Cianchi and Maz'ya

The result broadens understanding of regularity properties of p-harmonic functions

Abstract

We prove a global version of the classical result that $p$-harmonic functions belong to $W^{2,2}_{loc}$ for $1<p<3+\frac{2}{n-2}$. The proof relies on Cordes' matrix inequalities [7] and techniques from the work of Cianchi and Maz'ya [5,6].