Local Lipschitz regularity for degenerate elliptic systems

TL;DR

This paper establishes an $L^{ abla}$-gradient bound for solutions to degenerate elliptic systems, revealing a $p$-independent Lipschitz regularity criterion and demonstrating existence of locally Lipschitz solutions in geometric analysis contexts.

Contribution

It introduces a novel $L^{ abla}$-gradient bound for degenerate elliptic systems, leading to $p$-independent Lipschitz regularity results and existence theorems for critical growth systems.

Findings

Gradient bounds imply Lipschitz regularity independent of $p$

Lorentz space characterization of solutions

Existence of Lipschitz solutions for critical growth systems

Abstract

We start presenting an $L^{\infty}$-gradient bound for solutions to non-homogeneous $p$-Laplacean type systems and equations, via suitable non-linear potentials of the right hand side. Such a bound implies a Lorentz space characterization of Lipschitz regularity of solutions which surprisingly turns out to be independent of $p$, and that reveals to be the same classical one for the standard Laplacean operator. In turn, the a priori estimates derived imply the existence of locally Lipschitz regular solutions to certain degenerate systems with critical growth of the type arising when considering geometric analysis problems, as recently emphasized by Rivi\`ere