A pointwise differential inequality and second-order regularity for nonlinear elliptic systems

TL;DR

This paper introduces a sharp pointwise differential inequality for vectorial second-order PDEs with Uhlenbeck structure, leading to optimal second-order regularity results for nonlinear elliptic systems, including broader conditions for the p-Laplace system.

Contribution

It provides a new differential inequality and extends second-order regularity results to a wider range of exponents and minimal boundary assumptions.

Findings

Established local and global second-order regularity estimates.

Broadened admissible p-values for the p-Laplace system.

Derived optimal regularity properties under minimal boundary conditions.

Abstract

A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in $\mathbb{R}^n$ are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the $p$-Laplace system, our conclusions broaden the range of the admissible values of the exponent $p$ previously known.