Regularity for weak solutions to nondiagonal quasilinear degenerate elliptic systems

TL;DR

This paper establishes regularity results for weak solutions to nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, including gradient estimates and Hölder continuity, under bounded coefficients with vanishing mean oscillation.

Contribution

It provides new regularity results for weak solutions to complex degenerate elliptic systems with nondiagonal structure and bounded oscillation coefficients, extending previous theories.

Findings

Proved $L^p$ estimates for gradients of weak solutions.

Established higher Morrey and Campanato estimates.

Achieved Hölder continuity for weak solutions.

Abstract

The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to H\"{o}rmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove $L^p$($p \ge 2$) estimates for gradients of weak solutions by using a priori estimates and a known reverse H\"{o}lder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and H\"{o}lder estimates for weak solutions.