Existence of Strong Solutions to Degenerate or Singular Strongly Coupled Elliptic Systems

TL;DR

This paper proves the existence of strong solutions for a broad class of strongly coupled elliptic systems that can be degenerate or singular, using a new unified approach and relaxed assumptions.

Contribution

It introduces a unified proof for degenerate and singular systems, replacing the VMO assumption with a more versatile local weighted Gagliardo-Nirenberg inequality involving BMO norms.

Findings

Established existence of strong solutions for degenerate or singular elliptic systems

Improved upon previous results by relaxing regularity assumptions

Provided examples from physical models

Abstract

A general class of strongly coupled elliptic systems with quadratic growth in gradients is considered and the existence of their strong solutions is established. The results greatly improve those in a recent paper \cite{dleJFA} as the systems can be either degenerate or singular when their solutions become unbounded. A unified proof for both cases is presented. Most importantly, the VMO assumption in \cite{dleJFA} will be replaced by a much versatile one thanks to a new local weighted Gagliardo-Nirenberg involving BMO norms. Examples in physical models will be provided.