Optimal conditions for $L^\infty$-regularity and a priori estimates for elliptic systems, II: $n(\geq 3)$ components

TL;DR

This paper develops a bootstrap method combined with $L^p$-$L^q$ estimates to determine optimal regularity and a priori bounds for multi-component elliptic systems, improving existence results.

Contribution

It introduces a new bootstrap procedure for elliptic systems with multiple components, establishing optimal $L^ Infty$-regularity and a priori estimates for various weak solutions.

Findings

Established optimal $L^ Infty$-regularity conditions for solutions.

Derived optimal a priori estimates for $L^1_ Delta$-solutions.

Improved existence theorems for certain elliptic systems.

Abstract

In this paper, we present a bootstrap procedure for general elliptic systems with $n(\geq 3)$ components. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^\infty$-regularity conditions for the three well-known types of weak solutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_\delta$-solutions. Thanks to the linear theory in $L^p_\delta(\Omega)$, it also yields the optimal conditions for a priori estimates for $L^1_\delta$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.