Optimal conditions for $L^\infty$-regularity and a priori estimates for elliptic systems, I: two components

TL;DR

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

Contribution

It introduces a novel bootstrap procedure that, together with existing estimates, achieves optimal regularity conditions and enhances existence theorems for elliptic systems.

Findings

Optimal $L^ abla$-regularity conditions for weak solutions.

Optimal a priori estimates for $L^1_ abla$-solutions.

Improved existence theorems for certain elliptic systems.

Abstract

In this paper we present a new bootstrap procedure for elliptic systems with two unknown functions. 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.