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

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 $L_{\delta }^1$-solutions. Thanks to the linear theory in $L_{\delta }^p(\Omega )$, it also yields the optimal conditions for a priori estimates for $L_{\delta }^1$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.

21pages