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In this paper we present a new bootstrap procedure for elliptic systems withtwo unknown functions. Combining with the $L^p$-$L^q$-estimates, it yields theoptimal $L^\infty$-regularity conditions for the three well-known types of weaksolutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_\delta$-solutions.Thanks to the linear theory in $L^p_\delta(\Omega)$, it also yields the optimalconditions for a priori estimates for $L^1_\delta$-solutions. Based on the apriori estimates, we improve known existence theorems for some classes ofelliptic systems.

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