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

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

2008-05-29

arxiv.org/abs/0805.4551v1

Mathematics

Analysis of PDEs

29pages

Scientific paper

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.

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