On the Interior Regularity of Weak Solutions to Nonlinear Elliptic Systems of Second Order

Abstract

The interior $C^{1,\alpha }$-regularity for a weak solution (with gradient in the BMO-space) of a non-linear second order elliptic system is investigated. The positive answer is obtained on the assumption that the elliptic system satisfy the generalized Liouville condition considered in the BMO-space instead of the usually used $L^{\infty }$-space. Finally it is proved that the Liouville condition holds in the case of ${\mathbb{R}}^2$.