Nonlinear Elliptic Systems

Abstract

We present a maximum principle of W. Jäger for the H-surface system in Section 1. Then we prove the fundamental gradient estimate of E. Heinz for nonlinear elliptic systems of differential equations in Section 2. Global estimates are established in Section 3. In combination with the Leray-Schauder degree of mapping, we deduce an existence theorem for nonlinear elliptic systems in Section 4. Especially for the system Δx=2H x _u ∧x _v which was discovered by F. Rellich, this result was proved by E. Heinz already in 1954. In Section 5 we derive an inner distortion estimate for plane nonlinear elliptic systems, which implies a curvature estimate presented in Section 6. In the next Sections 7-8 we introduce conformal parameters into a Riemannian metric and establish a priori estimates up to the boundary in this context. Finally, we explain the uniformization method for quasilinear elliptic differential equations in Section 9.