Über quasilineare elliptische Differentialgleichungen in der Ebene

Abstract

This paper deals with a-priori estimates for Dirichlet's problem for quasilinear elliptic equations

$$a_{ij} (x,u, \triangledown u)u_{x_i x_j } = f(x,u, \triangledown u)$$

in the plane. We give an a-priori estimate for the C^2+α-norms of all solutions under the following assumptions only: The principal part is uniformly elliptic, f has quadratical growth with respect to ∇u, a_ij and f are Hölder-continuous and an a-priori estimate for sup|u| is known.