Interior curvature bounds for a class of curvature equations

Abstract

We derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data, generalizing a well-known estimate of Pogorelov for equations of Monge-Ampère type. For equations for which convexity of the solution is the natural ellipticity assumption, the curvature bound is proved for solutions with C^1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.