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 C1,1 Dirichlet data. We also use the curvature bounds to improve and extend various existence results for the Dirichlet and Plateau problems.
"Interior curvature bounds for a class of curvature equations." Duke Math. J. 123 (2) 235 - 264, 1 June 2004. https://doi.org/10.1215/S0012-7094-04-12321-8