The Pogorelov Estimates for Degenerate Curvature Equations

Abstract

We establish the Pogorelov-type estimates for degenerate prescribed $k$-curvature equations as well as $k$-Hessian equations. Furthermore, we investigate the interior $C^{1,1}$ regularity of the solutions for Dirichlet problems. These techniques also enable us to improve the existence theorem for an asymptotic Plateau-type problem in hyperbolic space.