The Pogorelov estimates for degenerate curvature equations

TL;DR

This paper develops Pogorelov estimates for degenerate curvature equations, improving regularity results and existence theorems for related geometric problems in hyperbolic space.

Contribution

It introduces Pogorelov type estimates for degenerate prescribed k-curvature and k-Hessian equations, advancing understanding of solution regularity and existence in geometric PDEs.

Findings

Established Pogorelov estimates for degenerate curvature equations

Proved interior C1,1 regularity of solutions

Improved existence results for asymptotic Plateau problems

Abstract

We establish the Pogorelov type estimates for degenerate prescribed k-curvature equations as well as k-Hessian equations. Furthermore,we investigate the interior C1,1 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.