Dirichlet problem for degenerate Hessian quotient type curvature equations

Abstract

In the paper, we prove the existence and uniqueness results of the $C^{1,1}$ regular graphic hypersurface for Dirichlet problem of a class of degenerate Hessian quotient type curvature equations under the condition $\ddot{I}{ˆ}^{\frac{1}{k-l}}\in C^{1,1}(\overline{\hat{I}}\text{c◯}\times \mathbb{R}\times {\mathbb{S}}^n)$. Specially, we also consider the second order derivative estimates for the corresponding degenerate Hessian type curvature equations under the optimal condition $\ddot{I}{ˆ}^{\frac{1}{k-1}}\in C^{1,1}(\overline{\hat{I}}\text{c◯}\times \mathbb{R}\times {\mathbb{S}}^n)$.

27 pages