Dirichlet problem for degenerate Hessian quotient type curvature equations

TL;DR

This paper establishes existence, uniqueness, and second-order derivative estimates for solutions to degenerate Hessian quotient curvature equations with specific regularity conditions, advancing understanding of geometric PDEs.

Contribution

It proves existence and uniqueness of $C^{1,1}$ solutions for degenerate Hessian quotient curvature equations and provides second-order derivative estimates under optimal conditions.

Findings

Existence and uniqueness of $C^{1,1}$ solutions for the Dirichlet problem.

Second order derivative estimates under optimal regularity conditions.

Extension of regularity theory to degenerate Hessian quotient 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 $\psi^{\frac{1}{k-l}}\in C^{1,1}(\overline{\Omega}\times\mathbb{R}\times\mathbb{S}^n)$. Specially, we also consider the second order derivative estimates for the corresponding degenerate Hessian type curvature equations under the optimal condition $\psi^{\frac{1}{k-1}}\in C^{1,1}(\overline{\Omega}\times\mathbb{R}\times\mathbb{S}^n)$.