Pogorelov type estimates for a class of Hessian quotient equations in Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$

TL;DR

This paper establishes Pogorelov type a priori estimates for k-convex solutions to a class of Hessian quotient equations on hyperbolic domains within Lorentz-Minkowski space, advancing understanding of geometric PDEs in this setting.

Contribution

It introduces Pogorelov type estimates for Hessian quotient equations in Lorentz-Minkowski space, a novel extension of classical estimates to a new geometric context.

Findings

Derived Pogorelov type estimates for solutions

Applied a priori estimates to Hessian quotient equations

Extended classical PDE estimates to Lorentz-Minkowski space

Abstract

Let $\Omega$ be a bounded domain (with smooth boundary) on the hyperbolic plane $\mathscr{H}^{n}(1)$, of center at origin and radius $1$, in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$. In this paper, by using a priori estimates, we can establish Pogorelov type estimates of $k$-convex solutions to a class of Hessian quotient equations defined over $\Omega\subset\mathscr{H}^{n}(1)$ and with the vanishing Dirichlet boundary condition.