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The Dirichlet problem for Hessian quotient type curvature equations in Minkowski space

Mengru Guo, Yang Jiao

TL;DR

The paper addresses the Dirichlet problem for a non-degenerate Hessian quotient type curvature equation on spacelike graphs in Minkowski space. By deriving global $C^1$ and boundary $C^2$ estimates and employing a barrier/linearization framework, the authors implement a continuity method to prove the existence of a unique smooth admissible solution under a positive, non-degenerate source term and the presence of an admissible subsolution. The approach extends prior Euclidean results to Lorentzian geometry, overcoming Minkowski-specific challenges and removing a priori boundary convexity requirements via cone properties. The results provide a rigorous existence/uniqueness theory for prescribed Hessian quotient curvature equations with homogeneous boundary data in the Minkowski setting, with potential implications for geometric analysis in Lorentzian manifolds.

Abstract

In this paper, we consider the Dirichlet problem for a class of prescribed Hessian quotient type curvature equations with homogeneous boundary data in Minkowski space. By establishing the a priori C2 estimates, we obtain the existence result in the non-degenerate case.

The Dirichlet problem for Hessian quotient type curvature equations in Minkowski space

TL;DR

The paper addresses the Dirichlet problem for a non-degenerate Hessian quotient type curvature equation on spacelike graphs in Minkowski space. By deriving global and boundary estimates and employing a barrier/linearization framework, the authors implement a continuity method to prove the existence of a unique smooth admissible solution under a positive, non-degenerate source term and the presence of an admissible subsolution. The approach extends prior Euclidean results to Lorentzian geometry, overcoming Minkowski-specific challenges and removing a priori boundary convexity requirements via cone properties. The results provide a rigorous existence/uniqueness theory for prescribed Hessian quotient curvature equations with homogeneous boundary data in the Minkowski setting, with potential implications for geometric analysis in Lorentzian manifolds.

Abstract

In this paper, we consider the Dirichlet problem for a class of prescribed Hessian quotient type curvature equations with homogeneous boundary data in Minkowski space. By establishing the a priori C2 estimates, we obtain the existence result in the non-degenerate case.

Paper Structure

This paper contains 5 sections, 9 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C^1$ estimates
  4. $C^2$ estimates
  5. Proof of the main result

Key Result

Theorem 1.2

Let $k \geqslant 2,~ 0 \leqslant l < k <n$. Assume that $\Omega\subset\mathbb{R}^n$ is bounded domain with smooth boundary $\partial \Omega$. Suppose $\psi = \psi (x,z) \in C^\infty (\overline \Omega \times \mathbb{R}) > 0$, $\psi_z \geqslant 0$. In addition, assume that there exists an admissible s Then there exists a unique admissible solution $u \in C^{\infty} (\overline \Omega)$ to GJ1.3.

Theorems & Definitions (13)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • ...and 3 more