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Interior Hessian estimates for Hessian quotient equations in dimension three

Heming Jiao, Zhenan Sui

Abstract

In this paper, we establish the interior Hessian estimates for $2$-convex solutions to $\frac{σ_2}{σ_1} (D^2 u) = ψ(x,u)$ in dimension three. In higher dimensions ($n \geq 4$), we prove the interior Hessian estimates for semi-convex solutions. We provide a new method to prove the doubling inequality for smooth solutions in dimensions three and four. In higher dimensions ($n\geq 5$) the doubling inequality is proved under an additional dynamic semi-convexity condition which is the same to that in \cite{SY2025}. The method also applies to the equation $σ_2 (D^2 u) = ψ(x, u, \nabla u)$.

Interior Hessian estimates for Hessian quotient equations in dimension three

Abstract

In this paper, we establish the interior Hessian estimates for -convex solutions to in dimension three. In higher dimensions (), we prove the interior Hessian estimates for semi-convex solutions. We provide a new method to prove the doubling inequality for smooth solutions in dimensions three and four. In higher dimensions () the doubling inequality is proved under an additional dynamic semi-convexity condition which is the same to that in \cite{SY2025}. The method also applies to the equation .
Paper Structure (5 sections, 6 theorems, 119 equations)

This paper contains 5 sections, 6 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The doubling inequality
  4. Proof of Theorem \ref{['thm1']} and Theorem \ref{['thm2']}
  5. Appendix

Key Result

Theorem 1.1

Let $u \in C^4 (B_1)$ be a $2$-convex solution to the equation Hessian in $B_1 \subset \mathbb{R}^3$ satisfying $\|u\|_{C^1(B_1(0))} \leq M < \infty$. Then where $C$ is a positive constant depending only on $\|\psi\|_{C^{1,1}}$, $\|\frac{1}{\psi}\|_{L^\infty}$ and $\|u\|_{C^1 (B_1(0))}$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 1 more