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Interior Hessian estimates for sum Hessian quotient equation

Changyu Ren, Ziyi Wang

TL;DR

This work addresses interior $C^2$ estimates for the sum Hessian quotient equation $\frac{\sigma_k(\eta)+\alpha\sigma_{k-1}(\eta)}{\sigma_l(\eta)+\alpha\sigma_{l-1}(\eta)}=f(x,u,Du)$. The authors develop an interior Hessian estimate via a maximum principle with a carefully chosen auxiliary function, and establish Pogorelov-type estimates for the Dirichlet problem using a transformed operator $U=(\Delta u)I-D^2u$ and a two-case analysis. The results cover the general regime $0\le l<k<n$ and provide a weaker Pogorelov-type bound when $k=n$ and $0\le l<n-1$, thereby extending existing Hessian-quotient theory to the sum form and furnishing robust a priori estimates for the solvability of fully nonlinear PDEs arising from $k$-convex and $(k-1)$-admissible frameworks. These estimates are essential for applying the continuity method to obtain existence and regularity results for related Dirichlet problems in geometric and PDE contexts.

Abstract

This paper is devoted to the interior $C^2$ estimates for a class of sum Hessian quotient equations. For $0\leq l<k<n$, we establish the interior estimates and the Pogorelov type estimates. In the case $k=n$, we obtain a weaker Pogorelov type estimate for $0\leq l<n-1$.

Interior Hessian estimates for sum Hessian quotient equation

TL;DR

This work addresses interior estimates for the sum Hessian quotient equation . The authors develop an interior Hessian estimate via a maximum principle with a carefully chosen auxiliary function, and establish Pogorelov-type estimates for the Dirichlet problem using a transformed operator and a two-case analysis. The results cover the general regime and provide a weaker Pogorelov-type bound when and , thereby extending existing Hessian-quotient theory to the sum form and furnishing robust a priori estimates for the solvability of fully nonlinear PDEs arising from -convex and -admissible frameworks. These estimates are essential for applying the continuity method to obtain existence and regularity results for related Dirichlet problems in geometric and PDE contexts.

Abstract

This paper is devoted to the interior estimates for a class of sum Hessian quotient equations. For , we establish the interior estimates and the Pogorelov type estimates. In the case , we obtain a weaker Pogorelov type estimate for .
Paper Structure (4 sections, 14 theorems, 106 equations)

This paper contains 4 sections, 14 theorems, 106 equations.

Table of Contents

  1. introduction
  2. Preliminary
  3. Proof of Theorem 2
  4. Proof of Theorem 4

Key Result

Theorem 2

Suppose that $u\in C^4(B_R(0))$ is a $(k-1)$-admissible solution of the sum Hessian equation (e1.1), where $B_R(0)$ is a ball centered at the origin with radius $R$ in ${\mathbb R}^n$, $0\leqslant l<k<n$ and $f\in C^2(B_R(0)\times\mathbb{R}\times\mathbb{R}^n)$ with $0<m\leqslant f\leqslant M$. Then where $C$ is a positive constant depending only on $n$, $k$, $l$, $m$, $M$, $R\sup|Df|$ and $R^2\su

Theorems & Definitions (26)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • proof
  • Lemma 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 16 more