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Interior $C^2$ estimates for a class of sum Hessian equation

Changyu Ren, Ziyi Wang

TL;DR

The paper addresses interior $C^2$ estimates for the sum Hessian equation $S_k(\lambda)=\sigma_k(\lambda)+\alpha\sigma_{k-1}(\lambda)=f(x,u,Du)$, with $0<k\le n$ and $f>0$, including a Pogorelov-type theory. It develops interior Hessian bounds for $0<k<n$ under $(k-1)$-admissibility and establishes Pogorelov-type estimates in the Dirichlet setting, while showing a weaker Pogorelov bound in the borderline case $k=n$. The analysis hinges on the algebraic structure of $S_k$, concavity properties of $S_k^{1/k}$ on admissible cones, and CTX-style second-derivative techniques, combining maximum-principle arguments with detailed derivative estimates. Collectively, these results extend regularity theory for gradient-dependent sum Hessian equations and provide robust a priori estimates for existence proofs via continuation methods in nonlinear elliptic PDEs with geometric content.

Abstract

In this paper, we mainly study the interior $C^2$ estimates for a class of sum Hessian equations. We establish the interior estimates and the Pogorelov type estimates for $0<k<n$. If $k=n$, we derive a weaker Pogorelov type estimates.

Interior $C^2$ estimates for a class of sum Hessian equation

TL;DR

The paper addresses interior estimates for the sum Hessian equation , with and , including a Pogorelov-type theory. It develops interior Hessian bounds for under -admissibility and establishes Pogorelov-type estimates in the Dirichlet setting, while showing a weaker Pogorelov bound in the borderline case . The analysis hinges on the algebraic structure of , concavity properties of on admissible cones, and CTX-style second-derivative techniques, combining maximum-principle arguments with detailed derivative estimates. Collectively, these results extend regularity theory for gradient-dependent sum Hessian equations and provide robust a priori estimates for existence proofs via continuation methods in nonlinear elliptic PDEs with geometric content.

Abstract

In this paper, we mainly study the interior estimates for a class of sum Hessian equations. We establish the interior estimates and the Pogorelov type estimates for . If , we derive a weaker Pogorelov type estimates.
Paper Structure (4 sections, 11 theorems, 104 equations)

This paper contains 4 sections, 11 theorems, 104 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 ${\mathbb R}^n$, $0<k<n$ 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 $n$, $m$, $M$, $R\sup|Df|$ and $R^2\sup|D^2f|$.

Theorems & Definitions (20)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 10 more