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Pogorelov interior estimates for general sum-type Hessian equations

Weisong Dong, Sirui Xu, Ruijia Zhang

Abstract

In this paper, we exploit the concavity of sums of Hessian operators to derive Pogorelov estimates for corresponding equations under the dynamic semi-convexity assumption, and we further obtain several Liouville-type results. Moreover, when k=n-1 and k=n we establish Pogorelov estimates in the admissible cone. As an application, we prove that any entire admissible solution in $\mathbb{R}^n$ with quadratic growth must be a quadratic polynomial.

Pogorelov interior estimates for general sum-type Hessian equations

Abstract

In this paper, we exploit the concavity of sums of Hessian operators to derive Pogorelov estimates for corresponding equations under the dynamic semi-convexity assumption, and we further obtain several Liouville-type results. Moreover, when k=n-1 and k=n we establish Pogorelov estimates in the admissible cone. As an application, we prove that any entire admissible solution in with quadratic growth must be a quadratic polynomial.
Paper Structure (7 sections, 14 theorems, 226 equations)

This paper contains 7 sections, 14 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. A Key Concavity Lemma
  4. Proof of Theorem \ref{['thm1']} for Case (B)
  5. Proof of Theorem \ref{['thm2']} for Case (B)
  6. Proof of Theorems \ref{['thm1']} and \ref{['thm2']} for Case (A)
  7. Condition \ref{['con2']} in Theorems \ref{['thm1']} and \ref{['thm2']}

Key Result

Theorem 1.1

Assume that $\psi \in C^2(\overline{\Omega} \times \mathbb{R} \times \mathbb{R}^n)$ satisfies $\psi > \psi_0 > 0$ for some constant $\psi_0 > 0$ in $\overline{\Omega}$. Then, for any smooth solution of the Dirichlet problem for equation eq1 satisfying Hypothesis (RR) and Condition con1 (or Condition under one of the following conditions: Here $\alpha > 0$ and $C > 0$ depend only on $n$, $k$, $m$,

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 13 more