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Uniform estimates and Brezis-Merle type inequalities for the $k$-Hessian equation

Jie Deng, Haibin Wang, Bin Zhou

Abstract

In this paper, we prove a Brezis-Merle type inequality for $k$-convex functions vanishing on the boundary. As an application, we establish an Alexandrov-Bakelman-Pucci type estimate for the intermediate Hessian equation. Furthermore, we establish a concentration-compactness principle for the blow-up behavior of solutions to the Liouville type $k$-Hessian equations.

Uniform estimates and Brezis-Merle type inequalities for the $k$-Hessian equation

Abstract

In this paper, we prove a Brezis-Merle type inequality for -convex functions vanishing on the boundary. As an application, we establish an Alexandrov-Bakelman-Pucci type estimate for the intermediate Hessian equation. Furthermore, we establish a concentration-compactness principle for the blow-up behavior of solutions to the Liouville type -Hessian equations.

Paper Structure

This paper contains 4 sections, 9 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. The Brezis-Merle type inequality
  3. An ABP type estimate
  4. Concentration-compactness principle

Key Result

Theorem 1.1

Let $u\in \mathcal{F}^k(\Omega)$. where

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • proof : Proof of Theorem \ref{['thm BM inequality']}
  • proof : Proof of Theorem \ref{['thm L infty estimate']}
  • Lemma 3.1: De Giorgi's lemma
  • Lemma 4.1
  • ...and 6 more