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Minimum principles and a priori estimates for 2-Hessian problems

Cristian Enache

Abstract

In this paper we investigate a class of $2$-Hessian equations and establish a minimum principle for a $P$-function in the sense of L.E. Payne (see R. Sperb \cite{Sp81}). The analysis is based on a sharp matrix inequality providing an estimate for a suitable combination of second-order partial derivatives of the solution. Exploiting this estimate, we derive a differential inequality for the associated $P$-function and obtain a minimum principle in higher dimensions under a convexity assumption. As an application of our results, together with convexity results established in X.-N. Ma and L. Xu \cite{MX08}, P. Liu, X.-N. Ma and L. Xu \cite{LMX10}, P. Salani \cite{Sa12}, and Y. Ye \cite{Ye13}, we derive a priori bounds for solutions of several classical $2$-Hessian boundary value problems.

Minimum principles and a priori estimates for 2-Hessian problems

Abstract

In this paper we investigate a class of -Hessian equations and establish a minimum principle for a -function in the sense of L.E. Payne (see R. Sperb \cite{Sp81}). The analysis is based on a sharp matrix inequality providing an estimate for a suitable combination of second-order partial derivatives of the solution. Exploiting this estimate, we derive a differential inequality for the associated -function and obtain a minimum principle in higher dimensions under a convexity assumption. As an application of our results, together with convexity results established in X.-N. Ma and L. Xu \cite{MX08}, P. Liu, X.-N. Ma and L. Xu \cite{LMX10}, P. Salani \cite{Sa12}, and Y. Ye \cite{Ye13}, we derive a priori bounds for solutions of several classical -Hessian boundary value problems.
Paper Structure (4 sections, 5 theorems, 75 equations)

This paper contains 4 sections, 5 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. A sharp matrix inequality and hessian estimates
  3. Proof of the main result
  4. Applications: a priori estimates

Key Result

Theorem 1

Assume that $f' \ge 0$ and that $u(\mathbf{x}) \in C^{3}(\Omega) \cap C^{2}(\overline{\Omega})$ is an admissible solution of eq:1.1 with convex level sets. Then the function $\Phi \left(\mathbf{x},(N(N-1)/2)^{-1/2}\right)$ attains its maximum on $\partial \Omega$.

Theorems & Definitions (10)

  • Theorem 1: C. Enache En10
  • Theorem 2: C. Enache En14
  • Theorem 3
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Remark 3