Minimum principles and a priori estimates for 2-Hessian problems

TL;DR

This paper develops a minimum principle for 2-Hessian equations using a sharp matrix inequality, leading to new a priori bounds for solutions of classical boundary value problems in higher dimensions.

Contribution

It introduces a novel minimum principle for 2-Hessian equations based on a sharp matrix inequality, extending analysis to higher dimensions under convexity assumptions.

Findings

Established a minimum principle for a P-function in 2-Hessian equations.

Derived a differential inequality leading to a priori bounds.

Extended results to classical boundary value problems in higher dimensions.

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.