Liouville theorem for a class of Hessian equations

TL;DR

This paper establishes a generalized Liouville theorem for a broad class of Hessian elliptic equations, including Monge-Ampère and k-Hessian equations, under new solution conditions, extending previous results in the field.

Contribution

The paper introduces a new general condition on solutions that unifies and extends existing Liouville theorems for various Hessian equations.

Findings

Proves Liouville theorem under the new condition for Hessian equations.

Unifies multiple known Liouville theorems as special cases.

Provides a framework for conditional interior estimates.

Abstract

In this paper, we study a general class of Hessian elliptic equations, including the Monge-Amp\`ere equation, the $k$-Hessian equation and $p$-Monge-Amp\`ere equations. We propose new additional condition on the solution and prove Liouville theorem under this assumption. We show that our general condition covers as special cases numerous sets of assumptions known in the literature which were tailored for specific equations. Thus we obtain a significant generalization of multiple isolated Liouville theorems and conditional interior estimates.