Liouville properties and critical value of fully nonlinear elliptic operators

TL;DR

This paper establishes Liouville properties for solutions of fully nonlinear degenerate elliptic equations, explores their implications for long-term behavior of parabolic equations, and determines a critical value for associated ergodic problems.

Contribution

It introduces new Liouville theorems for nonlinear elliptic equations with unbounded coefficients and applies these results to parabolic stabilization and ergodic control problems.

Findings

Liouville properties hold under sign conditions on coefficients.

Solutions stabilize over time for certain nonlinear parabolic equations.

A unique critical value for the ergodic Hamilton-Jacobi-Bellman equation is identified.

Abstract

We prove some Liouville properties for sub- and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in Ornstein- Uhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.