A Liouville theorem for fully nonlinear problems with infinite boundary conditions and applications

TL;DR

This paper establishes a Liouville classification theorem for fully nonlinear elliptic equations with infinite boundary conditions and applies it to analyze boundary blow-up rates and uniqueness of ergodic functions.

Contribution

It introduces a Liouville theorem for nonlinear problems with infinite boundary conditions and uses it to derive boundary blow-up rates and uniqueness results for ergodic functions.

Findings

Proved a Liouville type classification theorem for nonlinear elliptic problems.

Derived gradient boundary blow-up rates for ergodic functions.

Established the uniqueness of ergodic functions in bounded domains.

Abstract

We prove a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators. We then apply the result in order to obtain gradient boundary blow up rates for ergodic functions in bounded domains related to degenerate/singular operators, and, as a further consequence, we deduce the uniqueness of the ergodic functions.