Ergodic pairs for singular or degenerate fully nonlinear operators

TL;DR

This paper investigates the ergodic problem for fully nonlinear operators that can be singular or degenerate at points where the gradient vanishes, providing convergence results, characterizations, and key estimates.

Contribution

It introduces new convergence results for solutions of singular or degenerate fully nonlinear equations and characterizes the ergodic constant through bounded sub solutions.

Findings

Proves convergence of explosive and Dirichlet solutions

Characterizes the ergodic constant as an infimum of constants with bounded sub solutions

Establishes a priori Lipschitz estimates and a comparison principle for these equations

Abstract

We study the ergodic problem for fully nonlinear operators which may be singular or degenerate when the gradient of solutions vanishes. We prove the convergence of both explosive solutions and solutions of Dirichlet problems for approximating equations. We further characterize the ergodic constant as the infimum of constants for which there exist bounded sub solutions. As intermediate results of independent interest, we prove a priori Lipschitz estimates depending only on the norm of the zeroth order term, and a comparison principle for equations having no zero order terms.