Ergodic pairs for singular or degenerate fully nonlinear operators

¹ Dipartimento di Matematica, Sapienza Università di Roma, Rome, Italy.
² Département de Mathématiques, Université de Cergy-Pontoise, Cergy-Pontoise, France.

^* Corresponding author: leoni@mat.uniroma1.it

Received: 17 January 2018
Accepted: 8 December 2018

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.