Ergodic pairs for singular or degenerate fully nonlinear operators

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.

Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem

“…By estimates (6) and again by the results of [8], one has that the sequence {u δ } is converging, up to a subsequence, in C 1 loc (H). However, in the present case we cannot immediately detect from (8) the limit function v. We need to use the equations satisfied by u δ , in order to derive the equation satisfied by v and to apply Theorem 1.1.…”

“…In the fully nonlinear degenerate/singular setting, problem (4) has been recently studied in [6,7], where it has been proved in particular that if f is bounded and Lipschitz continuous, then ergodic pairs (c Ω , u) do exist.…”

“…Let us recall that the analysis performed in [6] yields that the ergodic constant c Ω is unique under the assumption that F (∇d(x) ⊗ ∇d(x)) is of class C 2 for x in a neighborhood of ∂Ω.…”

“…Namely, we consider the infinite boundary value problem (4) for a smooth bounded domain Ω ⊂ R N , and we refer to a solution (c Ω , u) as an ergodic pair. As recalled in the introduction, by assuming that f ∈ C(Ω) is bounded and Lipschitz continuous, F is positively homogeneous and satisfies (2), α > −1 and α + 1 < β ≤ α + 2, it is proved in [6] the existence of ergodic pairs (c Ω , u). Moreover, by further assuming the smoothness condition (5), the ergodic constant c Ω is unique and any ergodic function u satisfies estimates (6), see [6].…”

“…As recalled in the introduction, by assuming that f ∈ C(Ω) is bounded and Lipschitz continuous, F is positively homogeneous and satisfies (2), α > −1 and α + 1 < β ≤ α + 2, it is proved in [6] the existence of ergodic pairs (c Ω , u). Moreover, by further assuming the smoothness condition (5), the ergodic constant c Ω is unique and any ergodic function u satisfies estimates (6), see [6]. Furthermore, in the case β < α + 2 any ergodic function is proved to satisfy also (9) and, consequently, to be unique, see [7].…”