Lipschitz regularity for integro-differential equations with coercive hamiltonians and application to large time behavior

TL;DR

This paper adapts the Bernstein method to establish Lipschitz regularity for nonlinear integro-differential equations, enabling analysis of large time behavior even for degenerate or non-uniformly elliptic cases.

Contribution

It introduces a novel adaptation of the Bernstein method for Lipschitz regularity in nonlocal equations with coercive Hamiltonians, extending applicability to degenerate and non-uniformly elliptic cases.

Findings

Established Lipschitz regularity for a broad class of integro-differential equations.

Proved comparison principles enabling analysis of large time behavior.

Applied results to ergodic problems in periodic settings.

Abstract

In this paper, we provide suitable adaptations of the "weak version of Bernstein method" introduced by the first author in 1991, in order to obtain Lipschitz regularity results and Lipschitz estimates for nonlinear integro-differential elliptic and parabolic equations set in the whole space. Our interest is to obtain such Lipschitz results to possibly degenerate equations, or to equations which are indeed "uniformly el-liptic" (maybe in the nonlocal sense) but which do not satisfy the usual "growth condition" on the gradient term allowing to use (for example) the Ishii-Lions' method. We treat the case of a model equation with a superlinear coercivity on the gradient term which has a leading role in the equation. This regularity result together with comparison principle provided for the problem allow to obtain the ergodic large time behavior of the evolution problem in the periodic setting.