Regularity Results and Large Time Behavior for Integro-Differential Equations with Coercive Hamiltonians

TL;DR

This paper establishes regularity and large-time behavior results for elliptic integro-differential equations with coercive Hamiltonians, using novel methods that avoid traditional maximum principle techniques.

Contribution

It introduces new regularity results for coercive integro-differential equations and applies them to analyze the ergodic behavior of associated evolution problems.

Findings

H"older estimates for bounded subsolutions

A priori estimates for subsolutions in various cases

Ergodic asymptotic behavior for superlinear equations

Abstract

In this paper we obtain regularity results for elliptic integro-differential equations driven by the stronger effect of coercive gradient terms. This feature allows us to construct suitable strict supersolutions from which we conclude H\"older estimates for bounded subsolutions. In many interesting situations, this gives way to a priori estimates for subsolutions. We apply this regularity results to obtain the ergodic asymptotic behavior of the associated evolution problem in the case of superlinear equations. One of the surprising features in our proof is that it avoids the key ingredient which are usually necessary to use the Strong Maximum Principle: linearization based on the Lipschitz regularity of the solution of the ergodic problem. The proof entirely relies on the H\"older regularity.