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

Guy Barles (1, 2) , Shigeaki Koike (3) , Olivier Ley (4) , Erwin Topp (2, 5)

1 FDP - Fédération de recherche Denis Poisson
2 LMPT - Laboratoire de Mathématiques et Physique Théorique
3 Mathematical Institute
4 IRMAR - Institut de Recherche Mathématique de Rennes
5 Departamento de Ingenieria Matematica, Centro Modelamiento Matematico

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 Hölder 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 Hölder regularity.

Mots clés: Large Time Behavior; Strong Maximum Principles; Comparison Principles; Regularity; Parabolic Integro-Differential Equations

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Shigeaki Koike, Olivier Ley, Erwin Topp. Regularity Results and Large Time Behavior for Integro-Differential Equations with Coercive Hamiltonians. Calculus of Variations and Partial Differential Equations, 2015, 54 (1), pp.535-572. ⟨10.1007/s00526-014-0794-x⟩. ⟨hal-00980228⟩