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

Guy Barles 1, 2 Olivier Ley 3, * Erwin Topp 4
* Corresponding author
1 FDP - Fédération de recherche Denis Poisson
2 LMPT - Laboratoire de Mathématiques et Physique Théorique
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 Departamento de Matematica y C.C.
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.
Keywords : Lipschitz regularity nonlinear PDEs {Integro-differential equations strong maximum principle comparison principle large time behavior
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Guy Barles, Olivier Ley, Erwin Topp. Lipschitz regularity for integro-differential equations with coercive Hamiltonians and application to large time behavior. Nonlinearity, IOP Publishing, 2017, 30 (2), pp.703-734. ⟨10.1088/1361-6544/aa527f⟩. ⟨hal-01278603⟩

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