Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations

Guy Barles (1) , Emmanuel Chasseigne (1) , Adina Ciomaga (2) , Cyril Imbert (3, 4)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 CMLA - Centre de Mathématiques et de Leurs Applications
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 DMA - Département de Mathématiques et Applications - ENS Paris

We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local-nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one.

Mots clés: regularity of generalized solutions; viscosity solutions; nonlinear elliptic equations integro partial-differential equations

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Emmanuel Chasseigne, Adina Ciomaga, Cyril Imbert. Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations. Journal of Differential Equations, 2012, 252 (11), pp.6012-6060. ⟨10.1016/j.jde.2012.02.013⟩. ⟨hal-00608848v2⟩