Lipschitz regularity of solutions for mixed integro-differential equations

We establish new Holder 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 Holder 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. (C) 2012 Elsevier Inc. All rights reserved.