Lipschitz regularity of solutions for mixed integro-differential equations (Q424430)

The authors establish Hölder and Lipschitz estimates for viscosity solutions of a class of elliptic and parabolic nonlinear integro-differential equations by the classical Ishii-Lions's method. They extend the Hölder regularity results recently obtained by \textit{G. Barles} et al. in [J. Eur. Math. Soc. (JEMS) 13, No. 1, 1--26 (2011; Zbl 1207.35277)]. They extend their techniques to the case of nonlocal equations that they term mixed integro-differential equations. These equations are degenerate both in the local and nonlocal terms, and their overall behavior is driven by the local-nonlocal interaction. For instance, fractional diffusion may give ellipticity in one direction and the classical diffusion in the complementary one.