Here are the slides of the talk I gave during the workshop Nonlocal PDEs, Variational Problems and their Applications.
Abstract: In a joint work with G. Barles, E. Chasseigne and A. Ciomaga, we establish Holder and Lipschitz estimates for solutions of elliptic nonlinear integro-differential equations, by the classical Ishii-Lions’s method. The main novelty lies in the type of nonlocal equations we can deal with: they can be degenerate both in the local and nonlocal term, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one. Such equations are referred to as mixed integro-differential equations.