Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations

Abstract

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.

References in corpus (1)

• Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential Equations

Cited by in corpus (2)

• $C^{1,\alpha }$ regularity of solutions of degenerate fully non-linear elliptic equations
• $L^p$-maximal regularity of nonlocal parabolic equation and applications