Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations

TL;DR

This paper proves new Hölder and Lipschitz regularity estimates for solutions of a broad class of mixed local and nonlocal integro-differential equations, extending previous regularity results using viscosity solutions.

Contribution

It introduces regularity results for a novel class of mixed integro-differential equations where local and nonlocal effects interact, including degenerate cases.

Findings

Established Hölder and Lipschitz estimates for solutions.

Extended regularity results to mixed local-nonlocal equations.

Analyzed equations with degenerate local and nonlocal terms.

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.