Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations

Barles, Guy; Chasseigne, Emmanuel; Ciomaga, Adina; Imbert, Cyril

Barles, Guy; Chasseigne, Emmanuel; Ciomaga, Adina; Imbert, Cyril (2012), Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations, Journal of Differential Equations, 252, 11, p. 6012-6060. http://dx.doi.org/10.1016/j.jde.2012.02.013

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00608848/fr/

Date

2012

Journal name

Journal of Differential Equations

Volume

252

Number

Publisher

Elsevier

Pages

6012-6060

Abstract (EN)

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.

Subjects / Keywords

regularity of generalized solutions; viscosity solutions; nonlinear elliptic equations integro partial-differential equations