Semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in $\\mathbb R^n$

Abstract

This paper is concerned with semiconcavity of viscosity solutions for a class of degen- erate elliptic integro-differential equations in R n . This class of equations includes Bellman equations containing operators of Levy-Ito type. Holder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided. (u(x + j(x,ξ)) − u(x) − B1(0)(ξ)Du(x) · j(x,ξ))µ(dξ), where B1(0) denotes the indicator function of the unit ball B1(0), j(x,ξ) is a function that determines the size of the jumps for the diffusion related to the operator I, and µ is a Levy measure. The Levy measure µ is a Borel measure on R n {0} satisfying