10/5/2007

Luis Silvestre
Courant Institute, NY

Regularity results for nonlocal elliptic equations

We study nonlinear integro-differential equations. Typical examples are the ones that arise from control problems with discontinuous Levy processes. We can think of these as nonlinear equations of fractional order. Our aim is to extend the theory of fully nonlinear elliptic equations to this class of equations. We are able to obtain a result analogous to the Alexandroff estimate, Harnack inequality and $C^{1,\alpha}$ estimates. As the order of the equation approaches two, in the limit our estimates become the classical estimates for second order elliptic pdes.