Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics

In this paper we prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. This is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, that arises in the theory of dislocations dynamics. We show that if a mean curvature motion is approximated by this type of equations then it is always of variational type, whereas the converse is true only in dimension two. AMS Classification: 35F25, 35D05, 35Q99, 35B40, 35G25, 49L25.