Homogenization and Orowan's law for anisotropic fractional operators of any order

Abstract

We consider an anisotropic Lévy operator Is of any order s 2 (0, 1) and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain is different according to the cases s < 1/2 and s > 1/2. In the isotropic onedimensional case, we also prove a statement related to the so-called Orowans law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior.