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

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