We consider an anisotropic Lévy operator Is of any order 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<1/2 and s>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.
Homogenization and Orowan's law for anisotropic fractional operators of any order / S. Patrizi, E. Valdinoci. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 119(2015), pp. 3-36. [10.1016/j.na.2014.07.010]
Homogenization and Orowan's law for anisotropic fractional operators of any order
E. ValdinociUltimo
2015
Abstract
We consider an anisotropic Lévy operator Is of any order 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<1/2 and s>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.
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