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

TL;DR

This paper investigates the homogenization of anisotropic fractional operators of any order, revealing different effective behaviors depending on the order, and establishes a linear scaling law in a specific isotropic case.

Contribution

It introduces a comprehensive analysis of homogenization for anisotropic Lévy operators of arbitrary order, including a novel proof of Orowan's law in the isotropic one-dimensional setting.

Findings

Different effective Hamiltonians for s<1/2 and s>1/2

Homogenization results for anisotropic fractional operators

Validation of Orowan's law in the isotropic case

Abstract

We consider an anisotropic L\'evy operator $\mathcal{I}_s$ of any order $s\in(0,1)$ 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 Orowan's law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior.