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

We consider an anisotropic Levy operator I sub(s) of any order s [setmembership] (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 is...

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Bibliographic Details
Published in:Nonlinear analysis 2015-06, Vol.119, p.3-36
Main Authors: Patrizi, Stefania, Valdinoci, Enrico
Format: Article
Language:English
Subjects:
Anisotropy
Evolution
Homogenization
Homogenizing
Law
Mathematical analysis
Operators
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Summary:We consider an anisotropic Levy operator I sub(s) of any order s [setmembership] (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.
ISSN:0362-546X
DOI:10.1016/j.na.2014.07.010