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Homogenization of first order equations with (\(u/\varepsilon\))-periodic Hamiltonians part II: Application to dislocations dynamics. (English) Zbl 1143.35005
Summary: This paper is concerned with a result of homogenization of a non-local first order Hamilton-Jacobi equation describing the dislocations dynamics. Our model for the interaction between dislocations involves both an integro-differential operator and a (local) Hamiltonian depending periodically on \(u/\varepsilon\). The first two authors studied in part I [Arch. Ration. Mech. Anal. 187, No. 1, 49–89 (2008; Zbl 1127.70009)] homogenization problems involving such local Hamiltonians. Two main ideas of this previous work are used: on the one hand, we prove an ergodicity property of this equation by constructing approximate correctors which are necessarily non periodic in space in general; on the other hand, the proof of the convergence of the solution uses here a twisted perturbed test function for a higher dimensional problem. The limit equation is a nonlinear diffusion equation involving a first order Lévy operator; the nonlinearity keeps memory of the short range interaction, while the Lévy operator keeps memory of long ones. The homogenized equation is a kind of effective plastic law for densities of dislocations moving in a single slip plane.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B10 Periodic solutions to PDEs
35F20 Nonlinear first-order PDEs
45K05 Integro-partial differential equations
47G20 Integro-differential operators
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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