Homogenization of partial differential equations with Preisach operators

Abstract

The current work deals with initial boundary value parabolic problems with Preisach hysteresis whose the density functions are allowed to depend on the variable of space. The model contains nonlinear monotone operators in the diffusion term, arising from an energy. Thanks to the properties of Preisach hysteresis operators and to the sigma-convergence method, we obtain the convergence of the microscopic solutions to the solution of the homogenized problem. The effective operator is obtained in terms of a solution of a nonlinear corrector equation addressed in the usual sense of distributions, leading in an approximate scheme for the homogenized coefficient which is an important step towards the numerical implementation of the results from the homogenization theory beyond the periodic setting.