Periodic homogenization of monotone multivalued operators
(2007) Nonlinear Analysis: Theory, Methods & Applications — Vol. 67, n° 12, p. 3217-3239 (2007)
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- Using the unfolding method of Cioranescu, Damlamian and Griso [D. Cioranescu, A. Damlanuan, G. Griso, Periodic unfolding and homogenization, C. R. Acad. Sci. Paris Math. 335 (1) (2002) 99-104], we study the homogenization for equations of the form -divd(epsilon) = f, with (del u(epsilon)(x), d epsilon(x)) epsilon A(epsilon) (x) and where AE is a function whose values are maximal monotone graphs. Under appropriate growth and coercivity assumptions, if the sequence of unfolded maximal monotone graphs (T-epsilon(A(epsilon))(x, y)) converges in the graphical sense to a maximal monotone graph B(x, y) for almost every (x, y) epsilon Omega x Y, as epsilon -> 0, then (u(epsilon), d(epsilon)) converges weakly in a suitable Sobolev space to a solution (u(0), d(0)) of the problem -div d(0) = f, with (del u(0)(x), d(0)(x)) epsilon A(x) and A satisfies the same assumptions as AE. This result includes the case where A(epsilon) (x) is a monotone continuous function for almost every x epsilon Omega. (c) 2006 Elsevier Ltd. All rights reserved.
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Damlamian, A., Meunier, N., & Van Schaftingen, J. (2007). Periodic homogenization of monotone multivalued operators. Nonlinear Analysis: Theory, Methods & Applications, 67(12), 3217-3239. https://doi.org/10.1016/j.na.2006.10.007 (Original work published 2007)