Periodic homogenization of monotone multivalued operators

Using the unfolding method of Cioranescu, Damlamian and Griso [D. Cioranescu, A. Damlamian, 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 −divdε=f, with (∇uε(x),dε(x))∈Aε(x)(∇uε(x),dε(x))∈Aε(x) and where AεAε is a function whose values are maximal monotone graphs. Under appropriate growth and coercivity assumptions, if the sequence of unfolded maximal monotone graphs (Tε(Aε)(x,y))(Tε(Aε)(x,y)) converges in the graphical sense to a maximal monotone graph B(x,y)B(x,y) for almost every (x,y)∈Ω×Y(x,y)∈Ω×Y, as ε→0ε→0, then (uε,dε)(uε,dε) converges weakly in a suitable Sobolev space to a solution (u0,d0)(u0,d0) of the problem −divd0=f, with (∇u0(x),d0(x))∈A(x)(∇u0(x),d0(x))∈A(x) and AA satisfies the same assumptions as AεAε. This result includes the case where Aε(x)Aε(x) is a monotone continuous function for almost every x∈Ωx∈Ω.