Periodic homogenization of monotone multivalued operators

Abstract

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 − div d ε = f , with ( ∇ u ε ( x ) , d ε ( x ) ) ∈ A ε ( x ) and where A ε is a function whose values are maximal monotone graphs. Under appropriate growth and coercivity assumptions, if the sequence of unfolded maximal monotone graphs ( T ε ( A ε ) ( x , y ) ) converges in the graphical sense to a maximal monotone graph B ( x , y ) for almost every ( x , y ) ∈ Ω × Y , as ε → 0 , then ( u ε , d ε ) converges weakly in a suitable Sobolev space to a solution ( u 0 , d 0 ) of the problem − div d 0 = f , with ( ∇ u 0 ( x ) , d 0 ( x ) ) ∈ A ( x ) and A satisfies the same assumptions as A ε . This result includes the case where A ε ( x ) is a monotone continuous function for almost every x ∈ Ω .