The "strange term" in the periodic homogenization for multivalued Leray-Lions operators in perforated domains
Résumé
Using the periodic unfolding method of Cioranescu, Damlamian and Griso, we study the homogenization for equations of the form $$-\Div d_\varepsilon=f,\text{ with }\bigl(\nabla u_{\varepsilon , \delta }(x),d_{\varepsilon , \delta }(x)\bigr) \in A_\varepsilon(x)$$ in a perforated domain with holes of size $\varepsilon \delta $ periodically distributed in the domain, where $A_\varepsilon $ is a function whose values are maximal monotone graphs (on $\R^{N})$. Two different unfolding operators are involved in such a geometric situation. Under appropriate growth and coercivity assumptions, if the corresponding two sequences of unfolded maximal monotone graphs converge in the graph sense to the maximal monotone graphs $A(x,y)$ and $A_0(x,z)$ for almost every $(x,y,z)\in \Omega \times Y \times \R^N$, as $\varepsilon \to 0$, then every cluster point $(u_0,d_0)$ of the sequence $(u_{\varepsilon , \delta }, d_{\varepsilon , \delta } )$ for the weak topology in the naturally associated Sobolev space is a solution of the homogenized problem which is expressed in terms of $u_0$ alone. This result applies to the case where $A_{\varepsilon}(x)$ is of the form $B(x/\varepsilon)$ where $B(y)$ is periodic and continuous at $y=0$, and, in particular, to the oscillating $p$-Laplacian.
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