The “strange term” in the periodic homogenization for multivalued Leray–Lions operators in perforated domains

Abstract

Using the periodic unfolding method of Cioranescu, Damlamian and Griso, we study the homogenization for equations of the form $$-{\rm div}\,\,d_\varepsilon=f,\,\,{\rm with}\,\,\left(\nabla u_{\varepsilon , \delta }(x),d_{\varepsilon , \delta }(x)\right) \in A_\varepsilon(x)$$ 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 $${(x,y,z)\in \Omega \times Y \times {\rm {\bf R}}^N}$$ , 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.