The "strange term" in the periodic homogenization for multivalued Leray-Lions operators in perforated domains

International audienceUsing 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

The Periodic Unfolding Method in Homogenization

International audienceThe periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104] (with the basic proofs in [Proceedings of the Narvik Conference 2004, GAKUTO Internat. Ser. Math. Sci. Appl. 24, Gakkotosho, Tokyo, 2006, pp. 119-136]). In the present paper we go into all the details of the method and include complete proofs, as well as several new extensions and developments. This approach is based on two distinct ideas, each leading to a new ingredient. The first idea is the change of scale, which is embodied in the unfolding operator. At the expense of doubling the dimension, this allows one to use standard weak or strong convergence theorems in L(p) spaces instead of more complicated tools (such as two-scale convergence, which is shown to be merely the weak convergence of the unfolding; cf. Remark 2.15). The second idea is the separation of scales, which is implemented as a macro-micro decomposition of functions and is especially suited for the weakly convergent sequences of Sobolev spaces. In the framework of this method, the proofs of most periodic homogenization results are elementary. The unfolding is particularly well-suited for multiscale problems (a simple backward iteration argument suffices) and for precise corrector results without extra regularity on the data. A list of the papers where these ideas appeared, at least in some preliminary form, is given with a discussion of their content. We also give a list of papers published since the publication [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104], and where the unfolding method has been successfully applied

The periodic unfolding method in domains with holes

We give a comprehensive presentation of the periodic unfolding method for perforated domains, both when the unit hole is a compact subset of the open unit cell and when this is impossible to achieve. In order to apply the method to boundary-value problems with non homogeneous Neumann conditions on the boundaries of the holes, the properties of the boundary unfolding operator are also extensively studied. The paper concludes with applications to such problems and examples of reiterated unfolding

Periodic unfolding for countably many scales

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Periodic unfolding for countably many scales

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Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?

International audienceIn this paper we give a general presentation of the homogenization of Neumann type problems in periodically perforated domains, including the case where the shape of the reference hole varies with the size of the period (in the spirit of the construction of self-similar fractals). We shows that H-0-convergence holds under the extra assumption that there exists a bounded sequence of extension operators for the reference holes. The general class of Jones-domains gives an example where this result applies. When this assumption fails, another approach, using the Poincare-Wirtinger inequality is presented. A corresponding class where it applies is that of John-domains, for which the Poincare-Wirtinger constant is controlled. The relationship between these two kinds of assumptions is also clarified