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An elementary introduction to periodic unfolding. (English) Zbl 1204.35038
Damlamian, A. (ed.) et al., Multi-scale problems and asymptotic analysis. Proceedings of the midnight sun Narvik conference (satellite conference of the fourth European congress of mathematics), Narvik, Norway, June 22–26, 2004. Tokyo: Gakkōtosho (ISBN 4-7625-0433-5/hbk). GAKUTO International Series. Mathematical Sciences and Applications 24, 119-136 (2006).
The main purpose of the paper is to recall the notion of $$\varepsilon$$-unfolding $$T^\varepsilon$$ operator, as introduced by D. Cioranescu, A. Damlamian and G. Griso in [in: Multiple scales problems in biomathematics, mechanics, physics and numeric. Tokyo: Gakkotosho. GAKUTO International Series. Mathematical Sciences and Applications 31, 1–35 (2009; Zbl 1258.49013)], and to describe some properties of this operator $$T^{\varepsilon }$$, together with that of its adjoint.
Let $$Y$$ be a subset of $$\mathbb{R}^{N}$$ satisfying a paving property ($$Y=[0,1)^{N}$$ for example). Every $$x\in \mathbb{R}^{N}$$ can be written as $$x=\varepsilon [ x/\varepsilon ]_{Y}+\varepsilon \{x/\varepsilon \}_{Y}$$ where $$[x/\varepsilon ]_{Y}$$ (resp., $$\{x/\varepsilon \}_{Y}$$) is the “entire” (resp., “fractional”) part of $$x/\varepsilon$$. The $$\varepsilon$$-unfolding $$T^{\varepsilon }(u)$$ of $$u\in L^{p}(\mathbb{R}^{N})$$, $$p\in [ 1,\infty )$$, is defined as $$T^{\varepsilon }(u)(x,y)=u(\varepsilon [ x/\varepsilon ]_{Y}+\varepsilon y)$$. The adjoint of $$T^{\varepsilon }$$ is the averaging operator defined as $$U^{\varepsilon }(v(x))=\int_{Y}v(\varepsilon [ x/\varepsilon ]_{Y}+\varepsilon z,\{x/\varepsilon \}_{Y})\,dz/|Y|$$.
The main results of the paper establish the weak convergence properties of $$(T^{\varepsilon }(w_{\varepsilon }))_{\varepsilon }$$ imposing some boundedness properties of the sequence $$(w_{\varepsilon })_{\varepsilon }$$. The boundedness of $$(w_{\varepsilon })_{\varepsilon }$$ takes place in $$L^{p}$$ or in $$W^{1,p}$$. The author starts considering the whole domain $$\mathbb{R}^{N}$$, then a bounded domain $$\Omega$$ of $$\mathbb{R}^{N}$$. The author also considers the case of a bounded Jones-domain $$\Omega$$ such that there exists a linear continuous extension operator from $$W^{1,p}(\Omega )$$ to $$W^{1,p}(\mathbb{R}^{N})$$. Some convergence properties of the adjoint operator are also proved.
The last part of the paper presents applications of these convergence properties in a periodic homogenization situation for a scalar elliptic boundary value problem and in a multiscale problem obtained when decomposing $$\overline{Y}$$ as $$\overline{\omega }_{1}\cup \overline{\omega }_{2}$$ and considering a second scale in $$\omega _{2}$$.
For the entire collection see [Zbl 1089.74007].

MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 49J45 Methods involving semicontinuity and convergence; relaxation 35J25 Boundary value problems for second-order elliptic equations