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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
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