Two-scale convergence: some remarks and extensions (Q2438000)

The paper starts with the definition of the two-scale convergence as introduced by \textit{G. Nguetseng} [SIAM J. Math. Anal. 20, No. 3, 608--623 (1989; Zbl 0688.35007)]: a sequence \(\{u^{\varepsilon }\}\) in \(L^{2}(\Omega )\) two-scale converges to \(u_{0}\in L^{2}(\Omega \times Y)\) \((u^{\varepsilon } \overset{2}{\rightharpoonup }u_{0})\) if \(\int_{\Omega }u^{\varepsilon }(x)v(x,\frac{x}{\varepsilon })dx\rightarrow \int_{\Omega }\int_{Y}u_{0}(x,y)v(x,y)dxdy\) for every \(v\in L^{2}(\Omega ;C_{\sharp }(Y))\) . Here \(\Omega \) is a bounded domain in \(\mathbb{R}^{N}\) with Lipschitz boundary and \(Y\) is the unit cube in \(\mathbb{R}^{N}\), \(x\) and \(y\) being linked through \(y=\frac{x}{\varepsilon }\). The authors recall the compactness property of the two-scale convergence: every sequence \( \{u^{\varepsilon }\}\) which is bounded in \(L^{2}(\Omega )\) has a subsequence which two-scale converges. The main purpose of the paper is to extend this notion to that of very weak two-scale convergence. A sequence \( \{g^{\varepsilon }\}\) in \(L^{1}(\Omega )\) very weakly two-scale converges to \(g_{0}\in L^{1}(\Omega \times Y)\) (\(g^{\varepsilon }\overset{2}{\underset{vw} {\rightharpoonup }}g_{0}\)) if \(\int_{\Omega }g^{\varepsilon }(x)v_{1}(x)v_{2}(\frac{x}{\varepsilon })dx\rightarrow \int_{\Omega }\int_{Y}g_{0}(x,y)v_{1}(x)v_{2}(x,y)dxdy\) for every \(v_{1}\in D(\Omega )\) and every \(v_{2}\in L_{\sharp }^{2}(Y)\). In order to ensure the uniqueness of the very weak two-scale limit \(g_{0}\) one has to impose the condition \( \int_{\Omega }\int_{Y}g_{0}(x,y)dy=0\). The authors prove a compactness result for this new notion of convergence: every sequence \(\{u^{\varepsilon }\}\) which is bounded in \(H_{0}^{1}(\Omega )\) has a subsequence which satisfies \(u^{\varepsilon }\rightharpoonup u\) w-\(H_{0}^{1}(\Omega )\), \( u^{\varepsilon }\overset{2}{\rightharpoonup }u\), \(\nabla u^{\varepsilon } \overset{2}{\rightharpoonup }\nabla u\), \(u^{\varepsilon }\rightharpoonup \nabla u+\nabla _{y}u_{1}\), \(\frac{u^{\varepsilon }}{\varepsilon }\overset{2}{\underset{vw}{\rightharpoonup }}u_{1}\), with \(u\in H_{0}^{1}(\Omega )\) and \(u_{1}\in L^{2}(\Omega \times H_{\sharp }^{1}(Y))\). They apply this notion to the periodic homogenization case and they here characterize the weak limit of \(\frac{u^{\varepsilon }-u}{\varepsilon }\). Recalling then the notion of \((n+1)\)-scale convergence, as introduced by \textit{G. Allaire} and \textit{M. Briane} [Proc. R. Soc. Edinb., Sect. A, Math. 126, No. 2, 297--342 (1996; Zbl 0866.35017)], for separated or well separated scales, the main result of the last section of the paper proves a compactness result in the case of separated scales.