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\(\Sigma \)-convergence. (English) Zbl 1229.46035
The usual approach to the weak convergence of a product of functions is to require one of the sequences to converge strongly. For example, if \(u_n \to u\) in \(L^p(\Omega)\), \(v_n \rightharpoonup v\) in \(L^{p^{\prime}}(\Omega)\), then \(u_n v_n \rightharpoonup uv\) in \(L^1(\Omega)\), where \(\rightharpoonup\) means weak convergence. The authors generalize this by considering weak \(\Sigma\)-convergence, \(\Sigma\)-convergence, homogenization algebras and the Gelfand transform. The details are rather technical (see the paper). It is a generalization of the two scale convergence of [G. Nguetseng, SIAM J. Math. Anal. 20, No. 3, 608–623 (1989; Zbl 0688.35007)] to the non-periodic setting, using et al the Gelfand transform. The authors illustrate the techniques by applying the ideas to a few homogenization problems.

MSC:
46J10 Banach algebras of continuous functions, function algebras
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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