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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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