Banach Journal of Mathematical Analysis

\Sigma-convergence

Gabriel Nguetseng and Nils Svanstedt

Full-text: Open access

Abstract

We discuss two new concepts of convergence in $L^{p}$-spaces, the so-called weak $\Sigma $-convergence and strong $\Sigma $-convergence, which are intermediate between classical weak convergence and strong convergence. We also introduce the concept of $\Sigma $-convergence for Radon measures. Our basic tool is the classical Gelfand representation theory. Apart from being a natural generalization of well-known two-scale convergence theory, the present study lays the foundation of the mathematical framework that is needed to undertake a systematic study of deterministic homogenization problems beyond the usual periodic setting. A few homogenization problems are worked out by way of illustration.

Article information

Source
Banach J. Math. Anal. Volume 5, Number 1 (2011), 101-135.

Dates
First available in Project Euclid: 14 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1313362985

Digital Object Identifier
doi:10.15352/bjma/1313362985

Mathematical Reviews number (MathSciNet)
MR2738525

Zentralblatt MATH identifier
1229.46035

Subjects
Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 35B40: Asymptotic behavior of solutions 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx]

Keywords
homogenization homogenization algebras \Sigma-convergence Gelfand transformation

Citation

Nguetseng, Gabriel; Svanstedt, Nils. \Sigma-convergence. Banach J. Math. Anal. 5 (2011), no. 1, 101--135. doi:10.15352/bjma/1313362985. https://projecteuclid.org/euclid.bjma/1313362985.


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