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Homogenisation and \(\theta-2\) convergence. (English) Zbl 0886.35017

Summary: We introduce a new concept of convergence for bounded sequences of functions in \(L^2(\Omega)\), called \(\theta-2\) convergence, where \(\Omega\) is an open set of \(\mathbb{R}^n\) and \(\theta\) a \(C^2\) diffeomorphism of \(\mathbb{R}^n\). This tool enables us to deal with homogenisation problems in some nonperiodic perforated domains. In particular, it provides a simple proof, and extensions, of a recent result of M. Briane [J. Math. Pures Appl. 73, No. 1, 47-66 (1994; Zbl 0835.35016)].

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Citations:

Zbl 0835.35016
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References:

[1] Tartar, Probèmes d’homogénéisation dans les équations aux dérivées partielles (1977)
[2] Sanchez-Palencia, Non homogeneous media and vibration theory (1977) · Zbl 0432.70002
[3] Nguetseng, SIAM J. Appl. Math. 29 pp 608– (1990)
[4] DOI: 10.1137/0523084 · Zbl 0770.35005 · doi:10.1137/0523084
[5] DOI: 10.1016/0022-247X(79)90211-7 · Zbl 0427.35073 · doi:10.1016/0022-247X(79)90211-7
[6] Briane, J. Math. Pures Appl. pp 421– (1994)
[7] Bensoussan, Asymptotic Analysis in Periodic Structures (1979)
[8] Murat, (Seminaire d’analyse fonctionnelle et numérique (1978)
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