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Two-scale convergence: Some remarks and extensions
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics. (Tillämpad matematik)
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics. (Tillämpad matematik)ORCID iD: 0000-0001-6742-5781
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics. (Tillämpad matematik)ORCID iD: 0000-0003-2942-6841
Mid Sweden University, Faculty of Science, Technology and Media, Department of Quality Technology and Management, Mechanical Engineering and Mathematics. (Tillämpad matematik)ORCID iD: 0000-0001-9984-2424
2013 (English)In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 9, no 3, p. 461-486Article in journal (Refereed) Published
Abstract [en]

We first study the fundamental ideas behind two-scale conver-

gence to enhance an intuitive understanding of this notion. The classical

definitions and ideas are motivated with geometrical arguments illustrated

by illuminating figures. Then a version of this concept, very weak two-scale

convergence, is discussed both independently and brie°y in the context of

homogenization. The main features of this variant are that it works also

for certain sequences of functions which are not bounded in

L2  and at

the same time is suited to detect rapid oscillations in some sequences which

are strongly convergent in

L2 . In particular, we show how very weak

two-scale convergence explains in a more transparent way how the oscilla-

tions of the governing coe±cient of the PDE to be homogenized causes the

deviation of the

G-limit from the weak L2 NxN-limit for the sequence of

coe±cients. Finally, we investigate very weak multiscale convergence and

prove a compactness result for separated scales which extends a previous

result which required well-separated scales.

Place, publisher, year, edition, pages
International press of Boston , 2013. Vol. 9, no 3, p. 461-486
Keywords [en]
Two-scale convergence, multiscale convergence, very weak multiscale convergence, homogenization
National Category
Natural Sciences Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-20401DOI: 10.4310/PAMQ.2013.v9.n3.a4ISI: 000327544500004Scopus ID: 2-s2.0-84887587258OAI: oai:DiVA.org:miun-20401DiVA, id: diva2:668743
Available from: 2013-12-02 Created: 2013-12-02 Last updated: 2017-12-06Bibliographically approved

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Flodén, LiselottHolmbom, AndersOlsson Lindberg, MariannePersson, Jens

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