The Periodic Unfolding Method in Homogenization
International audience The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104] (with the basic proofs in [Proceedings of the Narvik Conference 2004, GAKUTO Internat. Ser. Math. Sci. Appl. 24, Gakkotosho, Tokyo...
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ftccsdartic:oai:HAL:hal-00693080v1 2023-05-15T17:14:08+02:00 The Periodic Unfolding Method in Homogenization Cioranescu, Doina Damlamian, Alain Griso, Georges Laboratoire Jacques-Louis Lions (LJLL) Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS) Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA) Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM) 2008 https://hal-upec-upem.archives-ouvertes.fr/hal-00693080 https://hal-upec-upem.archives-ouvertes.fr/hal-00693080/document https://hal-upec-upem.archives-ouvertes.fr/hal-00693080/file/CDGT.pdf https://doi.org/10.1137/080713148 en eng HAL CCSD Society for Industrial and Applied Mathematics info:eu-repo/semantics/altIdentifier/doi/10.1137/080713148 hal-00693080 https://hal-upec-upem.archives-ouvertes.fr/hal-00693080 https://hal-upec-upem.archives-ouvertes.fr/hal-00693080/document https://hal-upec-upem.archives-ouvertes.fr/hal-00693080/file/CDGT.pdf doi:10.1137/080713148 http://creativecommons.org/licenses/by/ info:eu-repo/semantics/OpenAccess ISSN: 0036-1410 SIAM Journal on Mathematical Analysis https://hal-upec-upem.archives-ouvertes.fr/hal-00693080 SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2008, 40 (4), pp.1585-1620. ⟨10.1137/080713148⟩ multiscale problems periodic unfolding homogenization [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] info:eu-repo/semantics/article Journal articles 2008 ftccsdartic https://doi.org/10.1137/080713148 2021-11-21T02:22:12Z International audience The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104] (with the basic proofs in [Proceedings of the Narvik Conference 2004, GAKUTO Internat. Ser. Math. Sci. Appl. 24, Gakkotosho, Tokyo, 2006, pp. 119-136]). In the present paper we go into all the details of the method and include complete proofs, as well as several new extensions and developments. This approach is based on two distinct ideas, each leading to a new ingredient. The first idea is the change of scale, which is embodied in the unfolding operator. At the expense of doubling the dimension, this allows one to use standard weak or strong convergence theorems in L(p) spaces instead of more complicated tools (such as two-scale convergence, which is shown to be merely the weak convergence of the unfolding; cf. Remark 2.15). The second idea is the separation of scales, which is implemented as a macro-micro decomposition of functions and is especially suited for the weakly convergent sequences of Sobolev spaces. In the framework of this method, the proofs of most periodic homogenization results are elementary. The unfolding is particularly well-suited for multiscale problems (a simple backward iteration argument suffices) and for precise corrector results without extra regularity on the data. A list of the papers where these ideas appeared, at least in some preliminary form, is given with a discussion of their content. We also give a list of papers published since the publication [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104], and where the unfolding method has been successfully applied. Article in Journal/Newspaper Narvik Narvik Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) Narvik ENVELOPE(17.427,17.427,68.438,68.438) SIAM Journal on Mathematical Analysis 40 4 1585 1620 |
institution |
Open Polar |
collection |
Archive ouverte HAL (Hyper Article en Ligne, CCSD - Centre pour la Communication Scientifique Directe) |
op_collection_id |
ftccsdartic |
language |
English |
topic |
multiscale problems periodic unfolding homogenization [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] |
spellingShingle |
multiscale problems periodic unfolding homogenization [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] Cioranescu, Doina Damlamian, Alain Griso, Georges The Periodic Unfolding Method in Homogenization |
topic_facet |
multiscale problems periodic unfolding homogenization [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] |
description |
International audience The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104] (with the basic proofs in [Proceedings of the Narvik Conference 2004, GAKUTO Internat. Ser. Math. Sci. Appl. 24, Gakkotosho, Tokyo, 2006, pp. 119-136]). In the present paper we go into all the details of the method and include complete proofs, as well as several new extensions and developments. This approach is based on two distinct ideas, each leading to a new ingredient. The first idea is the change of scale, which is embodied in the unfolding operator. At the expense of doubling the dimension, this allows one to use standard weak or strong convergence theorems in L(p) spaces instead of more complicated tools (such as two-scale convergence, which is shown to be merely the weak convergence of the unfolding; cf. Remark 2.15). The second idea is the separation of scales, which is implemented as a macro-micro decomposition of functions and is especially suited for the weakly convergent sequences of Sobolev spaces. In the framework of this method, the proofs of most periodic homogenization results are elementary. The unfolding is particularly well-suited for multiscale problems (a simple backward iteration argument suffices) and for precise corrector results without extra regularity on the data. A list of the papers where these ideas appeared, at least in some preliminary form, is given with a discussion of their content. We also give a list of papers published since the publication [Cioranescu, Damlamian, and Griso, C. R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99-104], and where the unfolding method has been successfully applied. |
author2 |
Laboratoire Jacques-Louis Lions (LJLL) Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS) Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA) Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM) |
format |
Article in Journal/Newspaper |
author |
Cioranescu, Doina Damlamian, Alain Griso, Georges |
author_facet |
Cioranescu, Doina Damlamian, Alain Griso, Georges |
author_sort |
Cioranescu, Doina |
title |
The Periodic Unfolding Method in Homogenization |
title_short |
The Periodic Unfolding Method in Homogenization |
title_full |
The Periodic Unfolding Method in Homogenization |
title_fullStr |
The Periodic Unfolding Method in Homogenization |
title_full_unstemmed |
The Periodic Unfolding Method in Homogenization |
title_sort |
periodic unfolding method in homogenization |
publisher |
HAL CCSD |
publishDate |
2008 |
url |
https://hal-upec-upem.archives-ouvertes.fr/hal-00693080 https://hal-upec-upem.archives-ouvertes.fr/hal-00693080/document https://hal-upec-upem.archives-ouvertes.fr/hal-00693080/file/CDGT.pdf https://doi.org/10.1137/080713148 |
long_lat |
ENVELOPE(17.427,17.427,68.438,68.438) |
geographic |
Narvik |
geographic_facet |
Narvik |
genre |
Narvik Narvik |
genre_facet |
Narvik Narvik |
op_source |
ISSN: 0036-1410 SIAM Journal on Mathematical Analysis https://hal-upec-upem.archives-ouvertes.fr/hal-00693080 SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2008, 40 (4), pp.1585-1620. ⟨10.1137/080713148⟩ |
op_relation |
info:eu-repo/semantics/altIdentifier/doi/10.1137/080713148 hal-00693080 https://hal-upec-upem.archives-ouvertes.fr/hal-00693080 https://hal-upec-upem.archives-ouvertes.fr/hal-00693080/document https://hal-upec-upem.archives-ouvertes.fr/hal-00693080/file/CDGT.pdf doi:10.1137/080713148 |
op_rights |
http://creativecommons.org/licenses/by/ info:eu-repo/semantics/OpenAccess |
op_doi |
https://doi.org/10.1137/080713148 |
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SIAM Journal on Mathematical Analysis |
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