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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The mathematics of thin structures

Authors: Jean-François Babadjian, Giovanni Di Fratta, Irene Fonseca, Gilles A. Francfort, Marta Lewicka and Cyrill B. Muratov
Journal: Quart. Appl. Math. 81 (2023), 1-64
MSC (2020): Primary 49J45, 74B20, 74K20, 74K35, 53A35
Published electronically: September 1, 2022
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Abstract: This article offers various mathematical contributions to the behavior of thin films. The common thread is to view thin film behavior as the variational limit of a three-dimensional domain with a related behavior when the thickness of that domain vanishes. After a short review in Section 1 of the various regimes that can arise when such an asymptotic process is performed in the classical elastic case, giving rise to various well-known models in plate theory (membrane, bending, Von Karmann, etc…), the other sections address various extensions of those initial results. Section 2 adds brittleness and delamination and investigates the brittle membrane regime. Sections 4 and 5 focus on micromagnetics, rather than elasticity, this once again in the membrane regime and discuss magnetic skyrmions and domain walls, respectively. Finally, Section 3 revisits the classical setting in a non-Euclidean setting induced by the presence of a pre-strain in the model.

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Additional Information

Jean-François Babadjian
Affiliation: Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405 Orsay, France

Giovanni Di Fratta
Affiliation: Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Napoli, Italy
MR Author ID: 864865
ORCID: 0000-0003-0254-2957

Irene Fonseca
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213
MR Author ID: 67965

Gilles A. Francfort
Affiliation: Université Paris-Nord, LAGA, Avenue J.-B. Clément, 93430 - Villetaneuse, France & Courant Institute, New York University, 251 Mercer St., NYC, New York 10012
MR Author ID: 68640
ORCID: 0000-0002-9943-919X

Marta Lewicka
Affiliation: Department of Mathematics, University of Pittsburgh, 139 University Place, Pittsburgh, Pennsylvania 15260
MR Author ID: 619488

Cyrill B. Muratov
Affiliation: Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102; and Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, Pisa 56127, Italy
MR Author ID: 606635
ORCID: 0000-0002-3337-6165

Received by editor(s): February 15, 2022
Received by editor(s) in revised form: July 1, 2022
Published electronically: September 1, 2022
Additional Notes: The second author was supported by the Austrian Science Fund (FWF) through the project Analysis and Modeling of Magnetic Skyrmions (grant P-34609). The third author’s research was partially funded by the NSF grant DMS-1906238. The fourth author’s research is partially funded by the NSF Grant DMREF-1921969. The fifth author was partially supported by the NSF award DMS-2006439. The work of the last author was supported, in part, by NSF via grants DMS-0908279, DMS-1313687, DMS-1614948 and DMS-1908709.
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