Minimizers of Kirchhoff's plate functional: Euler–Lagrange equations and regularity

Peter Hornung

doi:10.1016/j.crma.2009.03.031

Differential Geometry

Minimizers of Kirchhoff's plate functional: Euler–Lagrange equations and regularity
[Minimiseurs de la fonctionnelle de Kirchhoff : équations de Euler–Lagrange et régularité]

Présenté par : John Ball

Peter Hornung ¹

¹ Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 647-650.

Résumé
Abstract

Soit $S \subset R^{2}$ un $C^{\infty}$ -domaine borné. Dans cette Note on considère une immersion $W^{2, 2}$ -isométrique $u : S \to R^{3}$ qui minimise la fonctionnelle de Kirchhoff sous les conditions frontières imposant les valeurs de u et ∇u sur des partie de ∂S. On en déduit les équations de Euler–Lagrange satisfaites par u et un résultat de régularité pour u.

Let $S \subset R^{2}$ be a bounded $C^{\infty}$ -domain. In this Note we consider $W^{2, 2}$ isometric immersions $u : S \to R^{3}$ which minimize Kirchhoff's plate functional under boundary conditions prescribing the values of u and of ∇u on parts of ∂S. We derive the Euler–Lagrange equations satisfied by u and we derive regularity results for u.

Reçu le : 2008-03-13
Accepté le : 2009-03-31
Publié le : 2009-04-21

DOI : 10.1016/j.crma.2009.03.031

Affiliations des auteurs :

Peter Hornung ¹

¹ Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

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     author = {Peter Hornung},
     title = {Minimizers of {Kirchhoff's} plate functional: {Euler{\textendash}Lagrange} equations and regularity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {647--650},
     publisher = {Elsevier},
     volume = {347},
     number = {11-12},
     year = {2009},
     doi = {10.1016/j.crma.2009.03.031},
     language = {en},
}

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Peter Hornung. Minimizers of Kirchhoff's plate functional: Euler–Lagrange equations and regularity. Comptes Rendus. Mathématique, Volume 347 (2009) no. 11-12, pp. 647-650. doi : 10.1016/j.crma.2009.03.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.031/

Bibliographie
Cité par

[1] C. Bohle; G.P. Peters; U. Pinkall Constrained Willmore surfaces, Calc. Var., Volume 32 (2008), pp. 263-277

[2] G. Friesecke; R. James; S. Müller A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., Volume 55 (2002), pp. 1461-1506

[3] P. Hornung, A density result for $W^{2, 2}$ isometric immersions, MIS MPG Preprint, 2007

[4] P. Hornung Approximating $W^{2, 2}$ isometric immersions, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 189-192

[5] P. Hornung, Flat minimizers of the Willmore functional: Euler–Lagrange equations, Preprint, Universität Bonn, 2008

[6] P. Hornung, Regularity results for flat minimizers of the Willmore functional, Preprint, Universität Bonn, 2008

[7] B. Kirchheim, Geometry and Rigidity of Microstructures, Habilitation thesis, University of Leipzig, 2001

[8] S. Müller; M.R. Pakzad Regularity properties of isometric immersions, Math. Z., Volume 251 (2005), pp. 313-331

[9] M.R. Pakzad On the Sobolev space of isometric immersions, J. Differential Geom., Volume 66 (2004) no. 1, pp. 47-69

[10] E.L. Starostin; G.H.M. van der Heijden The shape of a Möbius strip, Nature Materials, Volume 6 (2007), pp. 563-567

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