The Monge–Ampère constraint: Matching of isometries, density and regularity, and elastic theories of shallow shells

Lewicka, Marta; Mahadevan, L.; Pakzad, Mohammad Reza

doi:10.1016/j.anihpc.2015.08.005

Lewicka, Marta ; Mahadevan, L. ; Pakzad, Mohammad Reza

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 45-67.

Résumé
Abstract

On démontre d'abord deux résultats indépendants, l'un sur la densité des fonctions régulières dans l'ensemble des solutions de l'équation de Monge–Ampère, l'autre sur la construction d'isométries exactes par continuation à partir d'isométries infinitésimales d'ordre 2, pour des surfaces bidimensionelles.

On dérive ensuite un modèle nouveau pour les coques minces peu profondes d'épaisseur h et profondeur de l'ordre de $h^{α}$ départant de la théorie trois-dimensionnelle de l'élasticité nonlinéaire. Le modèle limite obtenu par la Gamma-convergence consiste à minimiser une énergie biharmonique sous une contrainte de type Monge–Ampère. Ce résultat s'applique au cas où les forces sont de l'order de $h^{α + 2}$ et $1 / 2 < α < 1$ . On peut l'étendre pour $α \in (0, 1)$ dans certains cas spécifics, utilisant les résultats de la première partie de l'article.

The main analytical ingredients of the first part of this paper are two independent results: a theorem on approximation of $W^{2, 2}$ solutions of the Monge–Ampère equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometric embeddings of 2d surface in $R^{3}$ .

In the second part, we rigorously derive the Γ-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness h, where the depth of the shell scales like $h^{α}$ and the applied forces scale like $h^{α + 2}$ , in the limit when $h \to 0$ . We offer a full analysis of the problem in the parameter range $α \in (1 / 2, 1)$ . We also complete the analysis in some specific cases for the full range $α \in (0, 1)$ , applying the results of the first part of the paper.

MR Zbl | 1 citation dans Numdam

DOI : 10.1016/j.anihpc.2015.08.005

Classification : 35B65, 53C24, 74K25, 74B20
Mots clés : Monge–Ampère equation, Isometric continuation, Shallow shells, Nonlinear elasticity, Gamma convergence, Calculus of variations

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     title = {The {Monge{\textendash}Amp\`ere} constraint: {Matching} of isometries, density and regularity, and elastic theories of shallow shells},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Lewicka, Marta; Mahadevan, L.; Pakzad, Mohammad Reza. The Monge–Ampère constraint: Matching of isometries, density and regularity, and elastic theories of shallow shells. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 45-67. doi : 10.1016/j.anihpc.2015.08.005. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.08.005/

Bibliographie
Cité par

[1] Ball, J.M. Strict convexity, strong ellipticity and regularity in the calculus of variations, Math. Proc. Camb. Philos. Soc., Volume 87 (1980), pp. 501 | MR | Zbl

[2] Caffarelli, L.A. Interior $W^{2, p}$ estimates for solutions of the Monge–Ampère equation, Ann. Math. (2), Volume 131 (1990) no. 1, pp. 135–150 | DOI | MR | Zbl

[3] Caffarelli, L.A.; Cabré, X. Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43, AMS, 1995 | DOI | MR | Zbl

[4] Calladine, C.R. Theory of Shell Structures, Cambridge University Press, UK, 1983 | DOI | MR | Zbl

[5] Ciarlet, P.G. Mathematical Elasticity, North-Holland, Amsterdam, 1993

[6] Ciarlet, P.G. An Introduction to Differential Geometry with Applications to Elasticity, Springer, Dordrecht, 2005 | DOI | MR | Zbl

[7] Dervaux, J.; Ciarletta, P.; Ben Amar, M. Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Föppl–von Kármán limit, J. Mech. Phys. Solids, Volume 57 (2009) no. 3, pp. 458–471 | DOI | MR | Zbl

[8] Friesecke, G.; James, R.; Müller, S. A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Commun. Pure Appl. Math., Volume 55 (2002), pp. 1461–1506 | DOI | MR | Zbl

[9] Friesecke, G.; James, R.; Müller, S. A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., Volume 180 (2006) no. 2, pp. 183–236 | DOI | MR | Zbl

[10] Han, Q.; Hong, J.X. Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, Providence, RI, 2006 | DOI | MR | Zbl

[11] Hornung, P. Continuation of infinitesimal bendings on developable surfaces and equilibrium equations for nonlinear bending theory of plates, Commun. Partial Differ. Equ., Volume 38 (2013) no. 11, pp. 1368–1408 | MR | Zbl

[12] Hornung, P.; Lewicka, M.; Pakzad, M.R. Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells, J. Elast. (2012) | DOI | MR | Zbl

[13] Huang, Q. Sharp regularity results on second derivatives of solutions to the Monge–Ampère equation with VMO type data, Commun. Pure Appl. Math., Volume 62 (2009) no. 5, pp. 677–705 | DOI | MR | Zbl

[14] Iwaniec, T.; Šverák, V. On mappings with integrable dilatation, Proc. Am. Math. Soc., Volume 118 (1993), pp. 181–188 | DOI | MR | Zbl

[15] von Kármán, T., Encyclopädie der Mathematischen Wissenschaften, Volume vol. IV/4 (1910), pp. 311–385 (Leipzig) | JFM

[16] Kirchheim, B. Geometry and rigidity of microstructures, University of Leipzig, 2001 http://www.mis.mpg.de/preprints/ln/index.html (Habilitation thesis see also: MPI-MIS Lecture Notes 16/2003, 2003)

[17] Lewicka, M.; Mahadevan, L.; Pakzad, M.R. The Föppl–von Kármán equations for plates with incompatible strains, Proc. R. Soc. A, Volume 467 (2011), pp. 402–426 | DOI | MR | Zbl

[18] Lewicka, M.; Mahadevan, L.; Pakzad, M.R. Models for elastic shells with incompatible strains, Proc. R. Soc. A, Volume 470 (2014), pp. 20130604 | DOI

[19] Lewicka, M.; Mora, M.G.; Pakzad, M.R. Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume IX (2010), pp. 1–43 | Numdam | MR | Zbl

[20] Lewicka, M.; Mora, M.G.; Pakzad, M.R. The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Ration. Mech. Anal. (3), Volume 200 (2011), pp. 1023–1050 | MR | Zbl

[21] Lewicka, M.; Pakzad, M.R. The infinite hierarchy of elastic shell models, some recent results and a conjecture, Infinite Dimensional Dynamical Systems, Fields Institute Communications, vol. 64, 2013, pp. 407–420 | DOI | MR | Zbl

[22] Liang, H.; Mahadevan, L. The shape of a long leaf, Proc. Natl. Acad. Sci. USA, Volume 106 (2009), pp. 22049–22054 | DOI | MR | Zbl

[23] Liang, H.; Mahadevan, L. Growth, geometry and mechanics of the blooming lily, Proc. Natl. Acad. Sci. USA, Volume 108 (2011), pp. 5516–5521 | DOI

[24] Malý, J.; Martio, O. Lusin's condition (N) and mappings of the class $W^{1, n}$ , J. Reine Angew. Math., Volume 458 (1995), pp. 19–36 | MR | Zbl

[25] Manfredi, J.J. Weakly monotone functions, J. Geom. Anal., Volume 4 (1994) no. 3, pp. 393–402 | DOI | MR | Zbl

[26] Mardare, S. On isometric immersions of a Riemannian space with little regularity, Anal. Appl., Volume 2 (2004) no. 3, pp. 193–226 | DOI | MR | Zbl

[27] Morrey, C.B. Multiple Integrals in the Calculus of Variations, Springer, 1966 | DOI

[28] Müller, S.; Pakzad, M.R. Regularity properties of isometric immersions, Math. Z., Volume 251 (2005) no. 2, pp. 313–331 | DOI | MR | Zbl

[29] Nirenberg, L. The Weyl and Minkowski problems in differential geometry in the large, Commun. Pure Appl. Math., Volume 6 (1953), pp. 337–394 | DOI | MR | Zbl

[30] Schmidt, B. Plate theory for stressed heterogeneous multilayers of finite bending energy, J. Math. Pures Appl., Volume 88 (2007) no. 1, pp. 107–122 | DOI | MR | Zbl

[31] Šverák, V. On re gularity for the Monge–Ampère equation without convexity assumptions, Heriot-Watt University, 1991 (preprint)

[32] Šverák, V. Regularity properties of deformations with finite energy, Arch. Ration. Mech. Anal., Volume 100 (1988), pp. 105–127 | DOI | MR | Zbl

[33] Trudinger, N.S.; Wang, X.J., Handbook of Geometric Analysis, Volume vol. I, International Press (2008), pp. 467–524 | MR | Zbl

[34] Vodopyanov, S.K.; Goldstein, V.M. Quasiconformal mappings and spaces with generalized derivatives, Sib. Mat. Zh., Volume 17 (1976), pp. 399–411 | MR | Zbl

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