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Scaling laws for non-Euclidean plates and theW2,2 isometric immersions of Riemannian metrics

Published online by Cambridge University Press:  28 October 2010

Marta Lewicka  and
Mohammad Reza Pakzad
Marta Lewicka
Affiliation:
University of Minnesota, Department of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA. [email protected]
Mohammad Reza Pakzad
Affiliation:
University of Pittsburgh, Department of Mathematics, 139 University Place, Pittsburgh, PA 15260, USA. [email protected]
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Abstract

Recall that a smooth Riemannian metric on a simply connected domain canbe realized as the pull-back metric of an orientation preserving deformation ifand only if the associated Riemann curvature tensor vanishes identically.When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem byintroducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the properscaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W2,2 isometric immersion of a given 2d metricinto $\mathbb R^3$.

Keywords

Non-Euclidean platesnonlinear elasticityGamma convergencecalculus of variationsisometric immersions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 1158 - 1173
Copyright
© EDP Sciences, SMAI, 2010

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References

Calabi, E. and Hartman, P., On the smoothness of isometries. Duke Math. J. 37 (1970) 741–750. CrossRef
G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser (1993).
Efrati, E., Sharon, E. and Kupferman, R., Elastic theory of unconstrained non-Euclidean plates. J. Mech. Phys. Solids 57 (2009) 762–775. CrossRef
Friesecke, G., James, R. and Müller, S., A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure Appl. Math. 55 (2002) 1461–1506. CrossRef
Friesecke, G., James, R., Mora, M.G. and Müller, S., Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris 336 (2003) 697–702. CrossRef
Friesecke, G., James, R. and Müller, S., A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180 (2006) 183–236. CrossRef
M. Gromov, Partial Differential Relations. Springer-Verlag, Berlin-Heidelberg (1986).
Guan, P. and The Weyl, Y. Li problem with nonnegative Gauss curvature. J. Diff. Geometry 39 (1994) 331–342. CrossRef
Q. Han and J.-X. Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, Mathematical Surveys and Monographs 130. American Mathematical Society, Providence (2006).
Hong, J.-X. and Zuily, C., Isometric embedding of the 2-sphere with nonnegative curvature in $\mathbb R^3$. Math. Z. 219 (1995) 323–334. CrossRef
Iaia, J.A., Isometric embeddings of surfaces with nonnegative curvature in $\mathbb R^3$. Duke Math. J. 67 (1992) 423–459. CrossRef
Klein, Y., Efrati, E. and Sharon, E., Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315 (2007) 1116–1120. CrossRef
Kuiper, N.H., On C 1 isometric embeddings. I. Indag. Math. 17 (1955) 545–556. CrossRef
Kuiper, N.H., On C 1 isometric embeddings. II. Indag. Math. 17 (1955) 683–689. CrossRef
M. Lewicka, M.G. Mora and M.R. Pakzad, A nonlinear theory for shells with slowly varying thickness. C.R. Acad. Sci. Paris, Ser. I 347 (2009) 211–216.
M. Lewicka, M.G. Mora and M.R. Pakzad, Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. IX (2010) 1–43.
Liu, F.C., Lusin, A property of Sobolev functions. Indiana U. Math. J. 26 (1977) 645–651. CrossRef
Pogorelov, A.V., An example of a two-dimensional Riemannian metric which does not admit a local realization in E3. Dokl. Akad. Nauk. SSSR (N.S.) 198 (1971) 42–43. [Soviet Math. Dokl. 12 (1971) 729–730.]
M. Spivak, A Comprehensive Introduction to Differential Geometry. Third edition, Publish or Perish Inc. (1999).