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Scaling laws for non-Euclidean plates and the W2,2 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. lewicka@math.umn.edu
Mohammad Reza Pakzad
Affiliation:
University of Pittsburgh, Department of Mathematics, 139 University Place, Pittsburgh, PA 15260, USA. pakzad@pitt.edu
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Abstract

Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and 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 by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W2,2 isometric immersion of a given 2d metric into $\mathbb R^3$.

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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