Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics

Marta Lewicka; Mohammad Reza Pakzad

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 1158-1173
  • ISSN: 1292-8119

Abstract

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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 ℝ 3 .

How to cite

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Lewicka, Marta, and Reza Pakzad, Mohammad. "Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 1158-1173. <http://eudml.org/doc/221901>.

@article{Lewicka2011,
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 $\mathbb R^3$. },
author = {Lewicka, Marta, Reza Pakzad, Mohammad},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Non-Euclidean plates; nonlinear elasticity; Gamma convergence; calculus of variations; isometric immersions},
language = {eng},
month = {11},
number = {4},
pages = {1158-1173},
publisher = {EDP Sciences},
title = {Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics},
url = {http://eudml.org/doc/221901},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Lewicka, Marta
AU - Reza Pakzad, Mohammad
TI - Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/11//
PB - EDP Sciences
VL - 17
IS - 4
SP - 1158
EP - 1173
AB - 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 $\mathbb R^3$.
LA - eng
KW - Non-Euclidean plates; nonlinear elasticity; Gamma convergence; calculus of variations; isometric immersions
UR - http://eudml.org/doc/221901
ER -

References

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