# Scaling laws for non-euclidean plates and the ${W}^{2,2}$ 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

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topLewicka, Marta, and Reza Pakzad, Mohammad. "Scaling laws for non-euclidean plates and the $W^{2,2}$ isometric immersions of riemannian metrics." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 1158-1173. <http://eudml.org/doc/272774>.

@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},

number = {4},

pages = {1158-1173},

publisher = {EDP-Sciences},

title = {Scaling laws for non-euclidean plates and the $W^\{2,2\}$ isometric immersions of riemannian metrics},

url = {http://eudml.org/doc/272774},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Lewicka, Marta

AU - Reza Pakzad, Mohammad

TI - Scaling laws for non-euclidean plates and the $W^{2,2}$ isometric immersions of riemannian metrics

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2011

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/272774

ER -

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