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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 .
Keywords: non-euclidean plates, nonlinear elasticity, gamma convergence, calculus of variations, isometric immersions
@article{COCV_2011__17_4_1158_0,
author = {Lewicka, Marta and Reza Pakzad, Mohammad},
title = {Scaling laws for non-euclidean plates and the $W^{2,2}$ isometric immersions of riemannian metrics},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1158--1173},
publisher = {EDP-Sciences},
volume = {17},
number = {4},
year = {2011},
doi = {10.1051/cocv/2010039},
mrnumber = {2859870},
zbl = {1300.74028},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010039/}
}
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
SP - 1158
EP - 1173
VL - 17
IS - 4
PB - EDP-Sciences
UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010039/
DO - 10.1051/cocv/2010039
LA - en
ID - COCV_2011__17_4_1158_0
ER -
%0 Journal Article
%A Lewicka, Marta
%A Reza Pakzad, Mohammad
%T Scaling laws for non-euclidean plates and the $W^{2,2}$ isometric immersions of riemannian metrics
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2011
%P 1158-1173
%V 17
%N 4
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010039/
%R 10.1051/cocv/2010039
%G en
%F COCV_2011__17_4_1158_0
Lewicka, Marta; 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, Tome 17 (2011) no. 4, pp. 1158-1173. doi: 10.1051/cocv/2010039
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