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 .
Mots clés : 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://www.numdam.org/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://www.numdam.org/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://www.numdam.org/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. http://www.numdam.org/articles/10.1051/cocv/2010039/
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