Scaling laws for non-euclidean plates and the W 2,2 isometric immersions of riemannian metrics
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, p. 1158-1173

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 3 .

DOI : https://doi.org/10.1051/cocv/2010039
Classification:  74K20,  74B20
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},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {4},
     year = {2011},
     pages = {1158-1173},
     doi = {10.1051/cocv/2010039},
     zbl = {1300.74028},
     mrnumber = {2859870},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 17 (2011) no. 4, pp. 1158-1173. doi : 10.1051/cocv/2010039. http://www.numdam.org/item/COCV_2011__17_4_1158_0/

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