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

Lewicka, Marta; Reza Pakzad, Mohammad

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 4, pp. 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 W^2,2 isometric immersion of a given 2d metric into ${ℝ}^3$.

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