Scaling Laws for Non-euclidean Plates and the W 2,2 Isometric Immersions of Riemannian Metrics
نویسندگان
چکیده
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.
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