Scaling laws for non-Euclidean plates and the W^2,2 isometric immersions of Riemannian metrics

¹ University of Minnesota, Department of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA. lewicka@math.umn.edu
² University of Pittsburgh, Department of Mathematics, 139 University Place, Pittsburgh, PA 15260, USA. pakzad@pitt.edu

Received: 28 June 2010

Abstract

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 .