Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics
University of Minnesota, Department of Mathematics,
206 Church St. S.E., Minneapolis, MN 55455, USA. firstname.lastname@example.org
2 University of Pittsburgh, Department of Mathematics, 139 University Place, Pittsburgh, PA 15260, USA. email@example.com
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 .
Mathematics Subject Classification: 74K20 / 74B20
Key words: Non-Euclidean plates / nonlinear elasticity / Gamma convergence / calculus of variations / isometric immersions
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