ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2011 - Volume 17, Issue 04  

Research Article

Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics

Lewicka, Martaa1 and Reza Pakzad, Mohammada2

a1 University of Minnesota, Department of Mathematics, 206 Church St. S.E., Minneapolis, MN 55455, USA. lewicka@math.umn.edu

a2 University of Pittsburgh, Department of Mathematics, 139 University Place, Pittsburgh, PA 15260, USA. pakzad@pitt.edu

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 W2,2 isometric immersion of a given 2d metric into $\mathbb R^3$.

(Received June 28 2010)

(Online publication October 28 2010)

Key Words:

  • Non-Euclidean plates;
  • nonlinear elasticity;
  • Gamma convergence;
  • calculus of variations;
  • isometric immersions

Mathematics Subject Classification:

  • 74K20;
  • 74B20