Issue |
ESAIM: COCV
Volume 17, Number 4, October-December 2011
|
|
---|---|---|
Page(s) | 1158 - 1173 | |
DOI | https://doi.org/10.1051/cocv/2010039 | |
Published online | 28 October 2010 |
Scaling laws for non-Euclidean plates and the W2,2 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
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
© EDP Sciences, SMAI, 2010
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