A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity

The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a Gamma-limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v : U --> R-n, U subset of R-n. We show that the L-2-distance of delupsilon from a single rotation matrix is bounded by a multiple of the L-2-distance from the group SO(n) of all rotations. (C) 2002 Wiley Periodicals, Inc.