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 Γ‐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 ⊂ ℝ n . We show that the L 2 ‐distance of ∇ v from a single rotation matrix is bounded by a multiple of the L 2 ‐distance from the group SO( n ) of all rotations.