The Uniform Korn - Poincaré Inequality in Thin Domains

We study the Korn-Poincaré inequality: ‖u‖ W1,2(Sh) ≤ Ch‖D(u)‖L2(Sh), in domains S that are shells of small thickness of order h, around an arbitrary smooth and closed hypersurface S in R. By D(u) we denote the symmetric part of the gradient ∇u, and we assume the tangential boundary conditions: u · ~n = 0 on ∂S. We prove that Ch remains uniformly bounded as h → 0, for vector fields u in any family of cones (with angle < π/2, uniform in h) around the orthogonal complement of extensions of Killing vector fields on S. We show that this condition is optimal, as in turn every Killing field admits a family of extensions u, for which the ratio ‖u‖ W1,2(Sh)/‖D(u )‖ L2(Sh) blows up as h → 0, even if the domains S are not rotationally symmetric.