The uniform Korn - Poincar\'e inequality in thin domains

We study the Korn-Poincar\'e inequality: \|u\|_{W^{1,2}(S^h)} < C_h \|D(u)\|_{L^2(S^h)}, in domains S^h that are shells of small thickness of order h, around an arbitrary smooth and closed hypersurface S in R^n. By D(u) we denote the symmetric part of the gradient \nabla u, and we assume the tangential boundary conditions: u\vec n^h = 0 on \partial S^h. We prove that C_h remains uniformly bounded as h tends to 0, for vector fields u in any family of cones (with angle <\pi/2, uniform in h) around the orthogonal complement of extensions of Killing vector fields on S. We also show that this condition is optimal, as in turn every Killing field admits a family of extensions u^h, for which the ratio: \|u^h\|_{W^{1,2}(S^h)} / \|D(u^h)\|_{L^2(S^h)} blows up as h tends to 0, even if the domains S^h are not rotationally symmetric.