The uniform Korn–Poincaré inequality in thin domains

Abstract

We study the Korn–Poincaré inequality:‖u‖W1,2(Sh)⩽Ch‖D(u)‖L2(Sh), in domains Sh that are shells of small thickness of order h, around an arbitrary compact, boundaryless and smooth hypersurface S in Rn. By D(u) we denote the symmetric part of the gradient ∇u, and we assume the tangential boundary conditions:u⋅n→h=0on ∂Sh. 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 uh, for which the ratio ‖uh‖W1,2(Sh)/‖D(uh)‖L2(Sh) blows up as h→0, even if the domains Sh are not rotationally symmetric.