MiS Preprint Repository

We study the Korn-Poincaré inequality: $$\Vert u{\Vert }_{W^{1,2}(S^h)}\le C_h\Vert D(u){\Vert }_{L^2(S^h)},$$ in domains $S^h$ that are shells of small thickness of order $h$, around an arbitrary smooth and closed hypersurface $S$ in ${\mathbf{R}}^n$. By $D(u)$ we denote the symmetric part of the gradient $\mathrm{\nabla }u$, and we assume the tangential boundary conditions: $$u\cdot {\overrightarrow{n}}^h=0\text{ on }\partial S^h.$$ We prove that $C_h$ remains uniformly bounded as $h\to 0$, 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 show that this condition is optimal, as in turn every Killing field admits a family of extensions $u^h$, for which the ratio $\Vert u^h{\Vert }_{W^{1,2}(S^h)}/\Vert D(u^h){\Vert }_{L^2(S^h)}$ blows up as $h\to 0$, even if the domains $S^h$ are not rotationally symmetric.