MiS Preprint Repository

We study the Korn-Poincaré inequality: \begin{equation*} \|u\|_{W^{1,2}(S^h)} \leq C_h \|D(u)\|_{L^2(S^h)}, \end{equation*} 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 $\nabla u$, and we assume the tangential boundary conditions: \begin{equation*} u\cdot\vec n^h = 0 \quad \mbox{ on } \partial S^h. \end{equation*} 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 $\|u^h\|_{W^{1,2}(S^h)} / \|D(u^h)\|_{L^2(S^h)}$ blows up as $h\to 0$, even if the domains $S^h$ are not rotationally symmetric.