We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
102/2007
The uniform Korn-Poincaré inequality in thin domains
Marta Lewicka and Stefan Müller
Abstract
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.