The uniform Korn-Poincaré inequality in thin domains

Marta Lewicka and Stefan Müller

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Submission date: 19. Nov. 2007
Pages: 32
published in: Annales de l'Institut Henri Poincaré / C, 28 (2011) 3, p. 443-469 
DOI number (of the published article): 10.1016/j.anihpc.2011.03.003
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Abstract:
We study the Korn-Poincaré inequality:

in domains that are shells of small thickness of order h, around an arbitrary smooth and closed hypersurface S in . By D(u) we denote the symmetric part of the gradient , and we assume the tangential boundary conditions:

We prove that remains uniformly bounded as , for vector fields u in any family of cones (with angle , 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 , for which the ratio blows up as , even if the domains are not rotationally symmetric.