Preprint 102/2007

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
Bibtex
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Abstract:
We study the Korn-Poincaré inequality:
equation4
in domains formula19 that are shells of small thickness of order h, around an arbitrary smooth and closed hypersurface S in formula25. By D(u) we denote the symmetric part of the gradient formula29, and we assume the tangential boundary conditions:
equation9
We prove that formula31 remains uniformly bounded as formula33, for vector fields u in any family of cones (with angle formula37, 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 formula43, for which the ratio formula45 blows up as formula33, even if the domains formula19 are not rotationally symmetric.

23.06.2018, 02:11