The uniform Korn–Poincaré inequality in thin domains

By Marta Lewicka, S. Muller. 2008

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Abstract We study the Korn–Poincare inequality: ‖ u ‖ W 1 , 2 ( S h ) ⩽ C h ‖ D ( u ) ‖ L 2 ( S h ) , in domains S h that are shells of small thickness of order h, around an arbitrary compact, boundaryless and smooth hypersurface S in R n . By D ( u ) we denote the symmetric part of the gradient ∇u, and we assume the tangential boundary conditions: u ⋅ n → h = 0 on ∂ S h . We prove that C h remains uniformly bounded as h → 0 , for vector fields u in any family of cones (with angle π / 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 → 0 , even if the domains S h are not rotationally symmetric.

Subfields: Applied Mathematics

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