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The uniform Korn-Poincaré inequality in thin domains. (English) Zbl 1253.74055
The authors establish a condition for the constant \(C_h\) in the classic Korn-Poincaré inequality \[ \|u\|_{W^{1,2}(S^{h})} \leq C_{h} \|D(u)\|_{L^{2}(S_h)} \] to be uniformly bounded as \(h\to 0\). Here \(S^h\) are shells of small thickness of order \(h\) around an arbitrary compact boundaryless and smooth hypersurface \(S\) in \(\mathbb R^n\), \(D(u)=0.5 (\nabla u+\nabla u^T)\) and the boundary condition is \(u\cdot n^h=0\) on \(\partial S^h\). The established condition relates with Killing field and appears to be optimal.

MSC:
74K25 Shells
74B05 Classical linear elasticity
35Q74 PDEs in connection with mechanics of deformable solids
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