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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
##### Keywords:
shell; Killing vector field
Full Text:
##### References:
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