The uniform Korn–Poincaré inequality in thin domains  [ Lʼinégalité de Korn–Poincaré dans les domaines minces ]
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 3, p. 443-469
On étudie lʼinégalité de Korn–Poincaré : u W 1,2 (S h ) C h D(u) L 2 (S h ) , dans les domaines S h de type des coques dʼépaisseurs dʼordre h autour dʼune hypersurface compacte sans bord et regulière S de 𝐑 n . Par D(u), on réfère à la partie symétrique du gradient ∇u et on suppose la condition au bord : u·n h =0onS h . On démontre que C h reste uniformément borné car h0, pour tout champ de vecteurs dans une famille de cônes donnée (faisant un angle<π/2, uniforme en h) autour du complément orthogonal des extensions de champs de vecteurs de Killing sur S.On montre que cette condition est optimale comme tout champ de Killling u sur S admet une famille dʼextensions u h sur S h pour lesquelles le rapport u h W 1,2 (S h ) /D(u h ) L 2 (S h ) tend à lʼinfini comme h0, même si les S h ne possèdent pas de symmetrie axiale.
We study the Korn–Poincaré 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 =0onS h . We prove that C h remains uniformly bounded as h0, 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 h0, even if the domains S h are not rotationally symmetric.
@article{AIHPC_2011__28_3_443_0,
     author = {Lewicka, Marta and M\"uller, Stefan},
     title = {The uniform Korn--Poincar\'e inequality in thin domains},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {28},
     number = {3},
     year = {2011},
     pages = {443-469},
     doi = {10.1016/j.anihpc.2011.03.003},
     zbl = {1253.74055},
     mrnumber = {2795715},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2011__28_3_443_0}
}
Lewicka, Marta; Müller, Stefan. The uniform Korn–Poincaré inequality in thin domains. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 3, pp. 443-469. doi : 10.1016/j.anihpc.2011.03.003. http://www.numdam.org/item/AIHPC_2011__28_3_443_0/

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