The uniform Korn–Poincaré inequality in thin domains

Lewicka, Marta; Müller, Stefan

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 3, pp. 443-469.

On étudie lʼinégalité de Korn–Poincaré :

$$\parallel u{\parallel }_{W^{1,2}\left(S^h\right)}⩽C_h\parallel D\left(u\right){\parallel }_{L^2\left(S^h\right)},$$

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 $D\left(u\right)$, on réfère à la partie symétrique du gradient ∇u et on suppose la condition au bord :

$$u·{\overrightarrow{n}}^h=0\text{on}\partial S^h.$$

On démontre que $C_h$ reste uniformément borné car $h\to 0$, pour tout champ de vecteurs dans une famille de cônes donnée (faisant un $\text{angle}<\pi /2$, 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 $\parallel u^h{\parallel }_{W^{1,2}\left(S^h\right)}/\parallel D\left(u^h\right){\parallel }_{L^2\left(S^h\right)}$ tend à lʼinfini comme $h\to 0$, même si les $S^h$ ne possèdent pas de symmetrie axiale.

We study the Korn–Poincaré inequality:

$$\parallel u{\parallel }_{W^{1,2}\left(S^h\right)}⩽C_h\parallel D\left(u\right){\parallel }_{L^2\left(S^h\right)},$$

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 $D\left(u\right)$ we denote the symmetric part of the gradient ∇u, and we assume the tangential boundary conditions:

$$u·{\overrightarrow{n}}^h=0\text{on}\partial S^h.$$

We prove that $C_h$ remains uniformly bounded as $h\to 0$, for vector fields u in any family of cones (with $\text{angle}<\pi /2$, 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 $\parallel u^h{\parallel }_{W^{1,2}\left(S^h\right)}/\parallel D\left(u^h\right){\parallel }_{L^2\left(S^h\right)}$ blows up as $h\to 0$, even if the domains $S^h$ are not rotationally symmetric.

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