The uniform Korn–Poincaré inequality in thin domains

[Lʼinégalité de Korn–Poincaré dans les domaines minces]

Lewicka, Marta ; Müller, Stefan

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011), p. 443-469 / Harvested from Numdam

Résumé

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 $ {\mathbf{R}}^{n}$. Par $ D(u)$, on réfère à la partie symétrique du gradient ∇u et on suppose la condition au bord : u·n → h =0on∂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 $ \Vert {u}^{h}{\Vert }_{{W}^{1,2}({S}^{h})}/\Vert D({u}^{h}){\Vert }_{{L}^{2}({S}^{h})}$ tend à lʼinfini comme $ h\to 0$, même si les $ {S}^{h}$ ne possèdent pas de symmetrie axiale.

Abstract

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 $ {\mathbf{R}}^{n}$. By $ D(u)$ we denote the symmetric part of the gradient ∇u, and we assume the tangential boundary conditions: u·n → h =0on∂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 $ \Vert {u}^{h}{\Vert }_{{W}^{1,2}({S}^{h})}/\Vert D({u}^{h}){\Vert }_{{L}^{2}({S}^{h})}$ blows up as $ h\to 0$, even if the domains $ {S}^{h}$ are not rotationally symmetric.

Détail

Publié le : 2011-01-01

MR 2795715 | Zbl 1253.74055

DOI : https://doi.org/10.1016/j.anihpc.2011.03.003
Classification:  74B05

Bibliographie

[1] R. Chen, P. Li, On Poincaré type inequalities, Trans. Amer. Math. Soc. 349 no. 4 (1997), 1561-1585 | MR 1401517 | Zbl 0954.58022

[2] W. Chen, J. Jost, A Riemannian version of Kornʼs inequality, Calc. Var. Partial Differential Equations 14 (2002), 517-530 | MR 1911827 | Zbl 1006.74011

[3] P.G. Ciarlet, Mathematical Elasticity, vol. 1: Three Dimensional Elasticity, North-Holland, Amsterdam (1993)

[4] K.O. Friedrichs, On the boundary-value problems of the theory of elasticity and Kornʼs inequality, Ann. of Math. 48 no. 2 (1947), 441-471 | MR 22750 | Zbl 0029.17002

[5] G. Friesecke, R. James, S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure Appl. Math. 55 (2002), 1461-1506 | MR 1916989 | Zbl 1021.74024

[6] G. Friesecke, R. James, S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal. 180 no. 2 (2006), 183-236 | MR 2210909 | Zbl 1100.74039

[7] G. Geymonat, P. Suquet, Functional spaces for Norton–Hoff materials, Math. Methods Appl. Sci. 8 (1986), 206-222 | MR 845925 | Zbl 0616.73010

[8] G. Griso, Asymptotic behaviour of curved rods by the unfolding method, Math. Methods Appl. Sci. 27 (2004), 2081-2110 | MR 2099819 | Zbl 1174.74313

[9] G. Griso, Asymptotic behavior of structures made of plates, Anal. Appl. 3 (2005), 325-356 | MR 2181252 | Zbl 1111.74029

[10] G. Griso, Decompositions of displacements of thin structures, J. Math. Pures Appl. 89 (2008), 199-223 | MR 2391647 | Zbl 1132.74029

[11] C.O. Horgan, Kornʼs inequalities and their applications in continuum mechanics, SIAM Rev. 37 no. 4 (1995), 491-511 | MR 1368384 | Zbl 0840.73010

[12] D. Iftimie, G. Raugel, G. Sell, Navier–Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J. 56 no. 3 (2007), 1083-1156 | MR 2333468 | Zbl 1129.35056

[13] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. 1, Interscience Publishers (1963) | MR 152974 | Zbl 0175.48504

[14] R.V. Kohn, M. Vogelius, A new model for thin plates with rapidly varying thickness. II: A convergence proof, Quart. Appl. Math. 43 (1985), 1-22 | MR 782253 | Zbl 0565.73046

[15] A. Korn, Solution générale du problème dʼéquilibre dans la théorie de lʼélasticité dans le cas où les efforts sont donnés à la surface, Ann. Fac. Sci. Toulouse Ser. 2 10 (1908), 165-269 | JFM 39.0853.03 | MR 1508302

[16] A. Korn, Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen, Bull. Int. Cracovie Akademie Umiejet, Classe des Sci. Math. Nat. (1909), 705-724 | JFM 40.0884.02

[17] A. Kufner, Weighted Sobolev Spaces, Wiley and Sons (1985) | MR 802206 | Zbl 0567.46009

[18] V. Kondratiev, O. Oleinik, On Kornʼs inequalities, C. R. Acad. Sci. Paris Ser. I 308 (1989), 483-487 | MR 995908

[19] P. Petersen, Riemannian Geometry, Springer (2006) | MR 2243772 | Zbl 1220.53002

[20] G. Raugel, Dynamics of partial differential equations on thin domains, CIME Course, Montecatini Terme, Lecture Notes in Math. vol. 1609, Springer-Verlag (1995), 208-315 | MR 1374110 | Zbl 0851.58038

[21] G. Raugel, G.R. Sell, Navier–Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993), 503-568 | MR 1179539 | Zbl 0787.34039

[22] V.A. Solonnikov, V.E. Scadilov, A certain boundary value problem for the stationary system of Navier–Stokes equations, Boundary Value Problems of Mathematical Physics, 8 Tr. Mat. Inst. Steklova 125 (1973), 196-210 | MR 364910 | Zbl 0313.35063

[23] M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc. (1979) | Zbl 0439.53001