The uniform Korn–Poincaré inequality in thin domains

Abstract We study the Korn–Poincare 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 = 0 on ∂ S h . We prove that C h remains uniformly bounded as h → 0 , 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 h → 0 , even if the domains S h are not rotationally symmetric.

[1]  A. Korn.  Solution générale du problème d'équilibre dans la théorie de l'élasticité, dans le cas ou les efforts sont donnés à la surface , 1908 .

[2]  Robert V. Kohn,et al.  A new model for thin plates with rapidly varying thickness , 1984 .

[3]  Hans F. Weinberger,et al.  On Korn's inequality , 1961 .

[4]  正人 木村.  Max-Planck-Institute for Mathematics in the Sciences(海外,ラボラトリーズ) , 2001 .

[5]  R. Kohn,et al.  A New Model for Thin Plates with Rapidly Varying Thickness. II. A Convergence Proof. , 1985 .

[6]  G. Friesecke,et al.  Mathematik in den Naturwissenschaften Leipzig A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence , 2005 .

[7]  Geneviève Raugel,et al.  Dynamics of partial differential equations on thin domains , 1995 .

[8]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[9]  Cornelius O. Horgan,et al.  Korn's Inequalities and Their Applications in Continuum Mechanics , 1995, SIAM Rev..

[10]  G. Friesecke,et al.  A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity , 2002 .

[11]  Kurt Friedrichs,et al.  On the Boundary-Value Problems of the Theory of Elasticity and Korn's Inequality , 1947 .

[12]  K. Nomizu,et al.  Foundations of Differential Geometry , 1963 .

[13]  On Poincaré Type Inequalities , 1997 .

[14]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[15]  George R. Sell,et al.  Navier-Stokes equations on thin 3D domains. I. Global attractors and global regularity of solutions , 1993 .

[16]  A. Kufner.  Weighted Sobolev Spaces , 1985 .

[17]  G. Sell,et al.  Navier-Stokes equations in thin 3D domains with Navier boundary conditions , 2007 .

[18]  G. Griso.  Asymptotic behaviour of curved rods by the unfolding method , 2004 .

[19]  J. Jost,et al.  A Riemannian version of Korn's inequality , 2002 .

[20]  J. Nédélec,et al.  Functional spaces for norton‐hoff materials , 1986 .

[21]  Decompositions of displacements of thin structures , 2008 .

[22]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[23]  B. Nowakowski,et al.  Strong solutions to the Navier-Stokes equations on thin 3D domains , 2012, 1204.5988.