Let Ω be an open, bounded set in Rn (n ≥ 2) with Lipschitz boundary ∂Ω. Assume that Γ ⊆ ∂Ω has a positive Hn-1 measure and let R : Ω → SO(n) be given. We construct counterexamples to the following ...version of Korn’s inequality in the following two cases: (a) n = 2 and Γ ≠ ∂Ω; (b) n = 3 and Γ = ∂Ω. It is known that the above inequality holds in any dimension n and for any Γ ⊆ ∂Ω, provided that R ∈ C(Ω). However, the inequality is true for n = 2 and Γ = ∂Ω without any regularity assumptions on R.
We prove the Korn's inequality for the conformal Killing operator on pseudo-Euclidean space
R
p
,
q
, and an existence theorem for solutions to the non-homogeneous conformal Killing equation, which ...is a pseudo-Euclidean conformal generalization of Donati's theorem for Euclidean Killing operator.
A NEW APPROACH TO LINEAR SHELL THEORY
Mathematical models and methods in applied sciences/Mathematical models & methods in applied sciences
Journal Article
In a three-dimensional solid with arbitrary periodic Lipschitz perforation the Korn inequality is proved with a constant independent of the perforation size. The convergence rate of homogenization as ...a function of the Sobolev-Slobodetskii smoothness of data is also estimated. We improve foregoing results in elasticity dropping customary restrictions on the shape of the periodicity cell and superfluous smoothness and smallness assumptions on the external forces and traction.
Consider an elastic thin three-dimensional body made of a periodic distribution of elastic inclusions. When both the thickness of the beam and the size of the heterogeneities tend simultaneously to ...zero the authors obtain three different one-dimensional models of beam depending upon the limit of the ratio of these two small parameters.
We state and prove a Korn-like inequality for a vector field in a bounded open set of $\mathbb{R}^N$, satisfying a tangency boundary condition. This inequality, which is crucial in our study of the ...trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case are briefly discussed.
Starting with a short introduction of a new non-conforming linear quadrilateral
P
˜
1
-finite element which has been recently proposed by Park A study on locking phenomena in finite element methods, ...Ph.D. Thesis, Seoul National University, February 2002 and Park and Sheen
P
1
-Nonconforming quadrilateral finite element methods for second-order elliptic problems, SIAM J. Numer. Anal. 41(2) (2003) 624–640, we examine in detail the numerical behaviour of this element with special emphasis on the treatment of Dirichlet boundary conditions, efficient matrix assembly and solver aspects. Furthermore, we compare the numerical characteristics of
P
˜
1
with other low-order finite elements, also regarding its use for the incompressible Navier–Stokes equations. Several test examples show the efficiency and reliability of the proposed methods for elliptic second-order problems.
Appropriate weighted norms in
H
1
are presented such that the Korn type inequality is asymptotically sharp with respect to relative thickness and stiffness of the elastic plates. The weights depend ...crucially on the geometric structure of the plates' junction.
To cite this article: O.V. Izotova et al., C. R. Mecanique 334 (2006).
Des normes dans
H
1
avec poids appropriés sont présentées tels que l'inégalité du type Korn soit asymptotiquement exacte en ce qui concerne l'épaisseur et la rigidité des plaques élastiques. Les poids dépendent crucialement de la structure géométrique de la jonction des plaques.
Pour citer cet article : O.V. Izotova et al., C. R. Mecanique 334 (2006).
By applying the inequality of Korn's type without boundary conditions on a general surface, we prove that the scaled displacement of the two-dimensional linearly viscoelastic Koiter's shell converges ...to the solution of two-dimensional model system of linearly viscoelastic "membrane" shell.PUBLICATION ABSTRACT
A constructive approach is presented to treat an initial boundary value problem for isothermal Navier–Stokes equations. It is based on a characteristics (Lagrangean) approximation locally in time and ...a boundary integral equation method via nonstationary potentials. As a basic problem, the latter leads to a Volterra integral equation of first kind which is proved to be uniquely solvable and even coercive in some anisotropic Sobolev spaces. The solution depends continuously upon the data and can be constructed by a quasioptimal Galerkin procedure.
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