scholarly journals A nonlinear Korn inequality in $\protect \mathbb{R}^n$ with an explicitly bounded constant

2020 ◽  
Vol 358 (5) ◽  
pp. 621-626
Author(s):  
Maria Malin ◽  
Cristinel Mardare
Keyword(s):  
Korn Inequality

scholarly journals Boundary Korn Inequality and Neumann Problems in Homogenization of Systems of Elasticity

2017 ◽  
Vol 224 (3) ◽  
pp. 1205-1236 ◽  
Author(s):  
Jun Geng ◽  
Zhongwei Shen ◽  
Liang Song
Keyword(s):  
Neumann Problems ◽  
Korn Inequality

scholarly journals A Nonlinear Korn Inequality Based on the Green-Saint Venant Strain Tensor

2016 ◽  
Vol 126 (1) ◽  
pp. 129-134
Author(s):  
Alessandro Musesti
Keyword(s):  
Strain Tensor ◽  
Korn Inequality

Two low-order nonconforming finite element methods for the Stokes flow in three dimensions

2018 ◽  
Vol 39 (3) ◽  
pp. 1447-1470 ◽  
Author(s):  
Jun Hu ◽  
Mira Schedensack
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Stokes Flow ◽  
Three Dimensions ◽  
Nonconforming Finite Element ◽  
Discrete Problem ◽  
Piecewise Affine ◽  
Korn Inequality ◽  
Nonconforming Finite Element Methods ◽  
Low Order

Abstract In this paper, we propose two low-order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the nonconforming FEM of Kouhia & Stenberg (1995, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng, 124, 195–212). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf–sup condition hold uniformly in the mesh size and also for a nonempty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia–Stenberg FEM to three space dimensions: the finite element space with one nonconforming and two conforming piecewise affine components does not satisfy a discrete inf–sup condition with piecewise constant pressure approximations, while finite element functions with two nonconforming and one conforming component do not satisfy a discrete Korn inequality.

Korn Inequality for a Thin Periodic Corrugated Beam

2017 ◽  
Vol 226 (4) ◽  
pp. 375-387
Author(s):  
G. Leugering ◽  
S. A. Nazarov ◽  
A. S. Slutskii
Keyword(s):  
Korn Inequality ◽  
Corrugated Beam

A conformal Korn inequality on Hölder domains

2020 ◽  
Vol 481 (1) ◽  
pp. 123440
Author(s):  
Zongqi Ding ◽  
Bo Li
Keyword(s):  
Korn Inequality

scholarly journals Korn inequality on irregular domains

2015 ◽  
Vol 423 (1) ◽  
pp. 41-59 ◽  
Author(s):  
Renjin Jiang ◽  
Aapo Kauranen
Keyword(s):  
Irregular Domains ◽  
Korn Inequality

A difference analog of the korn inequality

1989 ◽  
Vol 46 (6) ◽  
pp. 2176-2182 ◽  
Author(s):  
V. D. Glushenkov
Keyword(s):  
Difference Analog ◽  
Korn Inequality

ON KORN'S INEQUALITIES IN CURVILINEAR COORDINATES

2001 ◽  
Vol 11 (08) ◽  
pp. 1379-1391 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
SORIN MARDARE
Keyword(s):  
Three Dimensional ◽  
Curvilinear Coordinates ◽  
Careful Study ◽  
Displacement Fields ◽  
Korn Inequality

We show how the inequality of Korn's type on a surface can be established as a corollary to the three-dimensional Korn inequality in curvilinear coordinates. The proof relies in particular on a careful study of the linearized Kirchhoff–Love displacement fields inside a "shell-like" body.

scholarly journals $$L^p$$-trace-free version of the generalized Korn inequality for incompatible tensor fields in arbitrary dimensions

2021 ◽  
Vol 72 (3) ◽  
Author(s):  
Peter Lewintan ◽  
Patrizio Neff
Keyword(s):  
Matrix Representation ◽  
Three Dimensional ◽  
Type Inequality ◽  
Cross Product ◽  
Dimensional Case ◽  
Tensor Fields ◽  
Korn Inequality ◽  
Free Part ◽  
Arbitrary Dimensions ◽  
Square Matrix

AbstractFor $$n\ge 3$$ n ≥ 3 and $$1<p<\infty $$ 1 < p < ∞ , we prove an $$L^p$$ L p -version of the generalized trace-free Korn-type inequality for incompatible, p-integrable tensor fields $$P:\Omega \rightarrow \mathbb {R}^{n\times n}$$ P : Ω → R n × n having p-integrable generalized $${\text {Curl}}_{n}$$ Curl n and generalized vanishing tangential trace $$P\,\tau _l=0$$ P τ l = 0 on $$\partial \Omega $$ ∂ Ω , denoting by $$\{\tau _l\}_{l=1,\ldots , n-1}$$ { τ l } l = 1 , … , n - 1 a moving tangent frame on $$\partial \Omega $$ ∂ Ω . More precisely, there exists a constant $$c=c(n,p,\Omega )$$ c = c ( n , p , Ω ) such that $$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{n\times n})}\le c\,\left( \Vert {\text {dev}}_n {\text {sym}}P \Vert _{L^p(\Omega ,\mathbb {R}^{n \times n})}+ \Vert {\text {Curl}}_{n} P \Vert _{L^p\left( \Omega ,\mathbb {R}^{n\times \frac{n(n-1)}{2}}\right) }\right) , \end{aligned}$$ ‖ P ‖ L p ( Ω , R n × n ) ≤ c ‖ dev n sym P ‖ L p ( Ω , R n × n ) + ‖ Curl n P ‖ L p Ω , R n × n ( n - 1 ) 2 , where the generalized $${\text {Curl}}_{n}$$ Curl n is given by $$({\text {Curl}}_{n} P)_{ijk} :=\partial _i P_{kj}-\partial _j P_{ki}$$ ( Curl n P ) ijk : = ∂ i P kj - ∂ j P ki and "Equation missing" denotes the deviatoric (trace-free) part of the square matrix X. The improvement towards the three-dimensional case comes from a novel matrix representation of the generalized cross product.

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