Korn's inequality

In mathematics, Korn's inequality is a result about the derivatives of Sobolev functions. Korn's inequality plays an important rôle in linear elasticity theory.

tatement of the inequality

Let Ω be an open, connected domain in "n"-dimensional Euclidean space R^"n", "n" ≥ 2. Let "H"¹(Ω) be the Sobolev space of all vector fields "v" = ("v"¹, ..., "v"^"n") on Ω that, along with their weak derivatives, lie in the Lebesgue space "L"²(Ω). Denoting the partial derivative with respect to the "i"^th component by ∂_"i", the norm in "H"¹(Ω) is given by

:$|\; v\; |\_\{H^\{1\}\; (Omega)\}\; :=\; left(\; int\_\{Omega\}\; sum\_\{i\; =\; 1\}^\{n\}\; |\; v^\{i\}\; (x)\; |^\{2\}\; ,\; mathrm\{d\}\; x\; ight)^\{1/2\}\; +\; left(\; int\_\{Omega\}\; sum\_\{i,\; j\; =\; 1\}^\{n\}\; |\; partial\_\{j\}\; v^\{i\}\; (x)\; |^\{2\}\; ,\; mathrm\{d\}\; x\; ight)^\{1/2\}.$

Then there is a constant "C" ≥ 0, known as the Korn constant of Ω, such that, for all "v" ∈ "H"¹(Ω),

:$|\; v\; |\_\{H^\{1\}\; (Omega)\}^\{2\}\; leq\; C\; int\_\{Omega\}\; sum\_\{i,\; j\; =\; 1\}^\{n\}\; left(\; |\; v^\{i\}\; (x)\; |^\{2\}\; +\; |\; (e\_\{ij\}\; v)\; (x)\; |^\{2\}\; ight)\; ,\; mathrm\{d\}\; x,\; quad\; (1)$

where "e" denotes the symmetrized gradient given by

:$e\_\{ij\}\; v\; =\; frac1\{2\}\; (\; partial\_\{i\}\; v^\{j\}\; +\; partial\_\{j\}\; v^\{i\}\; ).$

Inequality (1) is known as Korn's inequality.

References

* cite journal
last = Horgan
first = Cornelius O.
title = Korn's inequalities and their applications in continuum mechanics
journal = SIAM Rev.
volume = 37
year = 1995
number = 4
pages = 491–511
issn = 0036-1445
doi = 10.1137/1037123 MathSciNet|id=1368384