Korn's inequality

In mathematical analysis, Korn's inequality is an inequality concerning the gradient of a vector field that generalizes the following classical theorem: if the gradient of a vector field is skew-symmetric at every point, then the gradient must be equal to a constant skew-symmetric matrix. Korn's theorem is a quantitative version of this statement, which intuitively says that gradient of a vector field is on average not far from the space of skew-symmetric matrices, then the gradient must not be far from a particular skew-symmetric matrix. The statement that Korn's inequality generalizes thus arises as a special case of rigidity.

In (linear) elasticity theory, the symmetric part of the gradient is a measure of the strain that an elastic body experiences when it is deformed by a given vector-valued function. The inequality is therefore an important tool as an a priori estimate in linear elasticity theory.
Statement of the inequality

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

\( \| v \|_{H^{1} (\Omega)} := \left( \int_{\Omega} \sum_{i = 1}^{n} | v^{i} (x) |^{2} \, \mathrm{d} x+\int_{\Omega} \sum_{i, j = 1}^{n} | \partial_{j} v^{i} (x) |^{2} \, \mathrm{d} x \right)^{1/2} \) .

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

\( \| v \|_{H^{1} (\Omega)}^{2} \leq C \int_{\Omega} \sum_{i, j = 1}^{n} \left( | v^{i} (x) |^{2} + | (e_{ij} v) (x) |^{2} \right) \, \mathrm{d} x, \quad (1) \)

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

Horgan, Cornelius O. (1995). "Korn's inequalities and their applications in continuum mechanics". SIAM Rev. 37 (4): 491–511. doi:10.1137/1037123. ISSN 0036-1445. MR1368384

Mathematics Encyclopedia