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 if the 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. ‖ v ‖ H 1 ( Ω ) 2 ≤ C ∫ Ω ∑ i , j = 1 n ( | v i ( x ) | 2 + | ( e i j v ) ( x ) | 2 ) d x {displaystyle |v|_{H^{1}(Omega )}^{2}leq Cint _{Omega }sum _{i,j=1}^{n}left(|v^{i}(x)|^{2}+|(e_{ij}v)(x)|^{2} ight),mathrm {d} x}     (1) 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 if the 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. Let Ω be an open, connected domain in n-dimensional Euclidean space Rn, n ≥ 2. Let H1(Ω) be the Sobolev space of all vector fields v = (v1, ..., vn) on Ω that, along with their (first) weak derivatives, lie in the Lebesgue space L2(Ω). Denoting the partial derivative with respect to the ith component by ∂i, the norm in H1(Ω) is given by Then there is a constant C ≥ 0, known as the Korn constant of Ω, such that, for all v ∈ H1(Ω), where e denotes the symmetrized gradient given by Inequality (1) is known as Korn's inequality.