On a canonical extension of Korn's first and Poincaré's inequalities to H(CURL)

We prove a Korn-type inequality in H(Curl; Ω, ℝ 3×3) for tensor fields P mapping Ω to ℝ 3×3. More precisely, let Ω ⊂ ℝ 3 be a bounded domain with connected Lipschitz boundary ∂Ω. Then there exists a constant c > 0 such that, for all tensor fields P ∈ H(Curl; Ω, ℝ 3×3), i. e., all P ∈ H(Curl; Ω, ℝ 3×3) with vanishing tangential trace on ∂Ω. Here the rotation and tangential trace are defined row-wise. For compatible P of form P = ∇v, Curl P = 0, where v ∈ H 1(Ω, ℝ 3) is a vector field with components v n for which ∇v n are normal at ∂Ω, estimates (0. 1) is reduced to a non standard variant of Korn's first inequality:, For skew-symmetric P (with sym P = 0), estimates (0. 1) generates a nonstandard version of Poincaré's inequality. Therefore, the estimate is a generalization of two classical inequalities of Poincaré and Korn. Bibliography: 24 titles. © 2012 Springer Science+Business Media, Inc.