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

We prove a Korn-type inequality in $\mathop {\text{H}}\limits^\circ \left( {{\text{Curl}};\Omega, {\mathbb{R}^{3 \times 3}}} \right)$ for tensor fields P mapping Ω to ${\mathbb{R}^{3 \times 3}}$. More precisely, let $\Omega \subset {\mathbb{R}^3}$ be a bounded domain with connected Lipschitz boundary ∂Ω. Then there exists a constant c > 0 such that

$$ c{\left\| P \right\|_{{{\text{L}}^2}\left( {\Omega, {\mathbb{R}^{3 \times 3}}} \right)}} \leqslant {\left\| {{\text{sym}}\,P} \right\|_{{{\text{L}}^2}\left( {\Omega, {\mathbb{R}^{3 \times 3}}} \right)}} + {\left\| {{\text{Curl}}\,P} \right\|_{{{\text{L}}^2}\left( {\Omega, {\mathbb{R}^{3 \times 3}}} \right)}} $$

(0.1)

for all tensor fields P ∈ $\mathop {\text{H}}\limits^\circ \left( {{\text{Curl}};\Omega, {\mathbb{R}^{3 \times 3}}} \right)$, i.e., all P ∈ ${\text{H}}\left( {{\text{Curl}};\Omega, {\mathbb{R}^{3 \times 3}}} \right)$ 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 $v \in {{\text{H}}^1}\left( {\Omega, {\mathbb{R}^3}} \right)$ 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:

$$ c{\left\| {\nabla \upsilon } \right\|_{{{\text{L}}^2}\left( {\Omega, {\mathbb{R}^{3 \times 3}}} \right)}} \leqslant {\left\| {{\text{sym}}\,\nabla \upsilon } \right\|_{{{\text{L}}^2}\left( {\Omega, {\mathbb{R}^{3 \times 3}}} \right)}}. $$

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.