$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\ \begin{eqnarray} \sqrt{2}\left\|\triangledown u\right\|_{L^2(\Omega)}\leq \left\|\triangledown u+(\triangledown u)^T\right\|_{L^2(\Omega)} \end{eqnarray} for any $ u\in H_{0}^{1}(\Omega;\mathbb{R}^d) $, where $ (\triangledown u)^T $ denotes the transpose of $ \triangledown u $.

$\textbf{Theorem}.2$ (The second Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. If $ u\in H^{1}(\Omega,\mathbb{R}^d) $ is a function with the property that $ u\perp R $ in $ H^{1}(\Omega;\mathbb{R}^d) $, then
\begin{eqnarray} \int_{\Omega}|\triangledown u|^2dx\leq C\int_{\Omega}|\triangledown u+(\triangledown u)^T|^2dx \end{eqnarray} where $ R=\left\{\phi=Bx+b:B\in\mathbb{R}^{d\times d} \text{ is skew-symmetric and }b\in\mathbb{R}^d\right\} $ and $ C $ is a constant.

I recently see the two theroems in a book about elliptic equations. I tried to get the estimate for the second inequality by direct computation which works in the proof of the first Korn inequality, but for this inequality, I cannot combine the condition $ u\perp R $ with the final results. Can you give me some hints or references?

$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\ \begin{eqnarray} \sqrt{2}\left\|\triangledown u\right\|_{L^2(\Omega)}\leq \left\|\triangledown u+(\triangledown u)^T\right\|_{L^2(\Omega)} \end{eqnarray} for any $ u\in H_{0}^{1}(\Omega;\mathbb{R}^d) $, where $ (\triangledown u)^T $ denotes the transpose of $ \triangledown u $.

$\textbf{Theorem}.2$ (The second Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. If $ u\in H^{1}(\Omega,\mathbb{R}^d) $ is a function with the property that $ u\perp R $ in $ H^{1}(\Omega;\mathbb{R}^d) $, then
\begin{eqnarray} \int_{\Omega}|\triangledown u|^2dx\leq C\int_{\Omega}|\triangledown u+(\triangledown u)^T|^2dx \end{eqnarray} where $ R=\left\{\phi=Bx+b:B\in\mathbb{R}^{d\times d} \text{ is skew-symmetric and }b\in\mathbb{R}^d\right\} $ and $ C $ is a constant.

I recently see the two theorems in a book about elliptic equations. I tried to get the estimate for the second inequality by direct computation which works in the proof of the first Korn inequality, but for this inequality, I cannot combine the condition $ u\perp R $ with the final results. Can you give me some hints or references?

Suppose $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. If $ u\in H^{1}(\Omega,\mathbb{R}^d) $ is a function with the property that $ u\perp R $ in $ H^{1}(\Omega;\mathbb{R}^d) $, then
\begin{eqnarray} \int_{\Omega}|\triangledown u|^2dx\leq C\int_{\Omega}|\triangledown u+(\triangledown u)^T|^2dx \end{eqnarray} where $ R=\left\{\phi=Bx+b:B\in\mathbb{R}^{d\times d} \text{ is skew-symmetric and }b\in\mathbb{R}^d\right\} $ and $ C $ is a constant.

I recently see this statement in a book about elliptic equations. It is called the second Korn inequality. I tried to get the estimate by direct computation which works in the proof of the first Korn inequality, but for this inequality, I cannot combine the condition $ u\perp R $ with the final results. Can you give me some hints or references?

$\textbf{Theorem}.1$ (The first Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. Then\ \begin{eqnarray} \sqrt{2}\left\|\triangledown u\right\|_{L^2(\Omega)}\leq \left\|\triangledown u+(\triangledown u)^T\right\|_{L^2(\Omega)} \end{eqnarray} for any $ u\in H_{0}^{1}(\Omega;\mathbb{R}^d) $, where $ (\triangledown u)^T $ denotes the transpose of $ \triangledown u $.

$\textbf{Theorem}.2$ (The second Korn inequality) Suppose that $ \Omega $ is a bounded domain in $ \mathbb{R}^d $ with Lipschitz boundary. If $ u\in H^{1}(\Omega,\mathbb{R}^d) $ is a function with the property that $ u\perp R $ in $ H^{1}(\Omega;\mathbb{R}^d) $, then
\begin{eqnarray} \int_{\Omega}|\triangledown u|^2dx\leq C\int_{\Omega}|\triangledown u+(\triangledown u)^T|^2dx \end{eqnarray} where $ R=\left\{\phi=Bx+b:B\in\mathbb{R}^{d\times d} \text{ is skew-symmetric and }b\in\mathbb{R}^d\right\} $ and $ C $ is a constant.

I recently see the two theroems in a book about elliptic equations. I tried to get the estimate for the second inequality by direct computation which works in the proof of the first Korn inequality, but for this inequality, I cannot combine the condition $ u\perp R $ with the final results. Can you give me some hints or references?