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Following this question I am reading Oleinik "Mathematical problems in elasticity and homogenization". On page 13 she proves Korn's first inequality

(The first Korn inequality) Suppose that Ω is a bounded domain in $\mathbb{R}^{d}$ with Lipschitz boundary. Then $$\|\nabla u\|_{L^2}^{2}\leq2\|e(u)\|_{L^{2}}^2$$for any $u\in H^{1}_{0}(\Omega,\mathbb{R}^d)$ and $e(u)$ the matrix with entries $e(u)_{ih}=\frac{1}{2}(\frac{\partial u_{i}}{\partial x_{h}}+\frac{\partial u_{h}}{\partial x_{i}})$

She proves for $u\in C^{\infty}_{0}(\Omega,\mathbb{R}^{n})$ the following equality

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My question is - why at her last sentence she says the term $\frac{1}{2}\int_{\Omega}\frac{\partial u_{i}}{\partial x_{i}}\frac{\partial u_{h}}{\partial x_{h}}dx$ is non-negative?

I have tried to prove it by playing with the formula but I am trying to think there might be greater machinery at work here. Following this proof, she writes a remark

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which makes me think the reason this term is non-negative is due to simple properties but I am failing to see it.

Thanks in advance

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1 Answer 1

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I think it's because: $$\sum_{i,h}\int_\Omega \dfrac{\partial u_i}{\partial x_i}\cdot\dfrac{\partial u_h}{\partial x_h} = \int_\Omega\sum_{i,h} \dfrac{\partial u_i}{\partial x_i}\cdot\dfrac{\partial u_h}{\partial x_h} = \int_\Omega \left(\sum_i \dfrac{\partial u_i}{\partial x_i}\right)^2 \geq 0$$

In general, one has: $$\sum_i a_i\sum_jb_j = \sum_{i,j}a_ib_j$$

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