Let $\Omega \subset \subset \mathbb{R}^N$ have smooth boundary, $N \geqslant 2$ and
$ \mathcal{E} ( v) := \int_{\Omega} \sum_{i, j} \varepsilon_{i j} ( v) \varepsilon_{i j} ( v) \mathrm{d} x := \int_{\Omega} \sum_{i, j} \left( \frac{v_{i, j} + v_{j, i}}{2} \right)^2 \mathrm{d} x $
be defined in $H^1 ( \Omega, \mathbb{R}^N)$. Having just read for the umpteenth time that the reason that Korn's inequality
$ \mathcal{E} ( v) + \| v \|_{L^2}^2 \geqslant c \| v \|^2_{H^1} $
is not trivial is that the left hand side "involves only certain combinations of derivatives", I wonder whether this is actually true and if yes, whether I understand things correctly, because to me it's a matter of some products (the $v_{i,j}v_{j,i}$ for $i \neq j$ ) possibly being negative. Did I get lost among the indices?
Edit: in the context of linear elasticity it is often stated that this inequality is not a triviality (in the sense that it's not tautological), because of the different combinations of partial derivatives appearing at each side. Some authors affirm that only some (six) different partial derivatives appear at the left hand side (see [1], [2], [3]). However I see all partial derivatives at both sides, but combined differently (see my answer). I understand the actual difficulties in proving it and the implications for coercivity proofs for instance. It's this assertion that I find confusing.
[1] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, vol. 219. Springer-Verlag, 1976.
[2] P. G. Ciarlet, An introduction to differential geometry with applications to elasticity. Springer, 2005, .
[3] F. Demengel and G. Demengel, Functional spaces for the theory of elliptic partial differential equations, vol. 8. Springer, 2012.