This post is about Korn's inequality as it appears for instance in this paper https://cedricvillani.org/sites/dev/files/old_images//2012/08/021.DV-Korn.pdf
Basically the inequality says that, on a bounded space $U$, writing $M^{as}_0$ for the space of anti-symmetric matrices, we have the existence of a constant $C>0$ such that, writing $D(u)$ for the symmetric gradient of $u$, for all $u \in H^1(U)$,
\begin{equation} \inf_{c \in \mathbf{R}} \inf_{M \in M^{as}_0} |u - c - Mx| \le C |D(u)|, \end{equation} where the norm on the left-hand side is the Sobolev norm $H^1(U)$ while the one on the right-hand side is the $L^2(U)$ norm.
My question is about the boundedness of the minimizing sequence associated with the left-hand side. Clearly as $c \to \infty$ we get away from the minimum, so I expect the minimizing sequence for $c$ to be bounded. Moreover, as stated in p.1 in the article mentioned above, the optimum for $M$ is also reached by a given constant (depending on the mean of some coordinates of $\nabla u$ in $U$), hence I expect boundedness for this minimizing sequence as well.
Am I missing something there ? This looks pretty straightforward, yet I can't find a paper mentioning this fact.