I am currently studying the proof of Korn inequality for $1 < p < \infty$ for an exam, and a counterexample for $p=1$. I've read in the article that my professor gave me that this result doesn't hold for $p= \infty$ either. Since I haven't found any article about this case online, I thought it was quite trivial and I started searching for a counterexample, only giving up after an hour or so. I admit that I am not that good at finding counterexamples and this isn't requested for my exam either, but I am still curious to find a counteraxample for $p= \infty$.
For those who don't know, Korn's inequality says that given $ \Omega \subset \mathbb{R}^n$ open, bounded with $C^2$ boundary (it's actually sufficient for it to be Lipshitz, but whatever) and $p \in (1, \infty)$ , there exists $C=C(\Omega, n, p)$ such that
$$||u||_{W^{1,p}} \leq C (||u||_p + ||D^{s} u||_p ) \quad \forall u \in W^{1,p}(\Omega, \mathbb{R}^n)$$
where $D^s u$ is the symmetric part of the jacobian, that is $(D^{s} u)_{ij} = \frac{\partial_i u_j + \partial_j u_i}{2}$.
P.S. If it really is that trivial, a hint only would be appreciated. It's a matter of honor now!
Taken from https://rdcu.be/b5cEt: The function $u(x) := Ax \ln(|x|)$, where $A$ is any constant anti-symmetric matrix, belongs to $W^{1,1}_0(B(0,1))$. $|D^s u(x)| \leq |A|$ for any $x \in B(0,1)$, but $Du$ does not belong to $L^{\infty}(B(0,1))$.