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The validity of Korn's first inequality in the fractional setting in boundeddomains has been open. We resolve this problem by proving that in fact Korn'sfirst inequality holds in the case $ps>1$ for fractional $W^{s,p}_0(\Omega)$Sobolev fields in open and bounded $C^{1}$-regular domains $\Omega\subset\mathbb R^n$. Also, in the case $ps<1,$ for any open bounded $C^1$ domain$\Omega\subset \mathbb R^n$ we construct counterexamples to the inequality,i.e., Korn's first inequality fails to hold in bounded domains. The proof ofthe inequality in the case $ps>1$ follows a standard compactness approachadopted in the classical case, combined with a Hardy inequality, and a recentlyproven Korn second inequality by Mengesha and Scott [\textit{Commun. Math.Sci.,} Vol. 20, N0. 2, 405--423, 2022]. The counterexamples constructed in thecase $ps<1$ are interpolations of a constant affine rigid motion inside thedomain away from the boundary, and of the zero field close to the boundary.

On the fractional Korn inequality in bounded domains: Counterexamples to the case $ps<1$