  This work establishes fractional analogues of Korn's first and second
inequalities for vector fields in fractional Sobolev spaces defined over a
bounded domain. The validity of the inequalities require no additional boundary
condition, extending existing fractional Korn's inequalities that are only
applicable for Sobolev vector fields satisfying zero Dirichlet boundary
conditions. The domain of definition is required to have a $C^1$-boundary or,
more generally, a Lipschitz boundary with small Lipschitz constant. We
conjecture that the inequalities remain valid for vector fields defined over
any Lipschitz domain. We support this claim by presenting a proof of the
inequalities for vector fields defined over planar convex domains.
