A fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations

In this paper we prove a fractional analogue of the classical Korn's first inequality. The inequality makes it possible to show equivalence function space vector field characterized by Gagliardo-type seminorm with 'projected difference' that corresponding Sobolev space. As an application, will use obtain Caccioppoli-type for nonlinear system nonlocal equations, which in turn is key ingredient applying known results higher differentiability result weak solutions equations. regularity demonstrate well-known self-improving property scalar equations hold strongly coupled systems as well.