Fractional div-curl quantities and applications to nonlocal geometric equations

Abstract

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman–Lions–Meyer–Semmes to fractional div-curl quantities, obtaining, in particular, a nonlocal version of Wente's lemma.We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a theory for fractional harmonic maps analogous to the local theory. Firstly, regarding fractional harmonic maps into spheres, we obtain a conservation law analogous to Shatah's conservation law and give a new regularity proof analogous to Hélein's for harmonic maps into spheres.Secondly, we prove regularity for solutions to critical systems with nonlocal antisymmetric potentials on the right-hand side. Since the half-harmonic map equation into general target manifolds has this form, as a corollary, we obtain a new proof of the regularity of half-harmonic maps into general target manifolds following closely Rivière's celebrated argument in the local case.Lastly, the fractional div-curl quantities provide also a new, simpler, proof for Hölder continuity of Ws,n/s-harmonic maps into spheres and we extend this to an argument for Ws,n/s-harmonic maps into homogeneous targets. This is an analogue of Strzelecki's and Toro–Wang's proof for n-harmonic maps into spheres and homogeneous target manifolds, respectively.