Fractional div-curl quantities and applications to nonlocal geometric equations

摘要

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 Helein's for harmonic maps into spheres.