Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fractional div-curl quantities and applications to nonlocal geometric
  equations   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 H\'elein'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 Rivi\`{e}re's celebrated argument in the local
case.
  Lastly, the fractional div-curl quantities provide also a new, simpler, proof
for H\"older continuity of $W^{s,n/s}$-harmonic maps into spheres and we extend
this to an argument for $W^{s,n/s}$-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.

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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: cond-mat
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The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: acc-phys