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 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.2010 Mathematics Subject Classification. 42B37, 42B30, 35R11, 58E20, 35B65.

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Abstract. We prove that for each positive integer N the set of smooth, zero degree maps ψ : S 2 → S 2 which have the following three properties:(i) there is a unique minimizing harmonic map u : B 3 → S 2 which agrees with ψ on the boundary of the unit ball;(ii) this map u has at least N singular points in B 3 ;(iii) the Lavrentiev gap phenomenon holds for ψ, i. e., the infimum of the Dirichlet energies E(w) of all smooth extensions w : B 3 → S 2 of ψ is strictly larger than the Dirichlet energy B 3 |∇u| 2 of the (irregular) minimizer u, is dense in the set S of all smooth zero degree maps φ : S 2 → S 2 endowed with the W 1,p -topology, where 1 ≤ p < 2. This result is sharp: it fails in the W 1,2 -topology of S.

We prove full boundary regularity for minimizing biharmonic maps with smooth Dirichlet boundary conditions. Our result, similarly as in the case of harmonic maps, is based on the nonexistence of nonconstant boundary tangent maps. With the help of recently derivated boundary monotonicity formula for minimizing biharmonic maps by Altuntas we prove compactness at the boundary following Scheven's interior argument. Then we combine those results with the conditional partial boundary regularity result for stationary biharmonic maps by Gong-Lamm-Wang.

We deal with the generalized Emden–Fowler equation f″(x) + g(x)f−θ(x) = 0, where θ∈double-struckR,1emx∈(a,b),1emg belongs to Lp((a,b)). We obtain a priori estimates for the solutions, as well as information about their asymptotic behavior near boundary points. As a tool, we derive new nonlinear variants of first‐order and second‐order Poincaré inequalities, which are based on strongly nonlinear multiplicative inequalities obtained recently by first author and Peszek. Copyright © 2014 John Wiley & Sons, Ltd.

We prove that for antisymmetric vectorfield Ω with small L 2 -norm there exists a gaugeThis extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.

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Let Σ a closed n-dimensional manifold, N ⊂ R M be a closed manifold, and u ∈ W s, n s (Σ, N ) for s ∈ (0, 1). We extend the monumental work of Sacks and Uhlenbeck by proving that if π n (N ) = {0} then there exists a minimizing W s, n s -harmonic map homotopic to u. If π n (N ) = {0}, then we prove that there exists a W s, n s -harmonic map from S n to N in a generating set of π n (N ).Since several techniques, especially Pohozaev-type arguments, are unknown in the fractional framework (in particular when n s = 2 one cannot argue via an extension method), we develop crucial new tools that are interesting on their own: such as a removability result for point-singularities and a balanced energy estimate for non-scaling invariant energies. Moreover, we prove the regularity theory for minimizing W s, n s -maps into manifolds. ContentsW s, n s -HARMONIC MAPS IN HOMOTOPY CLASSES 5• K.M.