\(L^p\)-gradient harmonic maps into spheres and \(SO(N)\). (English) Zbl 1363.58010

The author proposes a variant of the fractional harmonic maps that have been studied recently following the paper [F. Da Lio and T. Rivière, Anal. PDE 4, No. 1, 149–190 (2011; Zbl 1241.35035)]. The new variant, rather than using norms of the fractional Laplacian directly, involves the integral \(E_s(v):=\int_{R^n}|\nabla^sv|^{n/s}\,dx\) for mappings \(v:\Omega\to S^{N-1}\), where \(s\in(0,n)\) and \(\nabla^sv=({\mathcal R}_1[\Delta^{s/2}v],\dots ,{\mathcal R}_n[\Delta^{s/2}v])\) with \({\mathcal R}_\alpha\) being the \(\alpha\)-th Riesz transform, i.e., the operator with Fourier symbol \(i\xi_\alpha/|\xi|\).
Assuming that \(E_s(u)\) is finite and that \(u\) is a critical point of \(E_s\) under maps to \(S^{N-1}\), it is proved that \(u\) is Hölder-continuous. A variant of the proof shows the same for mappings to \(\mathrm{SO}(N)\subset R^{N\times N}\).
While these “fractional” results are interesting in their own right, one of the points made here is that for \(s=1\) the critical points are just \(n\)-harmonic maps. Even though regularity proofs for those have been known, the proofs given here specialize to new ones which the author claims may be more robust. This is featured in the broad and very readable introduction of the paper.


58E20 Harmonic maps, etc.
35B65 Smoothness and regularity of solutions to PDEs
35J60 Nonlinear elliptic equations
35S05 Pseudodifferential operators as generalizations of partial differential operators


Zbl 1241.35035
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