Intrinsic semiharmonic maps (Q555692)

The author presents and studies a variational functional for mappings from an \(m\)-dimensional domain to a compact Riemannian manifold, connected to the seminorm of the homogeneous fractional Sobolev space \(\dot{H}^{1/2}\). It is an ``intrinsic'' variation of the functional proposed in [\textit{F. Da Lio} and \textit{T. Rivière}, ``Three-term commutator estimates and the regularity of \(1/2\)-harmonic maps into spheres'', Anal.\ PDE 4, No. 1, 149--190 (2011; Zbl 1241.35035)]. The new feature of Moser's functional is the fact that it does not depend on how the target manifold is embedded into some Euclidean space. The first variation of the functional can be identified with a Dirichlet-to-Neumann map for the harmonic map problem. This fact is used to reduce questions about the new ``semiharmonic'' functional to (mostly answered) questions about harmonic maps. The semiharmonic critical points have regularity properties analogous to harmonic maps. They are smooth in the critical dimension \(m=1\), and if \(m\geq2\), the stationarity condition which has been well known from harmonic maps implies regularity outside a closed ``singular set'' of vanishing \((m-1)\)-dimensional Hausdorff measure.