Fractional harmonic maps into manifolds in odd dimension n > 1

Abstract

In this paper we consider critical points of the following nonlocal energy

$$\begin{array}{ll}{\mathcal{L}}_n(u) = \int_{{I\!\!R}^n}| ({-\Delta})^{n/4} u(x)|^2 dx, \qquad(1)\end{array}$$

where ${u \in \dot{H}^{n/2}({I\!\!R}^n,{\mathcal{N}}), {\mathcal{N}} \subset {I\!\!R}^m}$ is a compact k dimensional smooth manifold without boundary and n > 1 is an odd integer. Such critical points are called n/2-harmonic maps into ${{\mathcal{N}}}$. We prove that ${(-\Delta) ^{n/4} u\in L^p_{loc}({I\!\!R}^n)}$ for every p ≥  1 and thus ${u \in C^{0,\alpha}_{loc}({I\!\!R}^n)}$, for every 0 < α < 1. The local Hölder continuity of n/2-harmonic maps is based on regularity results obtained in [4] for nonlocal Schrödinger systems with an antisymmetric potential and on some new 3-terms commutators estimates.