Lp-gradient harmonic maps into spheres and SO(N)

Abstract

We consider critical points of the energy $E(v):={\int }_{{\mathbb{R}}^n}|{\nabla }^{s}v{|}^{\frac{n}{s}}$, where $v$ maps locally into the sphere or $SO(N)$, and ${\nabla }^s=({\partial }_1^s,\dots ,{\partial }_n^s)$ is the formal fractional gradient, i.e. ${\partial }_{\alpha }^s$ is a composition of the fractional laplacian with the $\alpha$-th Riesz transform. We show that critical points of this energy are Hölder continuous. As a special case, for $s=1$, we obtain a new, more stable proof of Fuchs and Strzelecki's regularity result of $n$-harmonic maps into the sphere, which is interesting on its own.