$L^p$-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 $\nabla^s = (\partial_1^s,\ldots,\partial_n^s)$ 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 [22, 9], which is interesting on its own.