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 \(\nabla^s = (\partial_1^s,\ldots,\partial_n^s)\) 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 H\"older 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.