Hölder continuity of Minimizing $W^{s,p}$-Harmonic Maps

Abstract

We show that the mappings $u\in {\dot{W}}^{s,p}({\mathbb{R}}^n,\mathcal{N})$ into manifolds $\mathcal{N}$ of a sufficiently simple topology that minimize the energy $${\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{n+sp}}dxdy$$ are locally Hölder continuous in a bounded domain $\Omega$ outside a singular set $\Sigma$ with Hausdorff dimension strictly smaller than $n-sp$. We avoid the use of a monotonicity formula (which is unknown if $p\ne 2$) by using a blow-up argument instead.