H\"older continuity of Minimizing $W^{s,p}$-Harmonic Maps

TL;DR

This paper proves that energy-minimizing maps in fractional Sobolev spaces into certain manifolds are locally Hölder continuous outside a small singular set, using a blow-up method instead of monotonicity formulas.

Contribution

It establishes Hölder continuity for minimizers of fractional $W^{s,p}$-energy maps into manifolds with simple topology, avoiding the need for a monotonicity formula.

Findings

Minimizers are locally Hölder continuous outside a small singular set.

The singular set has Hausdorff dimension less than $n - sp$.

The proof uses a blow-up argument instead of monotonicity formulas.

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 $$\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} \;dx\;dy$$ are locally H\"older 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 $p \neq 2$) by using a blow-up argument instead.