Fractional Harmonic Maps into Manifolds in odd dimension n>1

TL;DR

This paper studies fractional harmonic maps into manifolds in odd dimensions, proving local regularity and Hölder continuity of critical points of a nonlocal energy functional.

Contribution

It establishes regularity results for fractional harmonic maps in odd dimensions using nonlocal Schrödinger system techniques and commutator estimates.

Findings

Proves $ abla^{n/2} u otin L^p_{loc}$ for all $p \,\geq 1$.

Shows $u$ is locally Hölder continuous.

Extends regularity theory to nonlocal fractional harmonic maps.

Abstract

In this paper we consider critical points of the following nonlocal energy {equation} {\cal{L}}_n(u)=\int_{\R^n}| ({-\Delta})^{n/4} u(x)|^2 dx\,, {equation} where $u\colon H^{n/2}(\R^n)\to{\cal{N}}\,$ ${\cal{N}}\subset\R^m$ is a compact $k$ dimensional smooth manifold without boundary and $n>1$ is an odd integer. Such critical points are called $n/2$-harmonic maps into ${\cal{N}}$. We prove that $\Delta ^{n/2} u\in L^p_{loc}(\R^n)$ for every $p\ge 1$ and thus $u\in C^{0,\alpha}_{loc}(\R^n)\,.$ The local H\"older continuity of $n/2$-harmonic maps is based on regularity results obtained in \cite{DL1} for nonlocal Schr\"odinger systems with an antisymmetric potential and on suitable {\it 3-terms commutators} estimates.