One-dimensional half-harmonic maps into the circle and their degree

TL;DR

This paper investigates half-harmonic maps into the circle, demonstrating existence results for minimizers in different homotopy classes, analyzing the degree of fractional Sobolev maps, and contrasting one-dimensional and two-dimensional cases.

Contribution

It establishes the existence of multiple half-harmonic minimizers in different homotopy classes and studies the degree theory for fractional Sobolev maps, highlighting differences from higher dimensions.

Findings

Existence of multiple minimizers in different homotopy classes.

Degree estimates for fractional Sobolev maps.

Contrast with 2D case showing limitations of minimization.

Abstract

Given a half-harmonic map $u\in \dot H^{\frac{1}{2},2}(\mathbb{R},\mathbb{S}^1)$ minimizing the fractional Dirichlet energy under Dirichlet boundary conditions in $\mathbb{R}\setminus I$, we show the existence of a second half-harmonic map, minimizing the fractional Dirichlet energy in a different homotopy class. This is based on the study of the degree of fractional Sobolev maps and a sharp estimate \`a la Brezis-Coron. We give examples showing that it is in general not possible to minimize in every homotopy class and show a contrast with the 2-dimensional case.