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

Abstract

Given a half-harmonic map $u\in {\dot{H}}^{\frac{1}{2},2}(R,S^1)$ minimizing the fractional Dirichlet energy under Dirichlet boundary conditions in $R\setminus I$, 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 à 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.