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

Given a half-harmonic map u H^1{2, 2} (R, S¹) minimizing the fractional Dirichlet energy under Dirichlet boundary conditions in R I, we show the existence of a second half-harmonic map, minimizing the Dirichlet energy in a different homotopy class. This is a first step towards addressing a problem raised by Brezis and 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.