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

Luca Martinazzi; Ali Hyder

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arXiv · 2405.18037

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

Luca Martinazzi·Ali Hyder

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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.

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Luca Martinazzi, Ali Hyder. 2024-05-28. One-dimensional half-harmonic maps into the circle and their degree. https://arxiv.org/abs/2405.18037

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