Compactness and bubble analysis for $1/2$-harmonic maps

Abstract

In this paper we study compactness and quantization properties of sequences of $1/2$-harmonic maps $u_k:\mathbb{R}\to {\mathcal{S}}^{m-1}$ such that $\Vert u_k{\Vert }_{{\dot{H}}^{1/2}(\mathbb{R},{\mathcal{S}}^{m-1})}⩽C$. More precisely we show that there exist a weak $1/2$-harmonic map $u_{\infty }:\mathbb{R}\to {\mathcal{S}}^{m-1}$, a finite and possible empty set $\{a_1,\dots ,a_{\ell }\}\subset \mathbb{R}$ such that up to subsequences

as $k\to +\infty$, with ${\lambda }_i⩾0$.

The convergence of $u_k$ to $u_{\infty }$ is strong in ${\dot{W}}_{\mathrm{loc}}^{1/2,p}(\mathbb{R}\setminus \{a_1,\dots ,a_{\ell }\})$, for every $p⩾1$. We quantify the loss of energy in the weak convergence and we show that in the case of non-constant $1/2$-harmonic maps with values in ${\mathcal{S}}^1$ one has ${\lambda }_i=2\pi n_i$, with $n_i$ a positive integer.