Compactness and Bubbles Analysis for 1/2-harmonic Maps

In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps $u_k\text{\hspace{0.17em}⁣}:\mathbb{R}\to {\mathcal{S}}^{m-1}$ such that $|u_k|{\dot H^{1/2}(\R,{\cal{S}}^{m-1})}\le C.$ More precisely we show that there exist a weak 1/2-harmonic map $u_{\mathrm{\infty }}\text{\hspace{0.17em}⁣}:\mathbb{R}\to {\mathcal{S}}^{m-1}$, a possible empty set $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_{\mathrm{\ell }}$ in $\mathbb{R}$ such that up to subsequences $$(\mathrm{\mid }(-\mathrm{\Delta }{)}^{1\mathrm{/}4}{u}_k{\mathrm{\mid }}^2\rightharpoonup \mathrm{\mid }(-\mathrm{\Delta }{)}^{1\mathrm{/}4}{u}_{\mathrm{\infty }}{\mathrm{\mid }}^2)dx+\sum _{i=1}^{\mathrm{\ell }}{\lambda }_i{\delta }_{a_i},inRadonmeasure,$$ as $k\to +\mathrm{\infty }$, with ${\lambda }_i\ge 0.$ The convergence of $u_k$ to $u_{\mathrm{\infty }}$ is strong in $\dot W^{1/2,p}{loc}(\R\setminus{a_1,…,a_\ell})$,forevery$p\ge 1.$ 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 ${\mathcal{S}}^2$ one has ${\lambda }_i=2\pi n_i$, with $n_i$ a positive integer.