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 $u_k\colon\R\to {\cal{S}}^{m-1}$ 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 $u_\infty\colon\R\to {\cal{S}}^{m-1}$, a possible empty set ${a_1,...,a_\ell}$ in $\R$ such that up to subsequences $$(|(-\Delta)^{1/4}u_k|^2 \rightharpoonup |(-\Delta)^{1/4}u_{\infty}|^2)dx+\sum_{i=1}^{\ell}\lambda_i \delta_{a_i}, in Radon measure,$$ as $k\to +\infty$, with $\lambda_i\ge 0.$ The convergence of $u_k$ to $u_\infty$ is strong in $\dot W^{1/2,p}_{loc}(\R\setminus{a_1,...,a_\ell})$, for every $p\ge 1.$ 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 ${\cal{S}}^2\,$ one has $\lambda_i=2 \pi n_i$, with $n_i$ a positive integer.