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

31 pages

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Cited by in corpus (1)

• On energy-critical half-wave maps into ${\mathbb{S}}^2$