Compactness and bubble analysis for 1/2-harmonic maps
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 1, p. 201-224
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In this paper we study compactness and quantization properties of sequences of 1/2-harmonic maps ${u}_{k}:ℝ\to {𝒮}^{m-1}$ such that ${\parallel {u}_{k}\parallel }_{{\stackrel{˙}{H}}^{1/2}\left(ℝ,{𝒮}^{m-1}\right)}⩽C$. More precisely we show that there exist a weak 1/2-harmonic map ${u}_{\infty }:ℝ\to {𝒮}^{m-1}$, a finite and possible empty set $\left\{{a}_{1},\cdots ,{a}_{\ell }\right\}\subset ℝ$ such that up to subsequences ${\left|{\left(-\Delta \right)}^{1/4}{u}_{k}\right|}^{2}\phantom{\rule{0.166667em}{0ex}}dx⇀{\left|{\left(-\Delta \right)}^{1/4}{u}_{\infty }\right|}^{2}\phantom{\rule{0.166667em}{0ex}}dx+\sum _{i=1}^{\ell }{\lambda }_{i}{\delta }_{{a}_{i}},\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\text{Radon}\phantom{\rule{4pt}{0ex}}\text{measure},$ as $k\to +\infty$, with ${\lambda }_{i}⩾0$.The convergence of ${u}_{k}$ to ${u}_{\infty }$ is strong in ${\stackrel{˙}{W}}_{\mathrm{𝑙𝑜𝑐}}^{1/2,p}\left(ℝ\setminus \left\{{a}_{1},\cdots ,{a}_{\ell }\right\}\right)$, 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 ${𝒮}^{1}$ one has ${\lambda }_{i}=2\pi {n}_{i}$, with ${n}_{i}$ a positive integer.

DOI : https://doi.org/10.1016/j.anihpc.2013.11.003
Classification:  58E20,  35J20,  35B65,  35J60,  35S99
Keywords: Fractional harmonic maps, Nonlinear elliptic PDE's, Regularity of solutions, Commutator estimates
@article{AIHPC_2015__32_1_201_0,
author = {Da Lio, Francesca},
title = {Compactness and bubble analysis for 1/2-harmonic maps},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {32},
number = {1},
year = {2015},
pages = {201-224},
doi = {10.1016/j.anihpc.2013.11.003},
zbl = {1310.58011},
mrnumber = {3303947},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2015__32_1_201_0}
}

Da Lio, Francesca. Compactness and bubble analysis for 1/2-harmonic maps. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 1, pp. 201-224. doi : 10.1016/j.anihpc.2013.11.003. http://www.numdam.org/item/AIHPC_2015__32_1_201_0/

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