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 :𝒮 m-1 such that u k H ˙ 1/2 (,𝒮 m-1 ) C. More precisely we show that there exist a weak 1/2-harmonic map u :𝒮 m-1 , a finite and possible empty set {a 1 ,,a } such that up to subsequences (-Δ) 1/4 u k 2 dx(-Δ) 1/4 u 2 dx+ i=1 λ i δ a i ,inRadonmeasure, as k+, with λ i 0.The convergence of u k to u is strong in W ˙ 𝑙𝑜𝑐 1/2,p ({a 1 ,,a }), for every p1. 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 λ i =2πn 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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