Compactness and bubble analysis for 1/2-harmonic maps
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 201-224.

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 : 10.1016/j.anihpc.2013.11.003
Classification : 58E20, 35J20, 35B65, 35J60, 35S99
Mots-clés : Fractional harmonic maps, Nonlinear elliptic PDE's, Regularity of solutions, Commutator estimates
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     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},
     pages = {201--224},
     publisher = {Elsevier},
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Da Lio, Francesca. Compactness and bubble analysis for 1/2-harmonic maps. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 201-224. doi : 10.1016/j.anihpc.2013.11.003. http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.003/

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