Compactness and bubble analysis for 1/2-harmonic maps

Da Lio, Francesca

doi:10.1016/j.anihpc.2013.11.003

Da Lio, Francesca

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 201-224

Résumé

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 ${∥ u_{k} ∥}_{{\dot{H}}^{1 / 2} (ℝ, 𝒮^{m - 1})} ⩽ C$ . 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 ${a_{1}, \dots, a_{ℓ}} \subset ℝ$ such that up to subsequences

{|{(- Δ)}^{1 / 4} u_{k}|}^{2} d x ⇀ {|{(- Δ)}^{1 / 4} u_{\infty}|}^{2} d x + \sum_{i = 1}^{ℓ} λ_{i} δ_{a_{i}}, in Radon measure,

as

k \to + \infty

, with

λ_{i} ⩾ 0

.The convergence of

u_{k}

to

u_{\infty}

is strong in

{\dot{W}}_{𝑙𝑜𝑐}^{1 / 2, p} (ℝ ∖ {a_{1}, \dots, a_{ℓ}})

, 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

λ_{i} = 2 π n_{i}

, with

n_{i}

a positive integer.

MR Zbl | 1 citation dans Numdam

DOI : 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},
     pages = {201--224},
     year = {2015},
     publisher = {Elsevier},
     volume = {32},
     number = {1},
     doi = {10.1016/j.anihpc.2013.11.003},
     mrnumber = {3303947},
     zbl = {1310.58011},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2013.11.003/}
}

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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

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