Compactness and bubble analysis for 1/2-harmonic maps

Da Lio, Francesca

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 201-224 / Harvested from Numdam

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.

Publié le : 2015-01-01

MR 3303947 | Zbl 1310.58011

DOI : https://doi.org/10.1016/j.anihpc.2013.11.003
Classification: 58E20, 35J20, 35B65, 35J60, 35S99

@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},
     volume = {32},
     year = {2015},
     pages = {201-224},
     doi = {10.1016/j.anihpc.2013.11.003},
     mrnumber = {3303947},
     zbl = {1310.58011},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 32 (2015) pp. 201-224. doi : 10.1016/j.anihpc.2013.11.003. http://gdmltest.u-ga.fr/item/AIHPC_2015__32_1_201_0/

Bibliographie

[1] D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory, Springer, Berlin (1996) | MR 1411441

[2] S. Alexakis, Rafe Mazzeo, The Willmore functional on complete minimal surfaces in H3: boundary regularity and bubbling, arXiv:1204.4955v2 | Zbl 1335.53076

[3] Y. Bernard, T. Rivière, Energy quantization for Willmore surfaces and applications, arXiv:1106.3780 (2011) | MR 3194812

[4] J.M. Bony, Cours d'analyse, Éditions de l'École polytechnique (2011)

[5] R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611 -635 | MR 412721 | Zbl 0326.32011

[6] F. Da Lio, Fractional harmonic maps into manifolds in odd dimension $n > 1$ , Calc. Var. Partial Differ. Equ. 48 no. 3–4 (2013), 421 -445 | MR 3116017 | Zbl 1281.58007

[7] F. Da Lio, Habilitation thesis, in preparation.

[8] F. Da Lio, in preparation.

[9] F. Da Lio, T. Riviere, 3-commutators estimates and the regularity of 1/2-harmonic maps into spheres, Anal. PDE 4 no. 1 (2011), 149 -190 , http://dx.doi.org/10.2140/apde.2011.4.149 | Zbl 1241.35035

[10] F. Da Lio, T. Riviere, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to 1/2-harmonic maps, Adv. Math. 227 (2011), 1300 -1348 | MR 2799607 | Zbl 1219.58004

[11] F. Da Lio, T. Riviere, Fractional harmonic maps and free boundaries problems, in preparation.

[12] F. Da Lio, A. Schikorra, $(n, p)$ -harmonic maps: regularity for the sphere case, Adv. Calc. Var. (2012), http://dx.doi.org/10.1515/acv-2012-0107, arXiv:1202.1151v1 | Zbl 1281.49034

[13] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, Stud. Adv. Math. , CRC Press, Boca Raton, FL (1992) | MR 1158660 | Zbl 0626.49007

[14] A. Fraser, R. Schoen, The first Steklov eigenvalue, conformal geometry, and minimal surfaces, Adv. Math. 226 no. 5 (2011), 4011 -4030 | MR 2770439 | Zbl 1215.53052

[15] A. Fraser, R. Schoen, Eigenvalue bounds and minimal surfaces in the ball, arXiv:1209.3789 | Zbl 1337.35099

[16] L. Grafakos, Classical Fourier Analysis, Grad. Texts Math. vol. 249 , Springer (2009) | MR 2445437

[17] L. Grafakos, Modern Fourier Analysis, Grad. Texts Math. vol. 250 , Springer (2009) | MR 2463316 | Zbl 1158.42001

[18] P. Laurain, T. Rivière, Angular energy quantization for linear elliptic systems with antisymmetric potentials and applications, Anal. PDE (2014), arXiv:1109.3599 (2011) | MR 3219498 | Zbl 1295.35204

[19] P. Laurain, T. Rivière, Energy quantization for biharmonic maps, Adv. Calc. Var. 6 no. 2 (2013), 191 -216 | MR 3043576 | Zbl 1275.35098

[20] F.H. Lin, T. Rivière, Quantization property for moving Line vortices, Commun. Pure Appl. Math. 54 (2001), 826 -850 | MR 1823421 | Zbl 1029.35127

[21] T. Rivière, Bubbling, quantization and regularity issues in geometric non-linear analysis, ICM, Beijing (2002) | MR 1957532 | Zbl 1136.35334

[22] T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin (1996) | MR 1419319 | Zbl 0873.35001

[23] A. Schikorra, Regularity of $n / 2$ harmonic maps into spheres, J. Differ. Equ. 252 (2012), 1862 -1911 | MR 2853564 | Zbl 1237.58018

[24] A. Schikorra, Epsilon-regularity for systems involving non-local, antisymmetric operators, preprint. | MR 3426086

[25] H.C. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969), 318 -344 | MR 243467 | Zbl 0181.11501