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,

k \to + \infty

, with

λ_{i} ⩾ 0

.The convergence of

u_{k}

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
Mots clés : 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},
     publisher = {Elsevier},
     volume = {32},
     number = {1},
     year = {2015},
     doi = {10.1016/j.anihpc.2013.11.003},
     mrnumber = {3303947},
     zbl = {1310.58011},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.003/}
}

TY  - JOUR
AU  - Da Lio, Francesca
TI  - Compactness and bubble analysis for 1/2-harmonic maps
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2015
SP  - 201
EP  - 224
VL  - 32
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.003/
DO  - 10.1016/j.anihpc.2013.11.003
LA  - en
ID  - AIHPC_2015__32_1_201_0
ER  -

%0 Journal Article
%A Da Lio, Francesca
%T Compactness and bubble analysis for 1/2-harmonic maps
%J Annales de l'I.H.P. Analyse non linéaire
%D 2015
%P 201-224
%V 32
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2013.11.003/
%R 10.1016/j.anihpc.2013.11.003
%G en
%F 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) 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/

Bibliographie
Cité par

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

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

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

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[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

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

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

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

Cité par Sources :