no. 3
On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 625-654.

We consider a family of positive solutions to the system of k components

−Δui,β=f(x,ui,β)−βui,β∑j≠iaijuj,β2in Ω,
where Ω⊂RN with N≥2. It is known that uniform bounds in L∞ of {uβ} imply convergence of the densities to a segregated configuration, as the competition parameter β diverges to +∞. In this paper we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of uβ in terms of entire solutions to the limit system
ΔUi=Ui∑j≠iaijUj2.
Moreover, we develop a uniform-in-β regularity theory for the interfaces.

DOI : 10.1016/j.anihpc.2016.04.001
Classification : 35B45, 35B65, 35R35, 35B08, 35B36, 35B25, 35J47
Mots clés : Nonlinear Schrödinger systems, Harmonic maps into singular manifolds, Competition and segregation, Point-wise asymptotic estimates, Regularity of free boundaries 
@article{AIHPC_2017__34_3_625_0,
     author = {Soave, Nicola and Zilio, Alessandro},
     title = {On phase separation in systems of coupled elliptic equations: {Asymptotic} analysis and geometric aspects},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {625--654},
     publisher = {Elsevier},
     volume = {34},
     number = {3},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.04.001},
     mrnumber = {3633738},
     zbl = {1417.35028},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.04.001/}
}
TY  - JOUR
AU  - Soave, Nicola
AU  - Zilio, Alessandro
TI  - On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2017
SP  - 625
EP  - 654
VL  - 34
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2016.04.001/
DO  - 10.1016/j.anihpc.2016.04.001
LA  - en
ID  - AIHPC_2017__34_3_625_0
ER  - 
%0 Journal Article
%A Soave, Nicola
%A Zilio, Alessandro
%T On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects
%J Annales de l'I.H.P. Analyse non linéaire
%D 2017
%P 625-654
%V 34
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2016.04.001/
%R 10.1016/j.anihpc.2016.04.001
%G en
%F AIHPC_2017__34_3_625_0
Soave, Nicola; Zilio, Alessandro. On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 625-654. doi : 10.1016/j.anihpc.2016.04.001. http://www.numdam.org/articles/10.1016/j.anihpc.2016.04.001/

[1] Akhmediev, N.; Ankiewicz, A. Partially coherent solitons on a finite background, Phys. Rev. Lett., Volume 82 (1999), pp. 2661 | DOI

[2] Berestycki, H.; Lin, T.-C.; Wei, J.; Zhao, C. On phase-separation models: asymptotics and qualitative properties, Arch. Ration. Mech. Anal., Volume 208 (2013) no. 1, pp. 163–200 | DOI | MR | Zbl

[3] Berestycki, H.; Terracini, S.; Wang, K.; Wei, J. On entire solutions of an elliptic system modeling phase separations, Adv. Math., Volume 243 (2013), pp. 102–126 | DOI | MR | Zbl

[4] Caffarelli, L.; Lin, F.-H. Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Am. Math. Soc., Volume 21 (2008) no. 3, pp. 847–862 | DOI | MR | Zbl

[5] Caffarelli, L.A.; Lin, F.H. Analysis on the junctions of domain walls, Discrete Contin. Dyn. Syst., Volume 28 (2010) no. 3, pp. 915–929 | DOI | MR | Zbl

[6] Chang, S.-M.; Lin, C.-S.; Lin, T.-C.; Lin, W.-W. Segregated nodal domains of two-dimensional multispecies Bose–Einstein condensates, Phys. D, Volume 196 (2004) no. 3–4, pp. 341–361 | MR | Zbl

[7] Conti, M.; Terracini, S.; Verzini, G. Nehari's problem and competing species systems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 19 (2002) no. 6, pp. 871–888 | DOI | Numdam | MR | Zbl

[8] Conti, M.; Terracini, S.; Verzini, G. An optimal partition problem related to nonlinear eigenvalues, J. Funct. Anal., Volume 198 (2003) no. 1, pp. 160–196 | DOI | MR | Zbl

[9] Conti, M.; Terracini, S.; Verzini, G. Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., Volume 195 (2005) no. 2, pp. 524–560 | DOI | MR | Zbl

[10] Dancer, E.N.; Wang, K.; Zhang, Z. The limit equation for the Gross–Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal., Volume 262 (2012) no. 3, pp. 1087–1131 | DOI | MR | Zbl

[11] Dipierro, S. Geometric inequalities and symmetry results for elliptic systems, Discrete Contin. Dyn. Syst., Volume 33 (2013) no. 8, pp. 3473–3496 | DOI | MR | Zbl

[12] Farina, A. Some symmetry results for entire solutions of an elliptic system arising in phase separation, Discrete Contin. Dyn. Syst., Volume 34 (2014) no. 6, pp. 2505–2511 | DOI | MR | Zbl

[13] Farina, A.; Soave, N. Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose–Einstein condensation, Arch. Ration. Mech. Anal., Volume 213 (2014) no. 1, pp. 287–326 | DOI | MR | Zbl

[14] Han, Q.; Hardt, R.; Lin, F. Geometric measure of singular sets of elliptic equations, Commun. Pure Appl. Math., Volume 51 (1998) no. 11–12, pp. 1425–1443 | MR | Zbl

[15] Hong, G.; Wang, L. A geometric approach to the topological disk theorem of Reifenberg, Pac. J. Math., Volume 233 (2007) no. 2, pp. 321–339 | DOI | MR | Zbl

[16] Noris, B.; Tavares, H.; Terracini, S.; Verzini, G. Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition, Commun. Pure Appl. Math., Volume 63 (2010) no. 3, pp. 267–302 | DOI | MR | Zbl

[17] Ramos, M.; Tavares, H.; Terracini, S. Existence and regularity of solutions to optimal partitions problems involving Laplacian eigenvalues, Arch. Ration. Mech. Anal., Volume 220 (2016) no. 1, pp. 363–443 | DOI | MR

[18] Soave, N.; Tavares, H.; Terracini, S.; Zilio, A. Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping, Nonlinear Anal., Theory Methods Appl., Volume 138 (2016), pp. 388–427 | DOI | MR | Zbl

[19] Soave, N.; Terracini, S. Liouville theorems and 1-dimensional symmetry for solutions of an elliptic system modelling phase separation, Adv. Math., Volume 279 (2015), pp. 29–66 | DOI | MR | Zbl

[20] N. Soave, A. Zilio, Multidimensional solutions for an elliptic system modelling phase separation, preprint, 2015. | MR

[21] Soave, N.; Zilio, A. Uniform bounds for strongly competing systems: the optimal Lipschitz case, Arch. Ration. Mech. Anal., Volume 218 (2015) no. 2, pp. 647–697 | DOI | MR | Zbl

[22] Soave, N.; Zilio, A. Entire solutions with exponential growth for an elliptic system modelling phase separation, Nonlinearity, Volume 27 (2014) no. 2, pp. 305–342 | DOI | MR | Zbl

[23] Tavares, H.; Terracini, S. Regularity of the nodal set of segregated critical configurations under a weak reflection law, Calc. Var. Partial Differ. Equ., Volume 45 (2012) no. 3–4, pp. 273–317 | MR | Zbl

[24] Tavares, H.; Terracini, S. Sign-changing solutions of competition–diffusion elliptic systems and optimal partition problems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 29 (2012) no. 2, pp. 279–300 | DOI | Numdam | MR | Zbl

[25] Timmermans, E. Phase separation of Bose–Einstein condensates, Phys. Rev. Lett., Volume 81 (1998), pp. 5718–5721 | DOI

[26] Wang, K. On the De Giorgi type conjecture for an elliptic system modeling phase separation, Commun. Partial Differ. Equ., Volume 39 (2014) no. 4, pp. 696–739 | DOI | MR | Zbl

[27] Wang, K. Harmonic approximation and improvement of flatness in a singular perturbation problem, Manuscr. Math., Volume 146 (2015) no. 1–2, pp. 281–298 | MR | Zbl

[28] Wei, J.; Weth, T. Asymptotic behavior of solutions of planar elliptic systems with strong competition, Nonlinearity, Volume 21 (2008) no. 2, pp. 305–317 | MR | Zbl

Cité par Sources :