On the De Giorgi Type Conjecture for an Elliptic System Modeling Phase Separation

Communications in Partial Differential Equations

Volumn 39, Issue 4, 2014, Pages 696-739

  Wang, Kelei ^a  

^a Wuhan National Laboratory for Optoelectronics   (China)

Author keywords

Elliptic systems; One dimensional symmetry; Phase separation; Sliding method

Indexed keywords

EID: 84897850015     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605302.2013.856916     Document Type: Article

References (18)

1. Alt, H., Caffarelli, L., Friedman, A. (1984). Variational problems with two phases and their free boundaries. Trans. Amer. Math. Soc. 282 (2): 431-461.
2. Berestycki, H., Hamel, F., Monneau, R. (2000). One dimensional symmetry of bounded entire solutions of some elliptic equations. Duke Math. J. 103 (3): 375-396.
3. Berestycki, H., Lin, T., Wei, J., Zhao, C. (2013). On phase-separation model: asymptotics and qualitative properties. Arch. Rat. Mech. Anal. 208: 163-200.
4. Berestycki, H., Terracini, S., Wang, K., Wei, J. (2013). Existence and stability of entire solutions of an elliptic system modeling phase separation, Adv. Math. 243: 102-126.
5. Caffarelli, L., Lin, F. (2008). Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Amer. Math. Soc. 21: 847-862.
6. Conti, M., Terracini, S. Verzini, G. (2005). Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195:524-560.
7. Dancer, E.N., Wang, K., Zhang, Z. (2011). Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species. J. Diff. Eqs. 251: 2737-2769.
8. Dancer, E.N., Wang, K., Zhang, Z. (2012). The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture, J. Funct. Anal. 262: 1087-1131.
9. N and related conjectures. Ricerche Mat. 48 (suppl.): 129-154.
10. Farina, A. (2014). Some symmetry results for entire solutions of an elliptic system arising in phase separation. Disc. Cont. Dyn. Sys. Ser. A 34: 2505-2511.
11. Farina, A., Soave, N. Monotonicity and 1-dimensional symmetry for solutions of an elliptic system arising in Bose-Einstein condensation, Preprint, arXiv:1303.1265.
12. Friedland, S., Hayman, W. (2001). Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions. Comm. Math. Helv. 51: 133-161.
13. Gilbarg, D., Trudinger, N. Elliptic Partial Differential Equations of Second Order. New York: Sprigner.
14. Helffer, B., Hoffmann-Ostenhof, T., Terracini, S. (2010). On Spectral Minimal Partitions: the case of the sphere. In: Around the Research of Vladimir Maz'ya III. International Mathematical Series, Vol. 13. New York: Springer 53-178.
15. Lieb, E., Loss, M. (2001). Analysis. Graduate Studies in Mathematics, Vol. 14. Providence, RI: American Mathematical Society.
16. Noris, B., Tavares, H., Terracini, S., Verzini, G. (2010). Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Comm. Pure Appl. Math. 63: 267-302.
17. Savin, O. (2009). Regularity of flat level sets in phase transitions. Ann. Math. 169: 41-78.
18. Tavares, H., Terracini, S. (2012). Regularity of the nodal set of segregated critical configurations under a weak reflection law. Calc. Var. PDEs 45: 273-317.