On entire solutions of an elliptic system modeling phase separations

Advances in Mathematics

Volumn 243, Issue , 2013, Pages 102-126

  Berestycki, Henri ^a   Terracini, Susanna ^b   Wang, Kelei ^c   Wei, Juncheng ^d  

^a ÉCOLE DES HAUTES ÉTUDES EN SCIENCES SOCIALES   (France)

^b UNIVERSITY OF TURIN   (Italy)

^c Wuhan National Laboratory for Optoelectronics   (China)

^d CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Almgren monotonicity formulas; Elliptic systems; Phase separations; Stable solutions

Indexed keywords

EID: 84877881696     PISSN: 00018708     EISSN: 10902082     Source Type: Journal    
DOI: 10.1016/j.aim.2013.04.012     Document Type: Article

References (32)

1. Alessio F., Calamai A., Montecchiari P. Saddle-type solutions for a class of semilinear elliptic equations. Adv. Differential Equations 2007, 12:361-380.
2. R3 and a conjecture of De Giorgi. J. Amer. Math. Soc. 2000, 13(4):725-739.
3. Ao P., Chui S.T. Binary Bose-Einstein condensate mixtures in weakly and strongly segregated phases. Phys. Rev. A 1998, 58:4836-4840.
4. Arapostathis A., Ghosh M., Marcus S. Harnack's inequality for cooperative weakly coupled elliptic systems. Comm. Partial Differential Equations 1999, 24(9-10):1555-1571.
5. Bao W. Ground states and dynamics of multi-component Bose-Einstein condensates. SIAM MMS 2004, 2:210-236.
6. Bao W., Du Q. Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comput. 2004, 25:1674-1697.
7. Berestycki H., Caffarelli L., Nirenberg L. Further qualitative properties for elliptic equations in unbounded domains. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1997, 25(4):69-94.
8. Berestycki H., Lin T.-C., Wei J., Zhao C. On phase-separation models: asymptotics and qualitative properties. Arch. Ration. Mech. Anal. 2013, 208:163-200.
9. Busca J., Sirakov B. Harnack type estimates for nonlinear elliptic systems and applications. Ann. I. H. Poincare-AN 2004, 21:543-590.
10. R2m. J. Eur. Math. Soc. 2009, 11(4):819-843.
11. Caffarelli L.A., Karakhanyan A.L., Lin F. The geometry of solutions to a segregation problem for non-divergence systems. J. Fixed Point Theory Appl. 2009, 5(2):319-351.
12. Caffarelli L.A., Lin F. Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Amer. Math. Soc. 2008, 21:847-862.
13. Caffarelli L.A., Salsa S. A Geometric Approach to Free Boundary Problems. Graduate Studies in Mathematics 2005, vol. 68:x+270. American Mathematical Society, Providence, RI.
14. Chang S.M., Lin C.S., Lin T.C., Lin W.W. Segregated nodal domains of two-dimensional multispecies Bose-Einstein condensates. Physica D 2004, 196(3-4):341-361.
15. Chen W., Li C. Classification of solutions of some nonlinear elliptic solutions. Duke Math. J. 1991, 63(3):615-622.
16. Conti M., Terracini S., Verzini G. Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 2005, 195(2):524-560.
17. Dancer E.N., Wang K., Zhang Z. The limit equation for the Gross-Pitaevskǐi equations and S. Terracini's conjecture. J. Funct. Anal. 2012, 262(3):1087-1131.
18. Dang H., Fife P.C., Peletier L.A. Saddle solutions of the bistable dffusion equation. Z. Angew. Math. Phys. 1992, 43(6):984-998.
19. del Pino M., Kowalczyk M., Pacard F., Wei J. Multiple-end solutions to the Allen-Cahn equation in R2. J. Funct. Anal. 2010, 258(2):458-503.
20. del Pino M., Kowalczyk M., Wei J. On de Giorgi conjecture in dimension n ≥ 9. Ann. of Math. 2011, 174(3):1485-1569.
21. Dipierro S. Geometric inequalities and symmetry results for elliptic systems. Discrete Contin. Dyn. Syst. 2013, 33(8):3473-3496.
22. Farina A., Sciunzi B., Vadinoci E. Bernstein and De Giorgi type problems: new results via a geometric approach. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2008, 7(4):741-791.
23. Ghoussoub N., Gui C. On a conjecture of De Giorgi and some related problems. Math. Ann. 1998, 311:481-491.
24. Hall D.S., Matthews M.R., Ensher J.R., Wieman C.E., Cornell E.A. Dynamics of component separation in a binary mixture of Bose-Einstein condensates. Phys. Rev. Lett. 1998, 81:1539-1542.
25. Myatt C.J., Burt E.A., Ghrist R.W., Cornell E.A., Wieman C.E. Production of two overlapping Bose-Einstein condensates by sympathetic cooling. Phys. Rev. Lett. 1997, 78:586-589.
26. Noris B., Tavares H., Terracini S., Verzini G. Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Comm. Pure Appl. Math. 2010, 63:267-302.
27. Pacard F., Wei J. Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones. J. Funct. Anal. 2013, 264(5):1131-1167.
28. Savin O. Regularity of flat level sets in phase transitions. Ann. of Math. (2) 2009, 169(1):41-78.
29. N. Soave, A. Zilio, Entire solutions with exponential growth for an elliptic system modeling phase-separation, Preprint 2013, arxiv:1303.3797.
30. Tavares H., Terracini S. Regularity of the nodal set of segregated critical configurations under a weak reflection law. Calc. Var. Partial Differential Equations 2012, 45(3-4):273-317.
31. K. Wang, On the De Giorgi type conjecture for an elliptic system modeling phase separation (submitted for publication).
32. Wei J., Weth T. Asymptotic behavior of solutions of planar elliptic systems with strong competition. Nonlinearity 2008, 21:305-317.