Free Boundary Problems, System of Equations for Bose–Einstein Condensate and Competing Species

  • Zhitao Zhang
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 29)

Abstract

In Chap. 11, we study free boundary problems, Schrödinger systems from Bose–Einstein condensates, and competing systems with many species. We prove the existence and uniqueness result of the Dirichlet boundary value problem of elliptic competing systems. We show that, for the singular limit, species are spatially segregated; they satisfy a remarkable system of differential inequalities as κ→+∞. We also introduce optimal partition problems related to eigenvalues and nonlinear eigenvalues. Finally, some recent new results on Schrödinger systems from Bose–Einstein condensates are presented.

Keywords

Comparison Principle Free Boundary Problem Einstein Condensate Solitary Wave Solution Singular Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 3.
    H.W. Alt, L.A. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282, 431–461 (1984) MathSciNetCrossRefGoogle Scholar
  2. 10.
    A. Ambrosetti, E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. 75(1), 67–82 (2007) MathSciNetMATHCrossRefGoogle Scholar
  3. 27.
    T. Bartsch, M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on \(\mathbb{R}^{N}\). Arch. Ration. Mech. Anal. 124(3), 261–276 (1993) MathSciNetMATHCrossRefGoogle Scholar
  4. 32.
    T. Bartsch, E.N. Dancer, Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Partial Differ. Equ. 37(3–4), 345–361 (2010) MathSciNetMATHCrossRefGoogle Scholar
  5. 38.
    L.A. Cafarelli, J.M. Roquejorffre, Uniform Hölder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames. Arch. Ration. Mech. Anal. 183(3), 457–487 (2007) MathSciNetCrossRefGoogle Scholar
  6. 40.
    L.A. Caffarelli, L. Karp, H. Shahgholian, Regularity of a free boundary with applications to the Pompeiu problem. Ann. Math. 151, 269–292 (2000) MathSciNetMATHCrossRefGoogle Scholar
  7. 41.
    L.A. Caffarelli, A.L. Karakhanyan, F. Lin, The geometry of solutions to a segregation problem for non-divergence systems. Fixed Point Theory Appl. 5(2), 319–351 (2009) MathSciNetMATHCrossRefGoogle Scholar
  8. 42.
    L.A. Cafferelli, F. Lin, An optimal partition problems for eigenvalues. J. Sci. Comput. 31(1/2), 5–18 (2007) MathSciNetMATHCrossRefGoogle Scholar
  9. 43.
    L.A. Cafferelli, F. Lin, Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Am. Math. Soc. 21(3), 847–862 (2008) CrossRefGoogle Scholar
  10. 45.
    A. Castro, J. Cossio, J.M. Neuberger, A minimax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems. Electron. J. Differ. Equ. 2, 1–18 (1998) MathSciNetGoogle Scholar
  11. 56.
    M. Conti, S. Terracini, G. Verzini, Neharis problem and competing species systems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19(6), 871–888 (2002) MathSciNetMATHCrossRefGoogle Scholar
  12. 57.
    M. Conti, S. Terracini, G. Verzini, An optimal partition problem related to nonlinear eigenvalues. J. Funct. Anal. 198, 160–196 (2003) MathSciNetMATHCrossRefGoogle Scholar
  13. 58.
    M. Conti, S.S. Terracini, G. Verzini, A variational problem for the spatial segregation of reaction diffusion systems. Indiana Univ. Math. J. 54(3), 779–815 (2005) MathSciNetMATHCrossRefGoogle Scholar
  14. 59.
    M. Conti, S.S. Terracini, G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195(2), 524–560 (2005) MathSciNetMATHCrossRefGoogle Scholar
  15. 72.
    E.N. Dancer, Z. Zhang, Dynamics of Lotka–Volterra competition systems with large interaction. J. Differ. Equ. 182(2), 470–489 (2002) MathSciNetMATHCrossRefGoogle Scholar
  16. 73.
    E.N. Dancer, J. Wei, T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27(3), 953–969 (2010) MathSciNetMATHCrossRefGoogle Scholar
  17. 74.
    E.N. Dancer, K. Wang, Z. Zhang, Uniform Hölder estimate for singularly perturbed parabolic systems of Bose–Einstein condensates and competing species. J. Differ. Equ. 251, 2737–2769 (2011) MathSciNetMATHCrossRefGoogle Scholar
  18. 75.
    E.N. Dancer, K. Wang, Z. Zhang, Dynamics of strongly competing systems with many species. Trans. Am. Math. Soc. 364(2), 961–1005 (2012) MathSciNetMATHCrossRefGoogle Scholar
  19. 76.
    E.N. Dancer, K. Wang, Z. Zhang, The limit equation for the Gross–Pitaevskii equations and S. Terracini’s conjecture. J. Funct. Anal. 262, 1087–1131 (2012) MathSciNetMATHCrossRefGoogle Scholar
  20. 79.
    D.G. de Figueiredo, O. Lopes, Solitary waves for some nonlinear Schrödinger systems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25(1), 149–161 (2008) MATHCrossRefGoogle Scholar
  21. 95.
    D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 2001) MATHGoogle Scholar
  22. 116.
    Q. Han, Nodal sets of harmonic functions. Pure Appl. Math. Q. 3(3), 647–688 (2007) MathSciNetMATHGoogle Scholar
  23. 120.
    B. Kawohl, Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Mathematics, vol. 1150 (Springer, Berlin, 1985) MATHGoogle Scholar
  24. 123.
    M.K. Kwong, Uniqueness of positive solutions of Δuu+u p=0 in \(\mathbb{R}^{n}\). Arch. Ration. Mech. Anal. 105, 243–266 (1989) MathSciNetMATHCrossRefGoogle Scholar
  25. 141.
    Z. Liu, Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282(3), 721–731 (2008) MATHCrossRefGoogle Scholar
  26. 146.
    Z. Nehari, Characteristic values associated with a class of nonlinear second order differential equations. Acta Math. 105, 141–175 (1961) MathSciNetMATHCrossRefGoogle Scholar
  27. 147.
    B. Noris, H. Tavares, S. Terracini, G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun. Pure Appl. Math. 63(3), 267–302 (2010) MathSciNetMATHGoogle Scholar
  28. 148.
    B. Noris, H. Tavares, S. Terracini, G. Verzini, Convergence of minimax and continuation of critical points for singularly perturbed systems. J. Eur. Math Soc. 14(3), 1245–1273 (2012) MathSciNetMATHCrossRefGoogle Scholar
  29. 166.
    S. Smale, On the differential equations of species in competition. J. Math. Biol. 3, 5–7 (1976) MathSciNetMATHCrossRefGoogle Scholar
  30. 177.
    H. Tavares, S. Terracini, Regularity of the nodal set of the segregated critical configurations under a weak reflection law. Calc. Var. Partial Differ. Equ. (2012). doi: 10.1007/s00526-011-0458-z MathSciNetGoogle Scholar
  31. 178.
    S. Terracini, G. Verzini, Multipulse phases in k-mixtures of Bose–Einstein condensates. Arch. Ration. Mech. Anal. 194(3), 717–741 (2009) MathSciNetMATHCrossRefGoogle Scholar
  32. 189.
    K. Wang, Z. Zhang, Some new results in competing systems with many species. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 27, 739–761 (2010) MATHCrossRefGoogle Scholar
  33. 191.
    J.C. Wei, T. Weth, Radial solutions and phase separation in a system of two coupled Schrodinger equations. Arch. Ration. Mech. Anal. 190, 83–106 (2008) MathSciNetMATHCrossRefGoogle Scholar
  34. 211.
    W. Ziemer, Weakly Differentiable Functions (Springer, Berlin, 1989) MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Zhitao Zhang
    • 1
  1. 1.Academy of Mathematics & Systems ScienceThe Chinese Academy of SciencesBeijingP.R. China

Personalised recommendations