Advertisement

Nodal set of strongly competition systems with fractional Laplacian

  • Shan Zhang
  • Zuhan Liu
Article

Abstract

This paper is concerned with the local structure of the nodal set of segregated configurations associated with a class of fractional singularly perturbed elliptic systems. We prove that the nodal set is a collection of smooth hyper-surfaces, up to a singular set with Hausdorff dimension not greater than n − 2. The proof relies upon a clean-up lemma and the classical dimension reduction principle by Federer.

Mathematics Subject Classification

35Q55 35A05 

Keywords

Competition system Spatial segregation Free boundary Clean up lemma 

References

  1. 1.
    Alt H.W., Caffarelli L.A., Friedman A.: A variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282(2), 431–461 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Caffarelli L.A., Lin F.: An optimal partition problem for eigenvalues. J. Sci. Comput. 31(1), 5–18 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caffarelli L.A., Lin F.: Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Am. Math. Soc. 21, 847–862 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caffarelli L.A., Karakhanyan A.L., Lin F.-H.: The geometry of solutions to a segregation problem for nondivergence systems. J. Fixed Point Theory Appl. 5(2), 319–351 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chang S.M., Lin C.S., Lin T.C., Lin W.W.: Segregated nodal domains of two-dimensional multispecies Bose–Einstein condensates. Phys. D 196(3–4), 341–361 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caffarelli, L.A., Salsa, S.: A Geometric Approach to the Free Boundary Problems. Graduate Studies in Mathematics, vol. 68. American Mathematical Society, Providence (2005)Google Scholar
  7. 7.
    Caffarelli L.A., Silvestre L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cabré X., Tan J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224, 2052–2093 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Conti M., Terracini S., Verzini G.: A variational problem for the spatial segregation of reaction diffusion systems. Indiana Univ. Math. J. 54(3), 779–815 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Conti M., Terracini S., Verzini G.: Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195(2), 524–560 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Conti M., Terracini S., Verzini G.: Uniqueness and least energy property for strongly competing systems. Interfaces Free Bound. 8, 437–446 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dancer E.N., Du Y.H.: Competing species equations with diffusion, large interactions, and jumping nonlinearities. J. Differ. Equ. 114, 434–475 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dancer E.N., Wang K., Zhang Z.: Uniform Hölder estimate for singulary perturbed parabolic systems of Bose–Einstein condensates and competing species. J. Differ. Equ. 251, 2737–2769 (2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    Dancer E.N., Wang K., Zhang Z.: Dynamics of strongly competing systems with many species. Trans. Am. Math. Soc. 364(2), 961–1005 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dancer E.N., Wang K., Zhang Z.: The limit equation for the Gross–Pitaevskii equations and S. Terracini’s conjecture. J. Funct. Anal. 262, 1087–1131 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dancer E.N., Zhang Z.: Dynamics of Lotka–Volterra competition systems with large interactions. J. Differ. Equ. 182, 470–489 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fuso N., Millot V., Morini M.: A quantitative isoperimetric inequality for fractional perimeters. J. Funct. Anal. 261, 697–715 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Han, Q., Lin, F.: Nodal sets of solutions of elliptic differential equations. (Books available on Han’s homepage)Google Scholar
  19. 19.
    Landis E.: Seconder Order Equations of Elliptic and Parabolic Type. America Mathematical Society, Providence (1998)Google Scholar
  20. 20.
    Landkof N.S.: Foundations of Modern Potential Theory. Springer, Heidelberg (1972)CrossRefzbMATHGoogle Scholar
  21. 21.
    Liu Z.: Phase separation of two component Bose–Einstein condensates. J. Math. Phys. 50, 102–104 (2009)MathSciNetGoogle Scholar
  22. 22.
    Liu Z.: The spatial behavior of rotating two-component Bose–Einstein condensates. J. Funct. Anal. 261, 1711–1751 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Noris B., Tavares H., Terracini S., Verzini G.: Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun. Pure Appl. Math. 63(3), 267–302 (2010)CrossRefzbMATHGoogle Scholar
  24. 24.
    Tavares H., Terracini S.: Regularity of the nodal set of the segregated critical configuration under a weak reflection law, preprint 2010. Calc. Var. Partial Differ. Equ. 45, 273–317 (2012)CrossRefzbMATHGoogle Scholar
  25. 25.
    Terracini, S., Verzini, G., Zilio, A.: Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian (preprint). arxiv:1211.6087v1
  26. 26.
    Terracini, S., Verzini, G., Zilio, A.: Uniform Hölder regularity with small exponent in competing fractional diffusion systems (preprint). arxiv:1303.6079v1
  27. 27.
    Verzini, G., Zilio, A.: strong competition versus fractional diffusion: the case of Lotka–Volterra interaction (preprint). arXiv:1310.7355v1
  28. 28.
    Wei J., Weth T.: Asymptotic behaviour of solutions of planar elliptic systems with strong competition. Nonlinearity 21(2), 305–317 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wang K., Zhang Z.: Some new results in competing systems with many species. Ann. Inst. H. Poincare Anal. Nonlinear Anal. 27(2), 739–761 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zhang, S., Liu, Z.: Singularities of the nodal set of segregated configurations. Calc. Var. Partial Differ. Equ. doi: 10.1007/s00526-015-0854-x
  31. 31.
    Zhang, S., Liu, Z., Zhou, L.: Spatial behavior of a strongly competition system with fractional Laplacian (2014, preprint)Google Scholar
  32. 32.
    Zhang S., Zhou L., Liu Z.: Global minimizers of coexistence for rotating N-component Bose–Einstein condensates. Nonlinear Anal. RWA 12, 2567–2578 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang S., Zhou L., Liu Z.: The spatial behavior of a competition diffusion advection system with strong competition. Nonlinear Anal. RWA 14, 976–989 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zhang S., Zhou L., Liu Z., Lin Z.: Spatial segregation limit of a non-autonomous competition diffusion system. J. Math. Anal. Appl. 389, 119–129 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.School of Mathematical ScienceNanjing Normal UniversityNanjingChina
  2. 2.School of Mathematical ScienceYangzhou UniversityYangzhouChina

Personalised recommendations