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Archive for Rational Mechanics and Analysis

, Volume 210, Issue 3, pp 857–908 | Cite as

A Free Boundary Problem Arising from Segregation of Populations with High Competition

  • Veronica Quitalo
Article

Abstract

In this work, we show how to obtain a free boundary problem as the limit of a fully nonlinear elliptic system of equations that models population segregation of the Gause–Lotka–Volterra type. We study the regularity of the solutions. In particular, we prove Lipschitz regularity across the free boundary. The problem is motivated by the work done by Caffarelli, Karakhanyan and Fang-Hua Lin for the linear case.

Keywords

Free Boundary Viscosity Solution Lipschitz Domain Comparison Principle Free Boundary Problem 
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.

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References

  1. 1.
    Alt H.W., Caffarelli L.A., Friedman 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., Xavier, C.: Fully Nonlinear Elliptic Equations. In: American Mathematical Society colloquium publications, vol. 43, 1995Google Scholar
  3. 3.
    Caffarelli L., Lin F.-H.: 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., Fang-Hua L.: The geometry of solutions to a segregation problem for nondivergence systems. J. Fixed Point Theory Appl. 5, 319–351 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caffarelli, L.A., Salsa, S.: A Geometric Approach to Free Boundary Problems, vol. 68. Amer Mathematical Society, New York, 2005Google Scholar
  6. 6.
    Caffarelli L.A., Souganidis P.E., Wang L.: Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Commun. Pure Appl. Math. 58(3), 319–361 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cristina Cerutti M., Escauriaza L., Fabes E.B.: Uniqueness in the dirichlet problem for some elliptic operators with discontinuous coefficients. Annali di Matematica pura ed applicata 163(1), 161–180 (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    Conti M., Terracini S., Verzini G.: Nehari’s problem and competing species systems. Ann. I. H. Poincare 19, 871–888 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Conti M., Terracini S., Verzini G.: Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195, 524–560 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Conti, M., Verzini, G., Terracini, S.: A regularity theory for optimal partition problems. In: SPT 2004 Symmetry and perturbation theory, pp. 91–98. World Sci. Publ., Hackensack, 2005Google Scholar
  11. 11.
    Conti, M., Felli, V.: Coexistence and segregation for strongly competing species in special domains. arXiv:math/0602334 (2006)Google Scholar
  12. 12.
    Conti, M., Felli, V.: Minimal coexistence configurations for multispecies systems. arXiv:0812.2376 (2008)Google Scholar
  13. 13.
    Conti, M., Terracini, S., Verzini, G.: A variational problem for the spatial segregation of reaction–diffusion systems. arXiv:math/0312210 (2003)Google Scholar
  14. 14.
    Dancer E.N., Yihing Du.: Positive solutions for a three-species competition system with diffusion ii.the case of equal birth rates. Nonlinear Anal. Theory, Methods Appl. Int Multidiscipl J. Ser. A Theory Methods, 24(3), 359–373 (1995)CrossRefzbMATHGoogle Scholar
  15. 15.
    Dancer E.N., Yihong Du.: Positive solutions for a three-species competition system with diffusion i. general existence results. Nonlinear Anal. Theory, Methods Appl. Int Multidiscipl J. Ser. A Theory Methods 24(3), 337–357 (1995)CrossRefzbMATHGoogle Scholar
  16. 16.
    Dancer E.N., Yihong Du.: On a free boundary problem arising from population biology. Indiana Univ. Math. J. 52(1), 51–67 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dancer E.N., Yihong Du.: A uniqueness theorem for a free boundary problem. Proc. Am. Math. Soc. 134(11), 3223–3230 (2006)CrossRefzbMATHGoogle Scholar
  18. 18.
    Dancer E.N., Hilhorst D., Mimura M., Peletier L.A.: Spatial segregation limit of a competition-diffusion system. Eur. J. Appl. Math. 10(2), 97–115 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Dancer N.: Competing species systems with diffusion and large interactions. Rendiconti del Seminario Matematico e Fisico di Milano, 65, 23–33 (1995)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Luis E.: w 2,n a priori estimates for solutions to fully nonlinear equations. Indiana Univ. Math. J. 42, 413–423 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fabes E.B., Strook D.W.: The lp-integrability of greens functions and fundamental solutions for elliptic and parabolic equations. Duke Math. J 51(4), 997–1016 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, vol. 224. Springer, Berlin, 2001Google Scholar
  23. 23.
    Mimura, M., Ei, S.-I., Fang, Q.: Effect of domain-shape on coexistence problems in a competition-diffusion system. J. Math. Biol. 219–237, (1991)Google Scholar
  24. 24.
    Shigesada N., Kawasaki K., Teramoto E.: The effects of interference competition on stability, structure and invasion of a multi-species system. J. Math. Biol. 21, 97–113 (1984)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas at AustinAustinUSA

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