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Connections

  • Nicholas D. Alikakos
  • Giorgio Fusco
  • Panayotis Smyrnelis
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 91)

Abstract

We begin by giving a concise proof of the existence of a heteroclinic connection (Theorem 2.1). The experienced reader then can move on to Sect. 2.6. In Sect. 2.4 we develop an alternative approach via constrained minimization. Most readers will find this easier and also good preparation for the polar form and the cut-off lemma in Chap.  4. In Sect. 2.6 we consider the connection problem for an unbalanced double-well potential, and handle it via the constrained method. Finally in Sect. 2.7 we investigate the failure of the existence of a connection when three or more global minima are present.

References

  1. 1.
    Alberti, G.: Variational models for phase transitions, an approach via Gamma convergence. In: Ambrosio, L., Dancer, N. (eds.) Calculus of Variations and Partial Differential Equations, pp. 95–114. Springer, Berlin (2000)CrossRefGoogle Scholar
  2. 2.
    Alikakos, N.D., Fusco, G.: On the connection problem for potentials with several global minima. Indiana Univ. Math. J. 57, 1871–1906 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alikakos, N.D., Betelú, S.I., Chen, X.: Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energies. Eur. J. Appl. Math. 17, 525–556 (2006)CrossRefGoogle Scholar
  4. 4.
    Alikakos, N.D., Katzourakis, N.: Heteroclinic travelling waves of gradient diffusion systems. Trans. Am. Math. Soc. 363, 1362–1397 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    André, N., Shafrir, I.: On a vector-valued singular perturbation problem on the sphere. In: Proceedings of the International Conference on Nonlinear Analysis, Recent Advances in Nonlinear Analysis, pp. 11–42. World Scientific Publishing, Singapore (2008)Google Scholar
  6. 6.
    Antonopoulos, P., Smyrnelis, P.: On minimizers of the Hamiltonian system u″ = ∇W(u), and on the existence of heteroclinic, homoclinic and periodic orbits. Indiana Univ. Math. J. 65(5), 1503–1524 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Colding, T.H., Minicozzi, W.P.: A Course in Minimal Surfaces. Graduate Studies in Mathematics, vol. 121. American Mathematical Society, Providence (2011)Google Scholar
  8. 8.
    Coti Zelati, V., Rabinowitz, P.H.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Am. Math. Soc. 4, 693–727 (1991)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fife, P.C., McLeod, J.B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65(4), 335–361 (1977)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fusco, G., Gronchi, G.F., Novaga, M.: On the existence of connecting orbits for critical values of the energy. J. Differ. Equ. 263, 8848–8872 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fusco, G., Gronchi, G.F., Novaga, M.: On the existence of heteroclinic connections. Sao Paulo J. Math. Sci. 12, 1–14 (2017)zbMATHGoogle Scholar
  12. 12.
    Heinze, S.: Travelling waves for semilinear parabolic partial differential equations in cylindrical domains. PhD thesis, Heidelberg University (1988)Google Scholar
  13. 13.
    Heinze, S., Papanicolaou, G., Stevens, A.: Variational principles for propagation speeds in inhomogeneous media. SIAM J. Appl. Math. 63(1), 129–148 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Katzourakis, N.: On the loss of compactness in the vectorial heteroclinic connection problem. Proc. Roy. Soc. Edinb. Sect. A 146(3), 595–608 (2016)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lin, F., Pan, X.B., Wang, C.: Phase transition for potentials of high-dimensional wells. Commun. Pure Appl. Math. 65(6), 833–888 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lucia, M., Muratov, C., Novaga, M.: Existence of traveling wave solutions for Ginzburg-Landau-type problems in infinite cylinders. Arch. Ration. Mech. Anal. 188(3), 475–508 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Modica, L.: A Gradient bound and a Liouville Theorem for nonlinear Poisson equations. Commun. Pure. Appl. Math. 38(5), 679–684 (1985)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Monteil, A., Santambrogio, F.: Metric methods for heteroclinic connections. Math. Methods Appl. Sci. 41(3), 1019–1024 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Muratov, C.B.: A global variational structure and propagation of disturbances in reacting-diffusion systems of gradient type. Discrete Contin. Dyn. Syst. Ser. B 4, 867–892 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rabinowitz, P.H.: Periodic and heteroclinic orbits for a periodic hamiltonian system. Ann. Inst. Henri Poincaré 6(5), 331–346 (1989)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Risler, R.E.: Global convergence towards travelling fronts in nonlinear parabolic systems with a gradient structure. Ann. Inst. Henri Poincaré (C) Non Linear Anal. 25(2), 381–424 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Smyrnelis, P.: Gradient estimates for semilinear elliptic systems and other related results. Proc. Roy. Soc. Edinb. Sect. A 145(6), 1313–1330 (2015)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Stefanopoulos, V.: Heteroclinic connections for multiple well potentials: the anisotropic case. Proc. Roy. Soc. Edinb. Sect. A 138, 1313–1330 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sourdis, C.: The heteroclinic connection problem for general double-well potentials. Mediterr. J. Math. 13, 4693–4710 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101(3), 209–260 (1988)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sternberg, P.: Vector-valued local minimizers of nonconvex variational problems. Rocky Mountain J. Math. 21, 799–807 (1991)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Sternberg, P., Zuniga, A.: On the heteroclinic problem for multi-well gradient systems. J. Differ. Equ. 261, 3987–4007 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Terman, D.: Infinitely many traveling wave solutions of a gradient system. Trans. Am. Math. Soc. 301(2), 537–556 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Nicholas D. Alikakos
    • 1
  • Giorgio Fusco
    • 2
  • Panayotis Smyrnelis
    • 3
  1. 1.Department of MathematicsNational and Kapodistrian UniversityAthensGreece
  2. 2.Department of MathematicsUniversity of L’AquilaCoppitoItaly
  3. 3.Center for Mathematical ModelingUniversity of ChileSantiagoChile

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