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Hierarchical Structure—Stratification

  • 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

In this chapter we extend the density estimate in Theorem  5.2 by replacing the constant solution a with a symmetric, minimal, hyperbolic connection e, (and more generally with any equivariant minimal hyperbolic solution), and then derive Liouville theorems and asymptotic information for minimal solutions under symmetry hypotheses. Utilizing the extended density estimate we give a proof of a result of Alama et al. (Calc Var 5:359–390, 1997) on the existence of stationary layered solutions in \({\mathbb R}^2\). The Alama, Bronsard and Gui example is revisited in Chap.  9 under no symmetry hypotheses. Our results were originally obtained by a different method in Alikakos and Fusco (Annali della Scuola Normale Superiore di Pisa XV:809–836, 2016).

Notes

Acknowledgements

We would like to thank Mihalis Nikolouzos for useful discussions on the proof of Corollary 8.2.

References

  1. 1.
    Agmon, S.: Lectures on Exponential Decay of Solutions of Second Order Elliptic Equations. Math. Notes Princeton University No. 29. Princeton University Press, Princeton (1982)Google Scholar
  2. 2.
    Alama, S., Bronsard, L., Gui, C.: Stationary layered solutions in \({\mathbb R}^2\) for an Allen–Cahn system with multiple well potential. Calc. Var. 5(4), 359–390 (1997)Google Scholar
  3. 3.
    Alessio, F.: Stationary layered solutions for a system of Allen-Cahn type equations. Indiana Univ. Math. J. 62(5), 1535–1564 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alikakos, N.D., Fusco, G.: Density estimates for vector minimizers and application. Discrete Cont. Dyn. Syst. 35(12), 5631–5663 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Alikakos, N.D., Fusco, G.: Asymptotic behavior and rigidity results for symmetric solutions of the elliptic system Δu = W u(u). Annali della Scuola Normale Superiore di Pisa, XV(issue special), 809–836 (2016)Google Scholar
  6. 6.
    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
  7. 7.
    Bronsard, L., Gui, C., Schatzman, M.: A three-layered minimizer in \({\mathbb R}^2\) for a variational problem with a symmetric three-well potential. Commun. Pure. Appl. Math. 49(7), 677–715 (1996)Google Scholar
  8. 8.
    Garofalo, N., Lin, F.H.: Monotonicity properties of variational integrals, A p weights and unique continuation. Ind. Univ. Math. J. 35(2), 245–268 (1986)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, vol. 224, revised 2nd edn. Springer, Berlin (1998)Google Scholar
  10. 10.
    Gui, C., Schatzman, M.: Symmetric quadruple phase transitions. Ind. Univ. Math. J. 57(2), 781–836 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hislop, P.D., Sigal, I.M.: Introduction to Spectral Theory with Applications to Schrödinger Operators. Applied Mathematical Sciences, vol. 113. Springer, New York (1996)CrossRefGoogle Scholar
  12. 12.
    Monteil, A., Santambrogio, F.: Metric methods for heteroclinic connections in infinite dimensional spaces. arXiv: 1709-02117v1 (To appear)Google Scholar
  13. 13.
    Schatzman, M.: Asymmetric heteroclinic double layers. ESAIM: Control Optim. Calc. Var. 8(A tribute to J. L. Lions), 965–1005 (electronic) (2002)MathSciNetzbMATHGoogle 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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