Hierarchical Structure—Stratification

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


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).



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


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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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