Symmetry and the Vector Allen–Cahn Equation: The Point Group in ℝn

  • 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 begin the study of entire solutions \(u:{\mathbb R}^n\rightarrow {\mathbb R}^n\) of the vector Allen–Cahn equation (6.1) that describe the coexistence of different phases in a neighborhood of a point. We work in a symmetry context where a finite reflection group G is acting both on the domain space \({\mathbb R}_x^n\) and on the target space \({\mathbb R}_u^n\), which are assumed to be of the same dimension. The scope of this chapter is to introduce the main ideas involved in the proof of Theorem  1.2 which invokes estimate ( 1.34) or alternatively the density estimate ( 1.28), but otherwise is self-contained. In Chap.  7 we present a systematic study of all symmetric entire solutions that can be obtained by a variational approach.


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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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