Symmetry and the Vector Allen–Cahn Equation: Crystalline and Other Complex Structures

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


We present a systematic study of entire symmetric solutions \(u:{\mathbb R}^n \to {\mathbb R}^m\) of the vector Allen–Cahn equation Δu − Wu(u) = 0, \(x \in {\mathbb R}^n\), where \(W:{\mathbb R}^m \to {\mathbb R}\) is smooth, symmetric, nonnegative with a finite number of zeros, and Wu := (∂W∂u1, …, ∂W∂um). We assume that W is invariant under a finite reflection group Γ acting on target space \({\mathbb R}^m\) and that there is a finite or discrete reflection group G acting on the domain space \({\mathbb R}^n\). G and Γ are related by a homomorphism f : G → Γ and a map u is said to be equivariant with respect to f if
$$\displaystyle u(gx)=f(g)u(x),\;\;\text{ for }\;g\in G,\;x\in {\mathbb R}^n. $$
We prove two abstract theorems, concerning the cases of G finite and G discrete, on the existence of equivariant solutions. Our approach is variational and based on a mapping property of the parabolic vector Allen–Cahn equation and on a pointwise estimate for vector minimizers. The abstract results are then applied for particular choices of G, Γ and f : G → Γ, and solutions with complex symmetric structure are described.


  1. 1.
    Alikakos, N.D.: A new proof for the existence of an equivariant entire solution connecting the minima of the potential for the system Δu − W u(u) = 0. Commun. Partial Diff. Equ. 37(12), 2093–2115 (2012)Google Scholar
  2. 2.
    Alikakos, N.D., Fusco, G.: Entire solutions to equivariant elliptic systems with variational structure. Arch. Ration. Mech. Anal. 202(2), 567–597 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alikakos, N.D., Smyrnelis, P.: Existence of lattice solutions to semilinear elliptic systems with periodic potential. Electron. J. Diff. Equ. 15, 1–15 (2012)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bates, P.W., Fusco, G., Smyrnelis, P.: Entire solutions with six-fold junctions to elliptic gradient systems with triangle symmetry. Adv. Nonlinear Stud. 13(1), 1–12 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bates, P.W., Fusco, G., Smyrnelis, P.: Multiphase solutions to the vector Allen-Cahn equation: crystalline and other complex symmetric structures. Arch. Ration. Mech. Anal. 225(2), 685–715 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Braun, R.J., Cahn, J.W., McFadden, G.B., Wheeler, A.A.: Anisotropy of interfaces in an ordered alloy: a multiple-order-parameter model. Philos. Trans. R. Soc. Lond. A 355, 1787–1833 (1997)CrossRefGoogle Scholar
  7. 7.
    Bronsard, L., Reitich, F.: On three-phase boundary motion and the singular limit of a vector-valued Ginzburg- Landau equation. Arch. Ration. Mech. Anal. 124(4), 355–379 (1993)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Fusco, G.: Equivariant entire solutions to the elliptic system Δu − W u(u) = 0 for general G-invariant potentials. Calc. Var. Part. Diff. Equ. 49(3), 963–985 (2014)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.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)CrossRefGoogle Scholar
  12. 12.
    Johnson, J.W.: The number of group homomorphisms from D m into D n. Collage Math. J. 44(3), 191–192 (2013)MathSciNetGoogle Scholar
  13. 13.
    Smyrnelis, P.: Solutions to elliptic systems with mixed boundary conditions. Phd Thesis (2012)Google Scholar

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© 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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