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

• 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 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.

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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