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

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.