Multyphase solutions to the vector Allen-Cahn equation: Crystalline and other complex symmetric structures

Original · EN

We present a systematic study of entire symmetric solutions u:Rⁿ→ Rᵐ of the vector Allen-Cahn equation Δu-Wᵤ(u)=0, x ∈ Rⁿ, where W:Rᵐ→ R is smooth, symmetric, nonnegative with a finite number of zeros and Wᵤ=(∂ W/∂ u₁,,∂ W/∂ uₘ). We introduce a general notion of equivariance with respect to a homomorphism f:G→Γ (G,Γ reflection groups) and prove two abstract results, concerning the cases of G finite and G discrete, for 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.