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

Peter W. Bates; Giorgio Fusco; Panayotis Smyrnelis

paper

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

arXiv:1411.4008

Abstract

We present a systematic study of entire symmetric solutions $u : R^{n} \to R^{m}$ of the vector Allen-Cahn equation $Δ u - W_{u} (u) = 0, x \in R^{n}$ , where $W : R^{m} \to R$ is smooth, symmetric, nonnegative with a finite number of zeros and $W_{u} = (\frac{\partial W}{\partial u _{1}}, \dots, \frac{\partial W}{\partial u _{m}})^{⊤}$ . We introduce a general notion of equivariance with respect to a homomorphism $f : G \to Γ$ ( $G, Γ$ 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.

27 pages, 12 figures