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

We present a systematic study of entire symmetric solutions $u:R^n\rightarrow R^m$ of the vector Allen-Cahn equation $\Delta u-W_u(u)=0, x \in R^n$, where $W:R^m\rightarrow R$ is smooth, symmetric, nonnegative with a finite number of zeros and $W_u=(\frac{\partial W}{\partial u_1},\ldots,\frac{\partial W}{\partial u_m})^\top$. We introduce a general notion of equivariance with respect to a homomorphism $f:G\rightarrow\Gamma$ ($G,\Gamma$ 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.