Symmetry and the Vector Allen–Cahn Equation: The Point Group in ℝ n

In this chapter we begin the study of entire solutions \(u:{\mathbb R}^n\rightarrow {\mathbb R}^n\) of the vector Allen–Cahn equation (6.1) that describe the coexistence of different phases in a neighborhood of a point. We work in a symmetry context where a finite reflection group G is acting both on the domain space \({\mathbb R}_x^n\) and on the target space \({\mathbb R}_u^n\), which are assumed to be of the same dimension. The scope of this chapter is to introduce the main ideas involved in the proof of Theorem 1.2 which invokes estimate ( 1.34) or alternatively the density estimate ( 1.28), but otherwise is self-contained. In Chap. 7 we present a systematic study of all symmetric entire solutions that can be obtained by a variational approach.