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

Abstract

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.