We characterize all minimizers of the vector-valued Allen–Cahn equation in ℝ 2 under the assumption that the potential W has three wells and that the associated degenerate metric does not satisfy the usual strict triangle inequality. These minimizers depend on one variable only in a suitable coordinate system. In particular, we show that no minimizing solution to Δ u = ∇ W ( u ) on ℝ 2 can approach the three distinct values of the potential wells.
Bronsard et al. (Mon,) studied this question.