A three‐layered minimizer in R2 for a variational problem with a symmetric three‐well potential

Let W be a potential on R2 which is equivariant by the symmetry group of the equilateral triangle and has three minima. We show that the elliptic system $$-\Delta U + D W (U)^T = 0,$$ possesses a nontrivial smooth solution U:R2 → R2. Here DW(U)T is the transpose of the derivative DW(U). The natural energy of the problem is unbounded and compactness techniques cannot be applied. The proof depends on careful energy estimates and asymptotics for …