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

Let W be a potential on R ² which is equivariant by the symmetry group of the equilateral triangle and has three minima. We show that the elliptic system possesses a nontrivial smooth solution U : R ² → R ². 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 several one-dimensional problems and for two-dimensional problems on bounded domains. © 1996 John Wiley & Sons, Inc.