Local minimisers and singular perturbations

Synopsis We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of Γ-convergence. The approach is demonstrated through the model problem It is shown that in certain nonconvex domains Ω ⊂ ℝ n and for ε small, there exist nonconstant local minimisers u ε satisfying u ε ≈ ± 1 except in a thin transition layer. The location of the layer is determined through the requirement that in the limit u ε → u 0 , the hypersurface separating the states u 0 = 1 and u 0 = −1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and “anisotropic” perturbations replacing |∇u| 2 .