Local minimisers and singular perturbations

We construct local minimisers to certain variational problems. The method is quite general and relies on the theory of F-convergence. The approach is demonstrated through the model problem [formula ommittes] It is shown that in certain nonconvex domains The approach is demonstrated through the model problem Ω ⊂Rⁿ and for ε small, there exist nonconstant localminimisers 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⁽⁾ the hypersurface separating the states u₀= 1 and u₀=-1 locally minimises surface area. Generalisations are discussed with, for example, vector-valued u and anisotropic perturbations replacing’∇_u’².