Local Minimizers Induced by Spatial Inhomogeneity with Inner Transition Layer

The main aim of this paper is to construct and provide the structure of local minimizers of the variational problem infv L1(Ω) F (v), where : small parameter and[formula]having only the functionsk1>0 andk2>0 as parameters. Given any closed simple smooth curveγmcontained in a level set ofa(X)=k1(X) k2(X),X Ω, we give necessary conditions relating the slope and concavity ofa(X),X γm, along the normaln[γm(X)], with the curvatureκ(X) ofγmso that the above problem possesses a nonconstant minimizerv (X) whose nodal curveγ mconverges uniformly toγm, as →0. Moreoverv →1, as →0, uniformly on compact sets contained in one of the two connected components ofΩ\γmandv →−1, as →0, uniformly on compact sets contained in the other one.