On a minimization problem with a mass constraint in dimension two

We continue our study that was begun in [1] of a singular perturbation-type minimization problem with a mass constraint, involving a potential vanishing on two curves in the plane. In the case of a two-dimensional nonconvex domain (and under some additional assumptions), we are able to prove a convergence result for the minimizers, and characterize the limit as a solution of a mixed Dirichlet-Neumann boundary condition problem with a mass constraint.