On a minimization problem with a mass constraint in dimension two Nelly AndreItai Shafrir Primary 35J20Secondary 35B2535J6058E50singular perturbationmass constraint We continue our study that was begun in [N. Andr\'e and I. Shafrir, \textit{On a minimization problem with a mass constraint involving a potential vanishing on two curves}, Israel J. Math. \textbf{186} (2011), 97--124.] 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. Indiana University Mathematics Journal 2014 text pdf 10.1512/iumj.2014.63.5201 10.1512/iumj.2014.63.5201 en Indiana Univ. Math. J. 63 (2014) 419 - 445 state-of-the-art mathematics http://iumj.org/access/