A convergence of the approximated free boundary of regularized functionals via -convergence

We treat a variational problem with a singular term by the method of approximation via Γ -convergence, which is a constructive way. It is known that the minimizers may have a free boundary in the interior of the domain. The main aim of this paper is to establish a criterion on the location of the free boundary in terms of the approximation. As a corollary, we obtain a non-degeneracy property which means that the convergence of the approximated free boundary is sharp in some sense, and show the corresponding examples in a one-dimensional linear case. In order to derive the non-degeneracy property we need a device to the choice of approximation to the singular term.