Approximation of topological singularities through free discontinuity functionals: the critical and super-critical regimes

We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in De Luca et al. (Indiana Univ Math J 73:723–779, 2024). We prove the variational equivalence between such energies, Ginzburg-Landau, and Core-Radius for anti-plane screw dislocations energies in dimension two, in the relevant energetic regimes,, where denotes the linear size of the process zone near the defects. Further, we remove the a priori restrictive assumptions that the approximating order parameters have compact jump set. This is obtained by proving a new density result for -valued functions, approximated through functions with essentially closed jump set, in the strong BV norm.