$\mathrm{\Gamma }$-convergence of free discontinuity problems for circle-valued maps in the linear regime

We investigate the $\mathrm{\Gamma }$-convergence of Ambrosio-Tortorelli type-functionals for circle valued functions, in the case of volume terms with linear growth. We show the emergence of a non-local $\mathrm{\Gamma }$-limit, which is due to the topological structure of the target space, and discuss compactness of minimal liftings. Our results extend the analysis of a previous work on the quadratic case.