$Γ$-convergence of free discontinuity problems for circle-valued maps in the linear regime

We investigate the $Γ$-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 $Γ$-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.

💡 Research Summary

The paper studies Γ‑convergence of Ambrosio‑Tortorelli type functionals for maps taking values in the unit circle S¹, focusing on the case where the bulk energy density has linear growth rather than the usual quadratic growth. The authors consider a pair (u,v) with u:Ω→S¹ and a phase‑field variable v:Ω→