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

TL;DR

This paper studies the $ ext{Γ}$-convergence of Ambrosio-Tortorelli functionals for circle-valued functions with linear growth, revealing a non-local limit influenced by topology and extending quadratic case results.

Contribution

It introduces the analysis of $ ext{Γ}$-convergence for circle-valued maps with linear growth, highlighting non-local effects due to topology, extending prior quadratic case studies.

Findings

Non-local $ ext{Γ}$-limit due to topology

Extension of quadratic case analysis

Discussion on compactness of minimal liftings

Abstract

We investigate the $\Gamma$-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 $\Gamma$-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.