Overview
The study examines the $\Gamma$-convergence of Ambrosio-Tortorelli type-functionals for circle-valued functions in the regime where volume terms exhibit linear growth.
Research Context
The analysis extends previous mathematical research originally conducted for the quadratic case, broadening the investigation of free discontinuity problems to include volume terms with linear growth.
Approach
The research evaluates the variational limits of Ambrosio-Tortorelli approximations applied to mappings taking values in the circle. As part of this variational analysis, the compactness of minimal liftings is examined under the linear growth assumption.
Findings
The investigation establishes the emergence of a non-local $\Gamma$-limit, which is driven by the topological structure of the target space. Additionally, the work provides results concerning the compactness of minimal liftings in this linear setting.
Potential Applications
The authors demonstrate applications of these convergence and compactness results to the area functional.