On jump minimizing liftings for $\mathbb S^1$-valued maps and connections with Ambrosio-Tortorelli-type $\Gamma$-limits

TL;DR

This paper investigates the $ ext{Gamma}$-limits of Ambrosio-Tortorelli-type functionals for maps into the circle, revealing two possible limits and introducing jump minimizing liftings with new compactness results.

Contribution

It introduces the concept of jump minimizing liftings for $ ext{S}^1$-valued maps and establishes compactness results to prove their existence.

Findings

Two different $ ext{Gamma}$-limits identified for the functionals.

Existence of jump minimizing liftings proved via new compactness results.

Connection established between nonlocal $ ext{Gamma}$-limits and liftings.

Abstract

This paper is concerned with the $\Gamma$-limits of Ambrosio-Tortorelli-type functionals, for maps $u$ defined on an open bounded set $\Omega\subset\mathbb R^n$ and taking values in the unit circle $\mathbb S^1\subset\mathbb R^2$. Depending on the domain of the functional, two different $\Gamma$-limits are possible, one of which is nonlocal, and related to the notion of jump minimizing lifting, i.e., a lifting of a map $u$ whose measure of the jump set is minimal. The latter requires ad hoc compactness results for sequences of liftings which, besides being interesting by themselves, also allow to deduce existence of a jump minimizing lifting.