On jump minimizing liftings for \mathbb S¹-valued maps and connections with Ambrosio-Tortorelli-type Γ-limits

2025/05/13 by Bellettini, Giovanni, Marziani, Roberta, Scala, Riccardo
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.08731

Abstract

This paper is concerned with the Γ-limits of Ambrosio-Tortorelli-type functionals, for maps u defined on an open bounded set Ω⊂\mathbb Rⁿ and taking values in the unit circle \mathbb S¹⊂\mathbb R². Depending on the domain of the functional, two different Γ-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.

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