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

Giovanni Bellettini, Roberta Marziani, Riccardo Scala

TL;DR

This work analyzes the Γ-convergence of Ambrosio–Tortorelli type functionals for circle-valued maps in the linear-growth regime, revealing a non-local Γ-limit tied to the topology of ${\rm S}^1$ and the need to optimize over liftings. By developing compactness results for liftings in the $BV/GBV$ framework and introducing a minimization problem $m_g[u]$ for liftings, the authors obtain two distinct Γ-limits depending on the domain: a local limit ${\rm F}^{\rm S^1}$ on a larger domain, and a lift-based nonlocal limit ${\rm F}_{\rm lift}$ on a restricted domain, with $m_g[u]$ attaining a minimum. The analysis intertwines lifting theory, measure-theoretic localization, and connections to optimal transport and area relaxation in ${\rm S}^1$, offering a robust foundation for linear-growth variational problems with topological constraints. The results extend prior quadratic-growth findings and pave the way for broader applications to Mumford–Shah-type energies for circle-valued maps and related dislocation models.

Abstract

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.

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

TL;DR

This work analyzes the Γ-convergence of Ambrosio–Tortorelli type functionals for circle-valued maps in the linear-growth regime, revealing a non-local Γ-limit tied to the topology of and the need to optimize over liftings. By developing compactness results for liftings in the framework and introducing a minimization problem for liftings, the authors obtain two distinct Γ-limits depending on the domain: a local limit on a larger domain, and a lift-based nonlocal limit on a restricted domain, with attaining a minimum. The analysis intertwines lifting theory, measure-theoretic localization, and connections to optimal transport and area relaxation in , offering a robust foundation for linear-growth variational problems with topological constraints. The results extend prior quadratic-growth findings and pave the way for broader applications to Mumford–Shah-type energies for circle-valued maps and related dislocation models.

Abstract

We investigate the -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 -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.
Paper Structure (12 sections, 11 theorems, 158 equations)

This paper contains 12 sections, 11 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. $BV$ and $GBV$ functions
  4. Approximation and compactness for the vector Mumford-Shah functional
  5. Liftings of $\mathbb{S}^1$-valued maps
  6. Setting of the problem and main results
  7. On $g$-minimizing liftings
  8. Proofs of Theorems \ref{['thm:compactness']} and \ref{['teo:compactness']}
  9. Proof of Theorem \ref{['thm:compactness']}
  10. Proof of Theorem \ref{['teo:compactness']}
  11. $\Gamma$-convergence of functionals on $\mathbb{S}^1$- valued maps
  12. Proof of Theorem \ref{['thm2']}

Key Result

Lemma 2.1

Let $\Omega\subset \mathbb{R}^n$ be open bounded with Lipschitz boundary and let $\mathcal{A}(\Omega)$ be the family of the open subsets of $\Omega$. Let $\mu\colon\mathcal{A}(\Omega)\to[0,+\infty)$ be a superadditive function on disjoint open sets, $\lambda$ be a positive measure on $\Omega$ and, f Then

Theorems & Definitions (18)

  • Lemma 2.1: Localisation
  • Remark 2.2: Equivalent definition of $GBV$ for $m=1$
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5: Davila-Ignat
  • proof
  • Theorem 3.1: $\Gamma$-convergence of $\widehat{{{\rm F}_\varepsilon^{{\mathbb S}^1}}}$
  • Theorem 3.2: $\Gamma$-convergence of ${{\rm F}_\varepsilon^{{\mathbb S}^1}}$
  • Definition 4.1: Local convergence modulo $2\pi$
  • Theorem 4.2: Compactness
  • ...and 8 more