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

Giovanni Bellettini, Roberta Marziani, Riccardo Scala

TL;DR

This work analyzes the Γ-convergence of Ambrosio–Tortorelli-type functionals for $\mathbb{S}^1$-valued maps on bounded domains. It reveals two distinct Γ-limits: a local Mumford–Shah-type functional ${\rm MS}_{\mathbb{S}^1}$ when using a larger admissible domain, and a nonlocal limit ${\rm MS}_{\rm lift}$ that incorporates a minimization over liftings via the quantity $m_2[u]$, reflecting topological obstructions. A central contribution is the development of compactness and lower semicontinuity results for sequences of liftings, yielding existence of jump-minimizing liftings and enabling the nonlocal Γ-limit derivation. The paper also connects the lifting problem to optimal transport and Steiner-type problems, provides detailed proofs of the two Γ-limits, and discusses density/approximation results essential for the analysis, with a vortex map serving as a guiding example of the topological phenomena involved. These insights enhance understanding of phase-field approximations for topologically constrained maps and suggest routes for applications to dislocations, vortices, and related free-discontinuity problems.

Abstract

This paper is concerned with the $Γ$-limits of Ambrosio-Tortorelli-type functionals, for maps $u$ defined on an open bounded set $Ω\subset\mathbb R^n$ and taking values in the unit circle $\mathbb S^1\subset\mathbb R^2$. Depending on the domain of the functional, two different $Γ$-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.

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

TL;DR

This work analyzes the Γ-convergence of Ambrosio–Tortorelli-type functionals for -valued maps on bounded domains. It reveals two distinct Γ-limits: a local Mumford–Shah-type functional when using a larger admissible domain, and a nonlocal limit that incorporates a minimization over liftings via the quantity , reflecting topological obstructions. A central contribution is the development of compactness and lower semicontinuity results for sequences of liftings, yielding existence of jump-minimizing liftings and enabling the nonlocal Γ-limit derivation. The paper also connects the lifting problem to optimal transport and Steiner-type problems, provides detailed proofs of the two Γ-limits, and discusses density/approximation results essential for the analysis, with a vortex map serving as a guiding example of the topological phenomena involved. These insights enhance understanding of phase-field approximations for topologically constrained maps and suggest routes for applications to dislocations, vortices, and related free-discontinuity problems.

Abstract

This paper is concerned with the -limits of Ambrosio-Tortorelli-type functionals, for maps defined on an open bounded set and taking values in the unit circle . Depending on the domain of the functional, two different -limits are possible, one of which is nonlocal, and related to the notion of jump minimizing lifting, i.e., a lifting of a map 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.
Paper Structure (16 sections, 13 theorems, 265 equations, 2 figures)

This paper contains 16 sections, 13 theorems, 265 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. $SBV$ and $GSBV$ functions
  4. $\Gamma$-approximation and compactness for the Mumford-Shah functional
  5. Liftings of $\mathbb{S}^1$-valued maps
  6. On jump minimizing liftings
  7. An example
  8. A connection with optimal transport: an example
  9. Main results on sequences of liftings
  10. Proofs of Theorems \ref{['thm:compactness&lsc']} and \ref{['teo:compactness']}
  11. Proof of Theorem \ref{['thm:compactness&lsc']}
  12. Proof of Theorem \ref{['teo:compactness']}
  13. $\Gamma$-convergence of functionals on $\mathbb{S}^1$- valued maps
  14. Some density and approximation results for ${\mathbb S}^1$-valued maps
  15. Proof of Theorem \ref{['thm1']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\Omega\subseteq\mathbb{R}^n$ be a connected bounded open set with Lipschitz boundary. We have where ${\rm MS}_{{\mathbb S}^1}\colon L^1(\Omega;\mathbb{S}^1)\times L^1(\Omega)\to[0,+\infty]$ is given by

Figures (2)

  • Figure 1: The dotted segment denotes the set $S^f_{\theta_\sigma}$ where $\text{\rm \textlbrackdbl}{\theta_\sigma}\text{\rm \textrbrackdbl}$ varies between 0 and $2\pi$, the continuous segment denotes the set $S^I_{\theta_\sigma}$ where $\text{\rm \textlbrackdbl}{\theta_\sigma}\text{\rm \textrbrackdbl}=2\pi$.
  • Figure 2: The rectangles $T_n$ and $B_n$ in a portion of $\Omega=(0,1)^2$.

Theorems & Definitions (22)

  • Theorem 1.1: $\Gamma$-convergence of ${\widehat{\rm AT}_\varepsilon^{{\mathbb S}^1}}$
  • Theorem 1.2: $\Gamma$-convergence of ${{\rm AT}_\varepsilon^{{\mathbb S}^1}}$
  • Remark 2.1: Equivalent definition of $GSBV$ for $m=1$
  • Remark 2.2: Approximation of a $BV$ function by smooth functions
  • Theorem 2.3: Ambrosio-Tortorelli
  • Remark 2.4
  • Theorem 2.5: Compactness in $SBV$
  • Theorem 2.6: Compactness
  • Theorem 2.7: Davila-Ignat
  • proof
  • ...and 12 more