Approximation of topological singularities through free discontinuity functionals: the critical and super-critical regimes

TL;DR

This paper explores the approximation of topological singularities using free discontinuity functionals, establishing variational equivalences in critical regimes and removing restrictive assumptions through new density results.

Contribution

It proves the variational equivalence between free discontinuity functionals, Ginzburg-Landau, and Core-Radius energies, and introduces a new density result for SBV functions.

Findings

Established variational equivalence in critical regimes

Removed restrictive assumptions on jump sets

Developed a new density approximation for SBV functions

Abstract

We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in \cite{DLSVG}. We prove the variational equivalence between such energies, Ginzburg-Landau, and Core-Radius for anti-plane screw dislocations energies in dimension two, in the relevant energetic regimes $|\log \varepsilon|^a$, $a\geq 1$, where $\varepsilon$ denotes the linear size of the process zone near the defects. Further, we remove the \emph{a priori} restrictive assumptions that the approximating order parameters have compact jump set. This is obtained by proving a new density result for $\mathbb S^1$-valued $SBV^p$ functions, approximated through functions with essentially closed jump set, in the strong $BV$ norm.