Preprint

Inserted: 11 jul 2024
Last Updated: 11 jul 2024

Year: 2024

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.

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