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<p>We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in De Luca et al. (Indiana Univ Math J 73:723–779, 2024). 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(|\log \varepsilon |^a\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(a\ge 1\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varepsilon\)</EquationSource> </InlineEquation> denotes the linear size of the process zone near the defects. Further, we remove the <i>a priori</i> restrictive assumptions that the approximating order parameters have compact jump set. This is obtained by proving a new density result for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {S}^1\)</EquationSource> </InlineEquation>-valued <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(SBV^p\)</EquationSource> </InlineEquation> functions, approximated through functions with essentially closed jump set, in the strong <i>BV</i> norm.</p>

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

  • 期刊论文

  • 发表日期：

    2025年12月4日

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  • V. Crismale,
  • L. De Luca,
  • R. Scala

摘要

We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in De Luca et al. (Indiana Univ Math J 73:723–779, 2024). 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 \(|\log \varepsilon |^a\) , \(a\ge 1\) , where \(\varepsilon\) denotes the linear size of the process zone near the defects. Further, we remove the 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.