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

2024/07/11 by Vito Crismale, Crismale, Vito, Lucia De Luca +3
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Advanced Thermodynamics and Statistical Mechanics #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2407.08285

openalex publication_date 2024/07/11 · openalex created_date 2024/07/14 · openalex updated_date 2026/07/28

Abstract

We further investigate the properties of an approach to topological singularities through free discontinuity functionals of Mumford-Shah type proposed in \citeDLSVG. 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 ε|^a, a≥ 1, where ε 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 S¹-valued SBV^p functions, approximated through functions with essentially closed jump set, in the strong BV norm.

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