Stationarity of the crack-front for the Mumford–Shah problem in 3D

Antoine Lemenant, Hayk Mikayelyan

Research output: Journal PublicationArticlepeer-review


In this paper we exhibit a family of stationary solutions of the Mumford–Shah functional in R3, arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet [4]. We also give a local version in a finite cylinder and prove an energy estimate for minimizers. Numerical illustrations indicate the stationary solutions are unlikely minimizers and show how the dependence on axial variable impacts the geometry of the discontinuity set. A self-contained proof of the stationarity of the cracktip function for the Mumford–Shah problem in 2D is presented.

Original languageEnglish
Pages (from-to)1555-1569
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Issue number2
Publication statusPublished - 15 Jun 2018


  • Calculus of variations
  • Cracktip function
  • Geometric measure theory
  • Mumford–Shah functional
  • Stationary solutions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics


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