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

2017/10/24 by Antoine Lemenant, Lemenant, Antoine, Hayk Mikayelyan +1
Computer Science · Engineering · #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #Elasticity and Material Modeling #FOS: Mathematics #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1710.08711

openalex publication_date 2017/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \citeb. 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 crack-tip function for the Mumford-Shah problem in 2D is presented.

Citations

• Optimal approximations by piecewise smooth functions and associated variational problems
• Regularity up to the Crack-Tip for the Mumford-Shah problem

Related

• Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer
• Regularity up to the Crack-Tip for the Mumford-Shah problem
• Endpoint regularity for 2d Mumford-Shah minimizers: On a theorem of\n Andersson and Mikayelyan
• The asymptotics of the curvature of the free discontinuity set near the\n cracktip for the minimizers of the Mumford-Shah functional in the plain
• Local calibrations for minimizers of the Mumford-Shah functional with a triple junction