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

2012/04/24 by John Andersson, Andersson, John, Hayk Mikayelyan +1 · 2 citations
Computer Science · Engineering · Mathematics · #35R35 #49Q20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1204.5328

openalex publication_date 2012/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider in 2D the following special case of the Mumford-Shah functional\nJ(u,
Gamma)=
int_{B1
backslash
Gamma} |
nabla u|² dx +
lambda²\n
frac
pi2
mathcalH¹(
Gamma). It is known that if the minimizer has a\ncrack-tip in the ball B₁ (assume at the origin), then u\≈ \λ Im\n\√(z) at this point. We calculate higher order terms in the asymptotic\nexpansion, where the homogeneity orders of those terms appear to be solutions\nto a certain trigonometric relation.\n

Cited by

• Fine numerical analysis of the crack-tip position for a Mumford-Shah minimizer
• Endpoint regularity of 2d Mumford-Shah minimizers

Related

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• Stationarity of the crack-front for the Mumford-Shah problem in 3D
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• Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets
• Endpoint regularity for 2d Mumford-Shah minimizers: On a theorem of\n Andersson and Mikayelyan