Endpoint regularity of 2d Mumford-Shah minimizers

2015/02/08 by Camillo De Lellis, De Lellis, Camillo, Matteo Focardi +1
Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Topology Optimization in Engineering

paper · pdf · doi:10.48550/arxiv.1502.02299

openalex publication_date 2015/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove an ε-regularity theorem at the endpoint of connected arcs for 2-dimensional Mumford-Shah minimizers. In particular we show that, if in a given ball B_r (x) the jump set of a given Mumford-Shah minimizer is sufficiently close, in the Hausdorff distance, to a radius of B_r (x), then in a smaller ball the jump set is a connected arc which terminates at some interior point y₀ and it is C^1,α up to y₀.

Citations

• Optimal approximations by piecewise smooth functions and associated variational problems
• 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

Related

• Endpoint regularity for 2d Mumford-Shah minimizers: On a theorem of\n Andersson and Mikayelyan
• The regularity theory for the Mumford-Shah functional on the plane
• Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture
• Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets
• Regularity up to the Crack-Tip for the Mumford-Shah problem