On the regularity of singular sets of minimizers for the Mumford-Shah energy

Abstract: The Mumford-Shah functional was introduced by Mumford and Shah in 1989 as a variational model for image reconstruction. The most important regularity problem is the famous Mumford-Shah conjecture, which states that (in 2 dimensions) the closure of the jump set can be described as the union of a locally finite collection of injective $C^1$ arcs that can meet only at the endpoints, in which case they have to form triple junctions. If a point is an endpoint of one (and only one) of such arcs, it is called cracktip. In this talk, I plan to survey some older results concerning the regularity of Mumford-Shah minimizers and their singular sets, and discuss more recent developments (the talk is based on joint work with Camillo De Lellis and Matteo Focardi).

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.