Calculus of Variations and Geometric Measure Theory

A. Lemenant - H. Mikayelyan

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

created by lemenant on 24 Oct 2017



Inserted: 24 oct 2017
Last Updated: 24 oct 2017

Year: 2017


In this paper we exhibit a family of stationary solutions of the Mumford-Shah problem in $\mathbb{R}^3$, arbitrary close to a crack-front. Unlike other examples, known in the literature, those are topologically non-minimizing in the sense of Bonnet.

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.