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

TL;DR

This paper constructs and analyzes stationary solutions near crack-fronts in the Mumford-Shah problem in 3D, revealing complex geometries and energy behaviors, and provides a detailed 2D proof of crack-tip stationarity.

Contribution

It introduces a family of stationary solutions close to crack-fronts in 3D, including topologically non-minimizing examples, and offers new energy estimates and numerical insights.

Findings

Stationary solutions near crack-fronts in 3D are topologically non-minimizing.

Energy estimates for Mumford-Shah minimizers are established.

Numerical illustrations suggest these solutions are unlikely to be minimizers.

Abstract

In this paper we exhibit a family of stationary solutions of the Mumford-Shah functional 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 \cite{b}. 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.