Regularity of the singular set for Mumford-Shah minimizers in R^3 near a minimal cone

TL;DR

This paper proves that near certain minimal cones, the singular set of Mumford-Shah minimizers in R^3 is smoothly regular, showing stability and regularity properties in a neighborhood of these cones.

Contribution

It establishes the regularity of the singular set near minimal cones of types P, Y, and T for Mumford-Shah minimizers in three dimensions.

Findings

K is C^1,alpha equivalent to the minimal cone in a smaller ball

Singular set regularity holds near minimal cones of types P, Y, and T

Results apply under Hausdorff distance closeness conditions

Abstract

We show that if (u;K) is a minimizer of the Mumford-Shah functional in an open set of R^3, and if x, K and r > 0 are such that K is close enough to a minimal cone of type P (a plane), Y (three half planes meeting with 120 degrees angles) or T (cone over a regular tetrahedron centered at the origin) in terms of Hausdorff distance in B(x; r), then K is C^1,alpha equivalent to the minimal cone in B(x; cr) where c < 1 is an universal constant.