Endpoint regularity of $2$d Mumford-Shah minimizers

TL;DR

This paper establishes an epsilon-regularity result for 2D Mumford-Shah minimizers, showing that near certain boundary points, the jump set is a smooth, connected arc terminating inside the domain.

Contribution

It proves a new epsilon-regularity theorem at the endpoint of connected arcs for 2D Mumford-Shah minimizers, detailing the regularity and structure of the jump set.

Findings

Jump set is a connected $C^{1,eta}$ arc near the endpoint.

If the jump set is close to a radius in Hausdorff distance, regularity is guaranteed.

The jump set terminates at an interior point with smooth regularity.

Abstract

We prove an $\varepsilon$-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 $y_0$ and it is $C^{1,\alpha}$ up to $y_0$.