A monotonicity formula for minimizers of the Mumford-Shah functional in 2d and a sharp lower bound on the energy density

TL;DR

This paper introduces a new monotonicity formula for Mumford-Shah minimizers in 2D using the David-Léger entropy, enabling sharp energy density bounds and distinguishing interface regularity from singularities.

Contribution

It develops a novel monotonicity formula based on the David-Léger entropy, providing a sharp threshold to differentiate interface regularity from singularities in Mumford-Shah minimizers.

Findings

Established a new monotonicity formula for Mumford-Shah minimizers in 2D.

Proved a sharp lower bound on the energy density at nonsmooth points.

Enabled discrimination between smooth interfaces and singularities via entropy gap.

Abstract

We establish a new monotonicity formula for minimizers of the Mumford-Shah functional in planar domains. Our formula follows the spirit of Bucur-Luckhaus, but works with the David-L\'eger entropy instead of the energy. Interestingly, this allows for a sharp truncation threshold. In particular, our monotonicity formula is able to discriminate between points at a $C^1$ interface and any other type of singularity in terms of a finite gap in the entropy. As a corollary, we prove an optimal lower bound on the energy density around any nonsmooth point for minimizers of the Mumford-Shah functional.