Fine regularity results for Mumford-Shah minimizers: porosity, higher integrability and the Mumford-Shah conjecture

TL;DR

This paper reviews classical and recent results on the regularity of Mumford-Shah minimizers, exploring their porosity, higher integrability, and implications for the Mumford-Shah conjecture, highlighting connections between geometric and energetic properties.

Contribution

It synthesizes known results and recent insights, establishing explicit links between porosity, higher integrability, and the Mumford-Shah conjecture, advancing understanding of the regularity theory.

Findings

Higher integrability relates to the Hausdorff dimension of the singular set.

Porosity of the jump set is connected to regularity properties.

Energetic characterizations provide new perspectives on the conjecture.

Abstract

We review some classical results and more recent insights about the regularity theory for local minimizers of the Mumford and Shah energy and their connections with the Mumford and Shah conjecture. We discuss in details the links among the latter, the porosity of the jump set and the higher integrability of the approximate gradient. In particular, higher integrability turns out to be related with an explicit estimate on the Hausdorff dimension of the singular set and an energetic characterization of the conjecture itself.