A second order local minimality criterion for the triple junction singularity of the Mumford-Shah functional

TL;DR

This paper introduces a second variation-based local minimality criterion for triple junction configurations in the Mumford-Shah functional, advancing understanding of their stability and optimality.

Contribution

It proposes a novel second variation approach to establish local minimality criteria for triple junctions in the Mumford-Shah functional.

Findings

First criterion for local minimality of triple junctions

Framework applicable to stability analysis of singularities

Lays groundwork for further analysis of Mumford-Shah minimizers

Abstract

This paper is the first part of an ongoing project aimed at providing a local minimality criterion, based on a second variation approach, for the triple point configurations of the Mumford-Shah functional.