The regularity theory for the Mumford-Shah functional on the plane

TL;DR

This work develops a complete, self-contained regularity theory for the Mumford–Shah functional in the plane, culminating in a detailed epsilon-regularity framework for jump sets and triple junctions. It builds a robust blow-up/compactness analysis for minimizers and generalized minimizers, derives powerful variational identities and monotonicity formulae (including the DLMS identity and Léger’s formula), and culminates in rigidity results for cracktips via the Bonnet–David approach. The combination of density bounds, Lipschitz approximations, and decay lemmas yields a structured description of K: regular jump segments, triple junctions, and elementary global minimizers, with global consequences such as gradient integrability and structural decompositions. The results inch toward the Mumford–Shah conjecture in the planar setting by classifying global blow-ups (elementary minimizers and cracktips) and proving sharp regularity near regular and singular points, thereby advancing both theory and potential applications in image segmentation and interface problems. The methods blend variational identities, harmonic extension estimates, and geometric measure techniques to obtain a cohesive, scale-invariant regularity theory.

Abstract

The aim of these notes is to give a complete self-contained account of the current state of the art in the regularity for planar minimizers and critical points of the Mumford-Shah functional.

Figures (17)

• Figure 1: On the left case (a) and on the right case (b) of Lemma \ref{['l:maximum']}. In these examples $U$ is a disk.
• Figure 2: The competitor $(v,J)$ in Lemma \ref{['l:upper-bound']}: we remove the dashed part of the set $K$, we add the circle $\partial B_r (x)$, we set $v=0$ in the shaded disk and we keep $v=u$ outside of it.
• Figure 3: A visual explanation of the admissibility in point (ii) of Definition \ref{['d:competitors']}. The set $K$ is given by the thick lines, while the open set $O$ is the interior of the circle. In $O$ we are allowed to change $K$ to a new set $J$ under the condition that any two points outside $O$ which belong to distinct components of $U\setminus K$ will still belong to distinct connected components of $U\setminus J$. For instance, we cannot remove from $K$ any arc which lies between the regions $A$ and $C$, since such operation would "connect" the points $p$ and $q$. However we are allowed to remove from $K$ an arc which lies between $A$ and $B$. The picture is also a good illustration of Lemma \ref{['l:ciao_componenti']}. As long as $J\cap O$ "separates" the four arcs which are the connected components of $\partial O\setminus K$ in $O$, the pair $(w, J)$ is certainly a topological competitor for $(v,K)$.
• Figure 4: The tangent vector $e(p) = \dot\gamma (t)$ (for an-arc length parametrization) and the normal vector $\nu (p)$. The picture illustrates the convention for the symbols $\pm$ on traces of functions over $\gamma$.
• Figure 5: The arc $\gamma$ is the set $\overline{U}\cap \partial B_r (x)$ and since its two endpoints belong to the same connected component of $K$, both conclusions of Corollary \ref{['c:Bonnet']} apply. On the left the case $U\subset B_r(x)$, on the right the case $U\subset\Omega\setminus \overline{B}_r(x)$.

Theorems & Definitions (198)

• Definition 1.2.1
• Remark 1.2.2
• Definition 1.2.3
• Conjecture 1.3.1: Mumford-Shah conjecture
• Definition 1.3.2
• Theorem 1.3.3
• Corollary 1.3.4
• Theorem 1.4.1
• Definition 1.4.2
• Remark 1.4.3