Finite number of traces for Mumford-Shah minimizers in dimension 2

TL;DR

The paper resolves De Giorgi's Conjecture 2 in dimension $N=2$ for Mumford–Shah minimizers by proving that the set of limit values of $u$ near the singular set $K$ contains at most three elements. The authors synthesize escaping-path (local-John) arguments from David, Bonnet, and Léger with gradient bounds and Hölder estimates, then employ a blow-up analysis together with the planar classification of global minimizers (David–Léger) to bound the number of components of $\mathbb{R}^2\setminus K$ that can accumulate at a point of $K$. A key gradient bound $|\nabla u|(x) \le 2\operatorname{dist}(x,K)^{-1/2} + 4(\|g\|_{\infty}\operatorname{dist}(x,K))^{1/2}$ and the Hölder control along escaping paths enable a precise comparison of $u$ across local components. This work fills a gap in De Giorgi's program for free-discontinuity problems in the plane and strengthens our understanding of the geometric structure of minimizers in 2D, with techniques that may inform partial progress in higher dimensions.

Abstract

In this short note, we answer a question raised by E. De Giorgi, showing that a Mumford-Shah minimizer in dimension 2 can only admit three maximum limit values as approaching the singular set. This result stems from tools developed in the early 2000's by G. David, A. Bonnet, and J.-C. Léger.

Theorems & Definitions (18)

• Conjecture 2: E. De Giorgi, Conjecture 2 page 56 of dg91
• Theorem 1.1
• Theorem 1.2
• Proposition 2.1: Proposition 16 page 473 of d
• Proposition 2.2
• proof
• Theorem 3.1: David-Léger dleger
• Proposition 3.2
• proof
• Remark 3.3