A growth estimate for the planar Mumford--Shah minimizers at a tip point: An alternative proof of David--Léger

TL;DR

The paper proves a local growth bound $|u(x)-u(0)| \le C r^{1/2}$ near a planar tip point for local Mumford–Shah minimizers $u \in SBV(\Omega)$ without assuming the connectedness of the discontinuity set. It introduces a new approach based on a dichotomy between tip and jump points and a John/carrot-domain decomposition, contrasting with prior BD2001 methods and avoiding the need for global connectedness. Central tools include Ahlfors regularity, epsilon-regularity, and a local Morrey-type growth estimate applied on finitely many John domains that cover the vicinity of the tip. The result provides a localized analogue of the global estimate of David–Léger and contributes to the understanding of tip behavior and local regularity in planar Mumford–Shah minimizers, with implications for the broader conjectural regularity picture.

Abstract

Let $Ω\subset \mathbb R^2$ be a bounded domain and $u\in SBV(Ω)$ be a local minimizer of the Mumford--Shah problem in the plane, with $0\in \overline{S}_u$ being a tip point and $B_1\subset Ω$. Then there exist absolute constants $C>0$ and $0<r_0<1$ such that $$|u(x)-u(0)|\le C r^{\frac 1 2} \quad \text{ for any } \ x\in B_r \ \text{ and } \ 0<r<r_0. $$ This estimate is a local version of the original one in \cite[Proposition 10.17]{DL2002}. Our result is based on a dichotomy and the John structure of $Ω\setminus \overline{S}_u$, different from the one by David--Léger \cite{DL2002} or Bonnet--David \cite[Lemma 21.3]{BD2001}.

Figures (1)

• Figure 1: The black lines represents the discontinuity set $K$ of $u$. Corollary \ref{['limit tip']} yields that the component $V$ is uniformly away from $0$.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1.2
• Proposition 1.3
• Corollary 1.4
• Lemma 2.1: F2003, BL2014
• Lemma 2.2: D2006
• proof : Proof of Proposition \ref{['dichotomy']}
• Definition 2.3
• Theorem 2.4
• Lemma 3.1