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

Abstract

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded domain and $u\in SBV(\mathrm{\Omega })$ be a local minimizer of the Mumford–Shah problem in the plane, with $0\in {\bar{S}}_u$ being a tip point and $B_1\subset \mathrm{\Omega }$. Then there exist absolute constants $C>0$ and $0<r_0<1$ such that
$|u(x)-u(0)|\le C{r}^{1/2}$ for any $x\in B_r$ and $0<r<r_0$.

This estimate is a local version of the original one in David–Léger (2002, Proposition 10.17). Our result is based on a dichotomy and the John structure of $\mathrm{\Omega }\setminus {\bar{S}}_u$, different from the one by David–Léger (2002) or Bonnet–David (2001, Lemma 21.3).