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

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 $$\mathrm{\mid }u(x)-u(0)\mathrm{\mid }\le C{r}^{\frac{1}{2}}\text{ for any }x\in B_r\text{ and }0<r<r_0\mathrm{.}$$ 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 $\mathrm{\Omega }\setminus {\bar{S}}_u$, different from the one by David–Léger \cite{DL2002} or Bonnet–David \cite[Lemma 21.3]{BD2001}.