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

Let $\Omega\subset \mathbb R^2$ be a bounded domain and $u\in SBV(\Omega)$ be a local minimizer of the Mumford--Shah problem in the plane, with $0\in \overline{S}_u$ being a tip point and $B_1\subset \Omega$. 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 $\Omega\setminus \overline{S}_u$, different from the one by David--L\'eger \cite{DL2002} or Bonnet--David \cite[Lemma 21.3]{BD2001}.