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

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 \(|u(x)-u(0)|\le C r^{1/2}\) 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 \(\Omega\setminus \overline{S}_u\), different from the one by David–Léger (2002) or Bonnet–David (2001, Lemma 21.3).