Free boundary regularity for the Poisson kernel below the continuous threshold (Q699281)

The main result is the following theorem: Assume that 1) \(\Omega\subset \mathbb{R}^{n+1}\) is a \(\delta\)-Reifenberg chord arc domain for some \(\delta\) small enough; 2) \(\log k_A\in \text{VMO} (\partial\Omega)\), where \(k_A\) is the Poisson kernel. Then \(\Omega\) is a chord arc domain with vanishing constant, i.e., \(n\in \text{VMO} (\partial\Omega)\). The above result proves a previous conjecture of the authors and shows that the ``weak'' regularity of the Poisson kernel fully determines the geometry of the boundary \(\partial\Omega\).