Free boundary problems in the spirit of Sakai’s theorem

A Schwarz function on an open domain Ω is a holomorphic function satisfying S ( ζ ) = ζ ¯ on Γ , which is part of the boundary of Ω . Sakai in 1991 gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if Ω is simply connected and Γ = ∂ Ω ∩ D ( ζ 0 , r ) , then Γ has to be regular real analytic (with possible cusps). Sakai’s result has natural applications to 1) quadrature domains, 2) free boundary problem for Δ u = 1 equation. In our scenarios Γ can be, respectively, from real-analytic to just C ∞ , regular except for a harmonic-measure-zero set, or regular except finitely many points.