Free boundary problems via Sakai's theorem

A Schwarz function on an open domain $\Omega$ is a holomorphic function satisfying $S(\zeta)=\overline{\zeta}$ on $\Gamma$, which is part of the boundary of $\Omega$. Sakai in 1991 gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if $\Omega$ is simply connected and $\Gamma=\partial \Omega\cap D(\zeta,r)$, then $\Gamma$ has to be regular real analytic. This paper is an attempt to describe $\Gamma$ when the boundary condition is slightly relaxed. In particular, three different scenarios over a simply connected domain $\Omega$ are treated: when $f_1(\zeta)=\overline{\zeta}f_2(\zeta)$ on $\Gamma$ with $f_1,f_2$ holomorphic and continuous up to the boundary, when $\mathcal{U}/\mathcal{V}$ equals certain real analytic function on $\Gamma$ with $\mathcal{U},\mathcal{V}$ positive and harmonic on $\Omega$ and vanishing on $\Gamma$, and when $S(\zeta)=\Phi(\zeta,\overline{\zeta})$ on $\Gamma$ with $\Phi$ a holomorphic function of two variables. It turns out that the boundary piece $\Gamma$ can be, respectively, anything from $C^\infty$ to merely $C^1$, regular except finitely many points, or regular except for a measure zero set.