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

Free boundary problems via Sakai's theorem