## Analysis and PDE

•  Dimitris Vardakis, MSU
•  Free boundary problems via Sakai's theorem
•  11/18/2020
•  1:00 PM - 2:00 PM
•  Online (virtual meeting) (Virtual Meeting Link)
•  Dapeng Zhan (zhan@msu.edu)

A Schwarz function on an open domain $\Omega$ is a holomorphic function satisfying $S(\zeta)=\overline{\zeta}$ on the boundary $\Gamma$ of $\Omega$. Sakai in 1991 managed to give a complete characterisation of the boundary of a domain admitting a Schwarz function. In fact, if $\Omega$ is simply connected $\Gamma$ has to be regular real analytic. Here we try to describe $\Gamma$ when the boundary condition is slightly relaxed. In particular, we are interested in three different conditions over a simply connected domain $\Omega$: When $f_1(\zeta)=\overline{\zeta}f_2(\zeta)$ with $f_1,f_2$ holomorphic, when $\mathcal{U}/\mathcal{V}$ equals some real analytic function on $\Gamma$ with $\mathcal{U},\mathcal{V}$ harmonic and when $S(\zeta)=\Phi(\zeta,\overline{\zeta})$ with $\Phi$ a holomorphic function of two variables. It turns out the boundary can be from analytic to just $C^1$, regular except finitely many points, or regular except for a measure zero set, respectively.

## Contact

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Michigan State University
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