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Research Papers
Free boundary problems via Sakai's theorem
D. Vardakisa, A. Volbergab a Department of Mathematics, Michigan State University, East Lansing, MI. 48823
b Hausdorff Center for Mathematics, Bonn, Germany
Abstract:
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.
Keywords:
free boundary problems, Schwarz function, real analytic curves, pseudocontinuation, positive harmonic functions, boundary Harnack principle, Nevanlinna domains.
Received: 01.06.2021
Citation:
D. Vardakis, A. Volberg, “Free boundary problems via Sakai's theorem”, Algebra i Analiz, 34:3 (2022), 252–275; St. Petersburg Math. J., 34:3 (2023), 497–514
Linking options:
https://www.mathnet.ru/eng/aa1818 https://www.mathnet.ru/eng/aa/v34/i3/p252
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Abstract page: | 121 | Full-text PDF : | 1 | References: | 24 | First page: | 11 |
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