Comptes Rendus
Complex analysis and geometry, Functional analysis
Free boundary problems in the spirit of Sakai’s theorem
Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1233-1238.

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=ΩD(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 Δu=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.

Dans le présent article, nous considérons la pléthore de résultats dans l’esprit de théorème de Sakai concernant les fonctions de Schwarz, c’est-à-dire les fonctions holomorphes dans un domaine ouvert Ω satisfaisant S(ζ)=ζ ¯ sur Γ, qui fait partie de la frontière de Ω. Sakai en 1991 a donné une caractérisation complète de la frontière d’un domaine admettant une fonction de Schwarz. Les résultats ci-dessous concernent trois scénarios de généralisation du résultat de Sakai, motivés plutôt par l’application au problème de dynamique complexe étudié dans [13]. À la fin de cette note, nous mentionnons quelques problèmes encore ouverts.

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DOI: 10.5802/crmath.259

Dimitris Vardakis 1; Alexander Volberg 1, 2

1 Department of Mathematics, Michigan State University, East Lansing MI. 48823, USA
2 Hausdorff Center for Mathematics, Universität Bonn, Bonn, Germany
License: CC-BY 4.0
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Dimitris Vardakis; Alexander Volberg. Free boundary problems in the spirit of Sakai’s theorem. Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1233-1238. doi : 10.5802/crmath.259. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.259/

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