Free boundary problems in the spirit of Sakai’s theorem

Dimitris Vardakis; Alexander Volberg

doi:10.5802/crmath.259

Analyse et géométrie complexes, Analyse fonctionnelle

Free boundary problems in the spirit of Sakai’s theorem

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

Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1233-1238.

Résumé
Abstract

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 $S (ζ) = \bar{ζ}$ 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.

A Schwarz function on an open domain $Ω$ is a holomorphic function satisfying $S (ζ) = \bar{ζ}$ 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 $Γ = \partial Ω \cap D (ζ_{0}, r)$ , 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 $Δ u = 1$ equation. In our scenarios $Γ$ can be, respectively, from real-analytic to just $C^{\infty}$ , regular except for a harmonic-measure-zero set, or regular except finitely many points.

Reçu le : 2021-06-07
Révisé le : 2021-07-25
Accepté le : 2021-08-15
Publié le : 2022-01-04

DOI : 10.5802/crmath.259

Affiliations des auteurs :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Free boundary problems in the spirit of {Sakai{\textquoteright}s} theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1233--1238},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {10},
     year = {2021},
     doi = {10.5802/crmath.259},
     language = {en},
}

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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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