Free boundary problems in the spirit of Sakai’s theorem

Vardakis, Dimitris; Volberg, Alexander

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

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 $\Omega$ satisfaisant $S\left(\zeta \right)=\overline{\zeta }$ sur $\Gamma$, qui fait partie de la frontière de $\Omega$. 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 $\Omega$ is a holomorphic function satisfying $S\left(\zeta \right)=\overline{\zeta }$ 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 $\Gamma =\partial \Omega \cap D({\zeta }_0,r)$, then $\Gamma$ has to be regular real analytic (with possible cusps). Sakai’s result has natural applications to 1) quadrature domains, 2) free boundary problem for $\Delta u=1$ equation. In our scenarios $\Gamma$ can be, respectively, from real-analytic to just $C^{\infty }$, regular except for a harmonic-measure-zero set, or regular except finitely many points.

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