A Schwarz function on an open domain is a holomorphic function satisfying on , which is part of the boundary of . Sakai in 1991 gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if is simply connected and , then has to be regular real analytic (with possible cusps). Sakai’s result has natural applications to 1) quadrature domains, 2) free boundary problem for equation. In our scenarios can be, respectively, from real-analytic to just , 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 satisfaisant sur , qui fait partie de la frontière de . 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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Dimitris Vardakis 1; Alexander Volberg 1, 2
@article{CRMATH_2021__359_10_1233_0, author = {Dimitris Vardakis and Alexander Volberg}, 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}, }
TY - JOUR AU - Dimitris Vardakis AU - Alexander Volberg TI - Free boundary problems in the spirit of Sakai’s theorem JO - Comptes Rendus. Mathématique PY - 2021 SP - 1233 EP - 1238 VL - 359 IS - 10 PB - Académie des sciences, Paris DO - 10.5802/crmath.259 LA - en ID - CRMATH_2021__359_10_1233_0 ER -
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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