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
Vardakis, Dimitris 1; Volberg, Alexander 1, 2

1 Department of Mathematics, Michigan State University, East Lansing MI. 48823, USA
2 Hausdorff Center for Mathematics, Universität Bonn, Bonn, Germany
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Vardakis, Dimitris; Volberg, Alexander. 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. http://www.numdam.org/articles/10.5802/crmath.259/

[1] Baranov, Anton D.; Belov, Yurii S.; Borichev, Alexander; Fedorovskiy, Konstantin Yu. Univalent functions in model spaces: revisited (2017) (https://arxiv.org/abs/1705.05930v2)

[2] Baranov, Anton D.; Fedorovskiy, Konstantin Yu. Boundary regularity of Nevanlinna domains and univalent functions in model subspaces, Sb. Math., Volume 202 (2011) no. 12, pp. 1723-1740 | DOI | MR | Zbl

[3] Belov, Yurii S.; Borichev, Alexander; Fedorovskiy, Konstantin Yu. Nevanlinna domains with large boundaries, J. Funct. Anal., Volume 277 (2019) no. 8, pp. 2617-2643 | DOI | MR | Zbl

[4] Belov, Yurii S.; Fedorovskiy, Konstantin Yu. Model spaces containing univalent functions, Russ. Math. Surv., Volume 73 (2018) no. 1, pp. 172-174 | DOI | MR | Zbl

[5] Carmona, Joan J.; Paramonov, Peter V.; Fedorovskiy, Konstantin Yu. On uniform approximation by polyanalytic polynomials and the Dirichlet problem for bianalytic functions, Sb. Math., Volume 193 (2002) no. 10, pp. 1469-1492 | DOI | MR | Zbl

[6] Cima, Joseph A.; Ross, William T. The Backward Shift on the Hardy Space, Mathematical Surveys and Monographs, 79, American Mathematical Society, 2000 | DOI

[7] Dyakonov, Konstantin; Khavinson, Dmitry Smooth functions in star-invariant subspaces, Recent advances in operator-related function theory (Contemporary Mathematics), Volume 393, American Mathematical Society, 2006 | DOI | MR | Zbl

[8] Fedorovskiy, Konstantin Yu. On some properties and examples of Nevanlinna domains, Proc. Steklov Inst. Math., Volume 253 (2006), pp. 186-194 | DOI | MR

[9] Jerison, David S.; Kenig, Carlos E. Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math., Volume 46 (1982), pp. 80-147 | DOI | Zbl

[10] Mazalov, Maksim Ya. An example of a nonconstant bianalytic function vanishing everywhere on a nowhere analytic boundary, Math. Notes, Volume 62 (1997) no. 4, pp. 524-526 | DOI | MR | Zbl

[11] Mazalov, Maksim Ya. Example of a non-rectifiable Nevanlinna contour, St. Petersbg. Math. J., Volume 27 (2016), pp. 625-630 | DOI

[12] Sakai, Makoto Regularity of a boundary having a Schwarz function, Acta Math., Volume 166 (1991) no. 3-4, pp. 263-297 | DOI | MR | Zbl

[13] Volberg, Alexander On the dimension of harmonic measure of Cantor repellers, Mich. Math. J., Volume 40 (1993) no. 2, pp. 239-258 | DOI | MR | Zbl

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