On the Compactness of One Class of Solutions for the Dirichlet Problem
DOI:
https://doi.org/10.15407/mag20.01.082Ключові слова:
рiвняння Бельтрамi, простi кiнцi, плоскi вiдображення зi скiнченним та обмеженим спотвореннямАнотація
Ми розглядаємо задачу Дiрiхле для рiвняння Бельтрамi у довiльнiй обмеженiй однозв’язнiй областi комплексної площини $\mathbb {C}$. Само, вивчається клас усiх регулярних розв’язкiв цiєї задачi з умовами нормування i теоретико-множинними обмеженнями на їх комплексну характеристику. Доведена компактнiсть цього класу в термiнах простих кiнцiв за наявностi довiльної неперервної функцiї в умовi Дiрiхле.
Mathematical Subject Classification 2020: 30C65, 35J70
Посилання
L.V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, Toronto, 1966.
O.P. Dovhopiatyi and E.A. Sevost'yanov, On the compactness of classes of the solutions of the Dirichlet problem, J. Math. Sci. 259 (2021), No. 1, 23--36. https://doi.org/10.1007/s10958-021-05598-7
Yu.P. Dybov, Compactness of classes of solutions of the Dirichlet problem for the Beltrami equations, Proc. Inst. Appl. Math. and Mech. of NAS of Ukraine 19 (2009), 81--89 (Russian).
V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach. Developments in Mathematics, 26, Springer, New York, 2012. https://doi.org/10.1007/978-1-4614-3191-6
A. Hurwitz and R. Courant, The Function Theory, Nauka, Moscow, 1968 (Russian).
A.A. Ignat'ev and V.I. Ryazanov, Finite mean oscillation in mapping theory, Ukr. Mat. Visn. 2 (2005), No. 3, 395--417, 443 (Russian) Engl. transl.: Ukr. Math. Bull. 2 (2005), No. 3, 403--424.
N.S. Ilkevych and E.A. Sevost'yanov, S.A. Skvortsov, On the global behavior of inverse mappings in terms of prime ends, Ann. Acad. Sci. Fenn. Math. 46 (2021), No. 2, 371--388. https://doi.org/10.5186/aasfm.2021.4630
D.P. Ilyutko and E.A. Sevost'yanov, On prime ends on Riemannian manifolds, J. Math. Sci. 241 (2019), No. 1, 47--63. https://doi.org/10.1007/s10958-019-04406-7
D.A. Kovtonyuk, I.V. Petkov, V.I. Ryazanov, and R.R. Salimov, The boundary behavior and the Dirichlet problem for the Beltrami equations, St. Petersburg Math. J. 25 (2014), No. 4, 587--603. https://doi.org/10.1090/S1061-0022-2014-01308-8
D.A. Kovtonyuk and V.I. Ryazanov, On the theory of prime ends for space mappings, Ukrainian Math. J. 67 (2015), No. 4, 528--541. https://doi.org/10.1007/s11253-015-1098-9
D.A. Kovtonyuk and V.I. Ryazanov, Prime ends and Orlicz-Sobolev classes, St. Petersburg Math. J. 27 (2016), No. 5, 765--788. https://doi.org/10.1090/spmj/1416
T. Lomako, On the theory of convergence and compactness for Beltrami equations, Ukrain. Math. J. 63 (2011), No. 3, 393--402. https://doi.org/10.1007/s11253-011-0510-3
T. Lomako, On the theory of convergence and compactness for Beltrami equations with constraints of set-theoretic type, Ukrain. Math. J. 63 (2012), No. 9, 1400--1414. https://doi.org/10.1007/s11253-012-0587-3
T. Lomako, V. Gutlyanskii, and V. Ryazanov, To the theory of variational method for Beltrami equations, J. Math. Sci. 182 (2012), No. 1, 37--54. https://doi.org/10.1007/s10958-012-0727-1
T. Lomako, R. Salimov, and E. Sevost'yanov, On equicontinuity of solutions to the Beltrami equations, Ann. Univ. Bucharest (Math. Series) LIX (2010), No. 2, 261--271.
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in modern mapping theory, Springer Science + Business Media, LLC, New York, 2009.
R. Näkki, Extension of Loewner's capacity theorem, Trans. Amer. Math. Soc. 180 (1973), 229--236. https://doi.org/10.1090/S0002-9947-1973-0328062-9
R. Näkki, Prime ends and quasiconformal mappings, J. Anal. Math. 35 (1979), 13--40. https://doi.org/10.1007/BF02791061
R. Näkki and B. Palka, Uniform equicontinuity of quasiconformal mappings, Proc. Amer. Math. Soc. 37 (1973), No. 2, 427--433. https://doi.org/10.2307/2039456
V. Ryazanov, R. Salimov, and E. Sevost'yanov, On Convergence Analysis of Space Homeomorphisms, Siberian Adv. Math. 23 (2013), No. 4, 263--293. https://doi.org/10.3103/S1055134413040044
V. Ryazanov, On the accuracy of some convergence theorems, Dokl. Akad. Nauk SSSR 315 (1990), no. 2, 317--319 (Russian) Engl. transl.: Soviet Math. Dokl. 42 (1991), No. 3, 793--795.
E.A. Sevost'yanov, Analog of the Montel theorem for mappings of the Sobolev class with finite distortion, Ukrain. Math. J. 67 (2015), No. 6, 938--947. https://doi.org/10.1007/s11253-015-1124-y
E.A. Sevost'yanov and O.P. Dovhopiatyi, On compact classes of solutions of the Dirichlet problem with integral restrictions, Complex Var. Elliptic Equ. 68 (2023), 1182--1203. https://doi.org/10.1080/17476933.2022.2040020
E.A. Sevost'yanov and S.A. Skvortsov, Equicontinuity of Families of Mappings with One Normalization Condition, Math. Notes 109 (2021), No. 4, 614--622. https://doi.org/10.1134/S0001434621030317
E.A Sevost'yanov, S.A. Skvortsov, and O.P. Dovhopiatyi, On non-homeomorphic mappings with inverse Poletsky inequality, J. Math. Sci. 252 (2021), no. 4, 541--557. https://doi.org/10.1007/s10958-020-05179-0
S. Stoilow, Principes Topologiques de la Théorie des Fonctions Analytiques, Gauthier-Villars, Paris, 1956.
J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin etc., 1971.
