On multivalent solutions of the Riemann–Hilbert problem in multiply connected domains

TitleOn multivalent solutions of the Riemann–Hilbert problem in multiply connected domains
Publication TypeJournal Article
Year of Publication2016
AuthorsYefimushkin, AS
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2016.08.007
Issue8
SectionMathematics
Pagination7-11
Date Published8/2016
LanguageRussian
Abstract

For the Beltrami equations in the domains bounded by a finite collection of smooth Jordan curves, the existence of multivalent solutions of the Riemann–Hilbert problem with coefficients of sigma–finite variation and with boundary data, which are measurable with respect to the logarithmic capacity, is proved. It is shown that these spaces of solutions have the infinite dimension.

Keywordsanalytic functions, Beltrami equation, logarithmic capacity, multivalent solutions, Riemann–Hilbert problem
References: 
  1. Efimushkin A. S., Ryazanov V. I. Ukr. mat. vestnik, 2015, 12, No 2: 190–209 (in Russian).
  2. Vekua I. N. Obobschennyie analiticheskie funktsii, Moscow: Fizmatgiz, 1959 (in Russian).
  3. Nosiro K. Predelnyie mnozhestva, Moscow: Izd-vo Inostr. lit., 1963 (in Russian).
  4. Collingwood E. F., Lohwater A. J. The theory of cluster sets, Cambridge Tracts in Math. and Math. Physics, No 56: Cambridge: Cambridge Univ. Press, 1966. https://doi.org/10.1017/CBO9780511566134
  5. Kusis P. Vvedenie v teoriyu prostranstv Hp, Moscow: Mir, 1984 (in Russian).
  6. Ryazanov V. I. On multivalent solutions of Riemann–Hilbert problem, arXiv:1506.08735v1 [math. CV] 29 Jun. 2015.
  7. Goluzin G. M. Geometricheskaya teoriya funktsiy kompleksnogo peremennogo, Moscow: Nauka, 1966 (in Russian).
  8. Alfors L. Lektsii po kvazikonformnyim otobrazheniyam, Moscow: Mir, 1969 (in Russian).
  9. Lehto O., Virtanen K. J. Quasiconformal mappings in the plane, Berlin, Heidelberg: Springer, 1973. https://doi.org/10.1007/978-3-642-65513-5
  10. Nevanlinna R. Odnoznachnyie analiticheskie funktsii, Moscow: OGIZ, 1941 (in Russian).
  11. Agard S. B., Gehring F. W. Proc. London Math. Soc. (3), 1965, 14A: 1–21.
  12. Yefimushkin A. S., Ryazanov V. I. Reports of the National Academy of Sciences of Ukraine, 2016, 2: 13–16 (in Russian). https://doi.org/10.15407/dopovidi2016.02.013