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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2015: 0.512

SCImago Journal Rank (SJR) 2015: 0.521
Source Normalized Impact per Paper (SNIP) 2015: 1.233
Impact per Publication (IPP) 2015: 0.546

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Open Access
Online
ISSN
2391-5455
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Volume 13, Issue 1 (May 2015)

Issues

Infinite dimension of solutions of the Dirichlet problem

Vladimir Ryazanov
  • Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 Roze Luxemburg Str., 83114, Donetsk, Ukraine
Published Online: 2015-05-20 | DOI: https://doi.org/10.1515/math-2015-0034

Abstract

It is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.

Keywords: Dirichlet problem; Harmonic functions; Laplace equation; Riemann-Hilbert problem

MSC: 31A05, 31A20, 31A25, 31B25, 35Q15, 30E25, 31C05, 34M50, 35F45

References

  • [1] Dovgoshey O., Martio O., Ryazanov V., Vuorinen M. The Cantor function, Expo. Math., 2006, 24, 1-37

  • [2] Efimushkin A., Ryazanov V. On the Riemann-Hilbert problem for the Beltrami equations, Contemporary Mathematics (to appear), see also preprint http://arxiv.org/abs/1402.1111v3 [math.CV] 30 July 2014, 1-25

  • [3] Garnett J.B., Marshall D.E. Harmonic Measure, Cambridge Univ. Press, Cambridge, 2005

  • [4] Gehring F.W., On the Dirichlet problem, Michigan Math. J., 1955-1956, 3, 201

  • [5] Goluzin G.M., Geometric theory of functions of a complex variable, Transl. of Math. Monographs, 26, American Mathematical Society, Providence, R.I., 1969

  • [6] Koosis P., Introduction to Hp spaces, 2nd ed., Cambridge Tracts in Mathematics, 115, Cambridge Univ. Press, Cambridge, 1998

  • [7] Ryazanov V., On the Riemann-Hilbert Problem without Index, Ann. Univ. Bucharest, Ser. Math. 2014, 5, 169-178

About the article

Received: 2014-10-29

Accepted: 2014-05-10

Published Online: 2015-05-20


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0034. Export Citation

©2015 Vladimir Ryazanov. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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[1]
Vladimir Gutlyanskii, Vladimir Ryazanov, and Artem Yefimushkin
Journal of Mathematical Sciences, 2016

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