Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access May 20, 2015

Infinite dimension of solutions of the Dirichlet problem

  • Vladimir Ryazanov
From the journal Open Mathematics


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.


[1] Dovgoshey O., Martio O., Ryazanov V., Vuorinen M. The Cantor function, Expo. Math., 2006, 24, 1-37 10.1016/j.exmath.2005.05.002Search in Google Scholar

[2] Efimushkin A., Ryazanov V. On the Riemann-Hilbert problem for the Beltrami equations, Contemporary Mathematics (to appear), see also preprint [math.CV] 30 July 2014, 1-25 Search in Google Scholar

[3] Garnett J.B., Marshall D.E. Harmonic Measure, Cambridge Univ. Press, Cambridge, 2005 10.1017/CBO9780511546617Search in Google Scholar

[4] Gehring F.W., On the Dirichlet problem, Michigan Math. J., 1955-1956, 3, 201 10.1307/mmj/1028990037Search in Google Scholar

[5] Goluzin G.M., Geometric theory of functions of a complex variable, Transl. of Math. Monographs, 26, American Mathematical Society, Providence, R.I., 1969 10.1090/mmono/026Search in Google Scholar

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

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

Received: 2014-10-29
Accepted: 2014-5-10
Published Online: 2015-5-20

©2015 Vladimir Ryazanov

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 3.12.2023 from
Scroll to top button