## 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.

- My Content (1)
### Recently viewed (1)

Menu

- Subjects
- Architecture and Design
- Arts
- Asian and Pacific Studies
- Business and Economics
- Chemistry
- Classical and Ancient Near Eastern Studies
- Computer Sciences
- Cultural Studies
- Engineering
- General Interest
- Geosciences
- History
- Industrial Chemistry
- Jewish Studies
- Law
- Library and Information Science, Book Studies
- Life Sciences
- Linguistics and Semiotics
- Literary Studies
- Materials Sciences
- Mathematics
- Medicine
- Music
- Pharmacy
- Philosophy
- Physics
- Social Sciences
- Sports and Recreation
- Theology and Religion

- Product Types

Show Summary Details# Open Mathematics

### formerly Central European Journal of Mathematics

# Infinite dimension of solutions
of the Dirichlet problem

#### Open Access

## Abstract

## References

## About the article

## Citing Articles

*Journal of Mathematical Sciences*, 2016## Related Content

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

Mathematical Citation Quotient (MCQ) 2015: 0.39

Open Access

- Online
- ISSN
- 2391-5455

See all formats and pricing

- Online
- Open Access

*Prices in US$ apply to orders placed in the Americas only. Prices in GBP apply to orders placed in Great Britain only. Prices in € represent the retail prices valid in Germany (unless otherwise indicated). Prices are subject to change without notice. Prices do not include postage and handling if applicable. RRP: Recommended Retail Price.

- Most Downloaded Articles
## Most Downloaded Articles

- On the use of semi-closed sets and functions in convex analysis by Zălinescu, Constantin
- Discrete limit theorems for general Dirichlet series. III by Laurinčikas, A. and Macaitienė, R.
- On the strong metric dimension of the strong products of graphs by Kuziak, Dorota / Yero, Ismael G. and Rodríguez-Velázquez, Juan A.
- Extending analyticK-subanalytic functions by Piękosz, Artur
- Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces by Yantir, Ahmet/ Kubiaczyk, Ireneusz and Sikorska-Nowak, Aneta
- Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition by Malinowski, Marek T.
- Nonrecursive solution for the discrete algebraic Riccati equation and X + A*X-1A=L by Adam, Maria and Assimakis, Nicholas
- Infinite dimension of solutions of the Dirichlet problem by Ryazanov, Vladimir
- Properties of k-beta function with several variables by Rehman, Abdur / Mubeen, Shahid / Safdar, Rabia and Sadiq, Naeem
- On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method by Baskonus, Haci Mehmet and Bulut, Hasan
- Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field by Melgaard, Michael
- Pointwise density topology by Górajska, Magdalena
- Invariance groups of finite functions and orbit equivalence of permutation groups by Horváth, Eszter K./ Makay, Géza/ Pöschel, Reinhard and Waldhauser, Tamás
- Boundary regularity of flows under perfect slip boundary conditions by Kaplický, Petr and Tichý, Jakub
- New properties of conformable derivative by Atangana, Abdon / Baleanu, Dumitru and Alsaedi, Ahmed
- New SOR-like methods for solving the Sylvester equation by Kierzkowski, Jakub
- Vietoris topology on spaces dominated by second countable ones by Islas, Carlos and Jardon, Daniel
- Complexity issues for the symmetric interval eigenvalue problem by Hladík, Milan
- Dynamics of differentiation operators on generalized weighted Bergman spaces by Zhang, Liang and Zhou, Ze-Hua
- Absolutely strongly star-Hurewicz spaces by Song, Yan-Kui

Volume 13, Issue 1 (May 2015)
## Issues

## Volume 14 (2016)

## Volume 13 (2015)

## Volume 12 (2014)

## Volume 11 (2013)

## Volume 10 (2012)

## Volume 9 (2011)

## Volume 8 (2010)

## Volume 7 (2009)

## Volume 6 (2008)

## Volume 5 (2007)

## Volume 4 (2006)

## Volume 3 (2005)

## Volume 2 (2004)

- Open AccessIssue 12 (Dec 2014) , pp. 1749-1889
- Open AccessIssue 11 (Nov 2014) , pp. 1615-1747
- Open AccessIssue 10 (Oct 2014) , pp. 1403-1614
- Open AccessIssue 9 (Sep 2014) , pp. 1285-1402
- Open AccessIssue 8 (Aug 2014) , pp. 1109-1284
- Open AccessIssue 7 (Jul 2014) , pp. 923-1108
- Open AccessIssue 6 (Jun 2014) , pp. 787-922
- Open AccessIssue 5 (May 2014) , pp. 659-786
- Open AccessIssue 4 (Apr 2014) , pp. 545-657
- Open AccessIssue 3 (Mar 2014) , pp. 395-544
- Open AccessIssue 2 (Feb 2014) , pp. 175-394
- Open AccessIssue 1 (Jan 2014) , pp. 1-174

- Open AccessIssue 12 (Dec 2013) , pp. 2053-2261
- Open AccessIssue 11 (Nov 2013) , pp. 1863-2051
- Open AccessIssue 10 (Oct 2013) , pp. 1713-1862
- Open AccessIssue 9 (Sep 2013) , pp. 1547-1712
- Open AccessIssue 8 (Aug 2013) , pp. 1359-1545
Topical Issue: Numeric...
- Open AccessIssue 7 (Jul 2013) , pp. 1159-1357
- Open AccessIssue 6 (Jun 2013) , pp. 985-1157
- Open AccessIssue 5 (May 2013) , pp. 817-984
- Open AccessIssue 4 (Apr 2013) , pp. 595-815
Topical Issue: Numeric...
- Open AccessIssue 3 (Mar 2013) , pp. 379-593
- Open AccessIssue 2 (Feb 2013) , pp. 197-377
- Open AccessIssue 1 (Jan 2013) , pp. 1-196

- Open AccessIssue 6 (Dec 2012) , pp. 1915-2295
Topics in Nonlinear An...
- Open AccessIssue 5 (Oct 2012) , pp. 1593-1913
Topical Issue: Finite ...
- Open AccessIssue 4 (Aug 2012) , pp. 1185-1591
Topical Issue: Instant...
- Open AccessIssue 3 (Jun 2012) , pp. 837-1184
- Open AccessIssue 2 (Apr 2012) , pp. 371-836
- Open AccessIssue 1 (Feb 2012) , pp. 1-369
Topical Issue: Advance...

- On the use of semi-closed sets and functions in convex analysis
- Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces
- Absolutely strongly star-Hurewicz spaces
- On the strong metric dimension of the strong products of graphs
- Erratum to “On the strong metric dimension of the strong products of graphs”
- Pointwise density topology
- Invariance groups of finite functions and orbit equivalence of permutation groups
- Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition
- Dynamics of differentiation operators on generalized weighted Bergman spaces
- Relative annihilator-preserving congruence relations and relative annihilator-preserving homomorphisms in bounded distributive semilattices
- Vietoris topology on spaces dominated by second countable ones
- Identities arising from higher-order Daehee polynomial bases
- A function space from a compact metrizable space to a dendrite with the hypo-graph topology
- Solutions of minus partial ordering equations over von Neumann regular rings
- New interval oscillation criteria for second-order functional differential equations with nonlinear damping
- Notes on monotonically metacompact generalized ordered spaces
- The classification of modular Lie superalgebras of type M
- On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem
- Two bounds on the noncommuting graph
- A new kind of the solution of degenerate parabolic equation with unbounded convection term
- A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings
- Properties of k-beta function with several variables
- Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms
- Infinite dimension of solutions of the Dirichlet problem
- An extended Prony’s interpolation scheme on an equispaced grid
- Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order
- On Poincaré duality for pairs (G,W)
- Some identities of degenerate special polynomials
- Chaotic and hypercyclic properties of the quasi-linear Lasota equation
- Semilinear problems for the fractional laplacian with a singular nonlinearity
- Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms
- On some ideal related to the ideal (v)
- The Umbral operator and the integration involving generalized Bessel-type functions
- Mild solution of fractional order differential equations with not instantaneous impulses
- On graded P-compactly packed modules
- The classification of partially symmetric 3-braid links
- Linear and nonlinear abstract differential equations of high order
- Real Hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting shape operator
- Exact solutions of supersymmetric Burgers equation with Bosonization procedure
- Only 3-generalized metric spaces have a compatible symmetric topology
- Restricted and quasi-toral restricted Lie-Rinehart algebras
- A class of zero divisor rings in which every graph is precisely the union of a complete graph and a complete bipartite graph
- Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds
- Laws of large numbers for ratios of uniform random variables
- The structure of finite groups and ɵ-pairs of general subgroups
- New criteria for H-tensors and an application
- Hom-structures on semi-simple Lie algebras
- Inequality for power series with nonnegative coefficients and applications
- Generalized co-annihilator of BL-algebras
- Linear relations between modular forms for Г(p)
- Weak amenability for the second dual of Banach modules
- On some dicardinal functions in ditopological texture spaces
- Boundedness of Toeplitz type operators associated to Riesz potential operator with general kernel on Orlicz space
- Inequalities of harmonic univalent functions with connections of hypergeometric functions
- Module Connes amenability of hypergroup measure algebras
- Three open problems and a conjecture
- Extremal properties of the set of vector-valued Banach limits
- The estimates of diagonally dominant degree and eigenvalues inclusion regions for the Schur complement of block diagonally dominant matrices
- Some properties of geodesic semi E-b-vex functions
- Computation of double Hopf points for delay differential equations
- Some extensions of a certain integral transform to a quotient space of generalized functions
- Parabolic variational inequalities with generalized reflecting directions
- New properties of conformable derivative
- Coupled fixed point theorems for (α, φ)-contractive type mappings in partially ordered G-metric spaces
- Double image multi-encryption algorithm based on fractional chaotic time series
- On numerical invariants for knots in the solid torus
- On V-semigroups
- W-perfect groups
- Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach
- All about the ⊥ with its applications in the linear statistical models
- Nonrecursive solution for the discrete algebraic Riccati equation and X + A*XA=L
- Generalizations of Nekrasov matrices and applications
- Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models
- Computing the numerical range of Krein space operators
- New SOR-like methods for solving the Sylvester equation
- Complexity issues for the symmetric interval eigenvalue problem
- Some fractional integral formulas for the Mittag-Leffler type function with four parameters
- On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method
- A finite difference method for fractional diffusion equations with Neumann boundary conditions
- Results for Mild solution of fractional coupled hybrid boundary value problems
- Third-order differential subordination and superordination involving a fractional operator
- Upper and lower bounds of integral operator defined by the fractional hypergeometric function
- A detailed analysis for the fundamental solution of fractional vibration equation
- Duplication in a model of rock fracture with fractional derivative without singular kernel
- System of fractional differential equations with Erdélyi-Kober fractional integral conditions
- The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1,2)
- Fractional derivative generalization of Noether’s theorem

Vladimir Ryazanov

- Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, 74 Roze Luxemburg Str., 83114, Donetsk, Ukraine

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

[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

**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)

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]

Vladimir Gutlyanskii, Vladimir Ryazanov, and Artem Yefimushkin

Loading ...

- On the use of semi-closed sets and functions in convex analysis
- Carathéodory solutions of Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces
- Absolutely strongly star-Hurewicz spaces
- On the strong metric dimension of the strong products of graphs
- Erratum to “On the strong metric dimension of the strong products of graphs”
- Pointwise density topology
- Invariance groups of finite functions and orbit equivalence of permutation groups
- Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition
- Dynamics of differentiation operators on generalized weighted Bergman spaces
- Relative annihilator-preserving congruence relations and relative annihilator-preserving homomorphisms in bounded distributive semilattices
- Vietoris topology on spaces dominated by second countable ones
- Identities arising from higher-order Daehee polynomial bases
- A function space from a compact metrizable space to a dendrite with the hypo-graph topology
- Solutions of minus partial ordering equations over von Neumann regular rings
- New interval oscillation criteria for second-order functional differential equations with nonlinear damping
- Notes on monotonically metacompact generalized ordered spaces
- The classification of modular Lie superalgebras of type M
- On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem
- Two bounds on the noncommuting graph
- A new kind of the solution of degenerate parabolic equation with unbounded convection term
- A signature-based algorithm for computing Gröbner-Shirshov bases in skew solvable polynomial rings
- Properties of k-beta function with several variables
- Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms
- Infinite dimension of solutions of the Dirichlet problem
- An extended Prony’s interpolation scheme on an equispaced grid
- Positivity and contractivity in the dynamics of clusters’ splitting with derivative of fractional order
- On Poincaré duality for pairs (G,W)
- Some identities of degenerate special polynomials
- Chaotic and hypercyclic properties of the quasi-linear Lasota equation
- Semilinear problems for the fractional laplacian with a singular nonlinearity
- Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms
- On some ideal related to the ideal (v0)
- The Umbral operator and the integration involving generalized Bessel-type functions
- Mild solution of fractional order differential equations with not instantaneous impulses
- On graded P-compactly packed modules
- The classification of partially symmetric 3-braid links
- Linear and nonlinear abstract differential equations of high order
- Real Hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting shape operator
- Exact solutions of supersymmetric Burgers equation with Bosonization procedure
- Only 3-generalized metric spaces have a compatible symmetric topology
- Restricted and quasi-toral restricted Lie-Rinehart algebras
- A class of zero divisor rings in which every graph is precisely the union of a complete graph and a complete bipartite graph
- Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds
- Laws of large numbers for ratios of uniform random variables
- The structure of finite groups and ɵ-pairs of general subgroups
- New criteria for H-tensors and an application
- Hom-structures on semi-simple Lie algebras
- Inequality for power series with nonnegative coefficients and applications
- Generalized co-annihilator of BL-algebras
- Linear relations between modular forms for Г0+(p)
- Weak amenability for the second dual of Banach modules
- On some dicardinal functions in ditopological texture spaces
- Boundedness of Toeplitz type operators associated to Riesz potential operator with general kernel on Orlicz space
- Inequalities of harmonic univalent functions with connections of hypergeometric functions
- Module Connes amenability of hypergroup measure algebras
- Three open problems and a conjecture
- Extremal properties of the set of vector-valued Banach limits
- The estimates of diagonally dominant degree and eigenvalues inclusion regions for the Schur complement of block diagonally dominant matrices
- Some properties of geodesic semi E-b-vex functions
- Computation of double Hopf points for delay differential equations
- Some extensions of a certain integral transform to a quotient space of generalized functions
- Parabolic variational inequalities with generalized reflecting directions
- New properties of conformable derivative
- Coupled fixed point theorems for (α, φ)g-contractive type mappings in partially ordered G-metric spaces
- Double image multi-encryption algorithm based on fractional chaotic time series
- On numerical invariants for knots in the solid torus
- On Vn-semigroups
- W-perfect groups
- Determinants of (–1,1)-matrices of the skew-symmetric type: a cocyclic approach
- All about the ⊥ with its applications in the linear statistical models
- Nonrecursive solution for the discrete algebraic Riccati equation and X + A*X-1A=L
- Generalizations of Nekrasov matrices and applications
- Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models
- Computing the numerical range of Krein space operators
- New SOR-like methods for solving the Sylvester equation
- Complexity issues for the symmetric interval eigenvalue problem
- Some fractional integral formulas for the Mittag-Leffler type function with four parameters
- On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method
- A finite difference method for fractional diffusion equations with Neumann boundary conditions
- Results for Mild solution of fractional coupled hybrid boundary value problems
- Third-order differential subordination and superordination involving a fractional operator
- Upper and lower bounds of integral operator defined by the fractional hypergeometric function
- A detailed analysis for the fundamental solution of fractional vibration equation
- Duplication in a model of rock fracture with fractional derivative without singular kernel
- System of fractional differential equations with Erdélyi-Kober fractional integral conditions
- The general solution for impulsive differential equations with Riemann-Liouville fractional-order q ∈ (1,2)
- Fractional derivative generalization of Noether’s theorem

## e-Products & Services

- eProducts
- Abstracting & Indexing
- Marketing & Sales Materials
- Services
- Advertising Rates
- Rights & Permissions

## Imprints and Publisher Partners

- Birkhauser
- De Gruyter Open
- De Gruyter Akademie Forschung
- De Gruyter Mouton
- De Gruyter Oldenbourg
- De Gruyter Saur
- De|G Press
- Publisher Partner

## Help & Contact Information

- Contact information
- FAQ
- Give Feedback

## News

- Conferences

Copyright © 2011–2017 by Walter de Gruyter GmbH

Powered by PubFactory

- [107.22.106.136|107.22.106.136]
- 107.22.106.136

## Comments (0)