Isaac Scientific Publishing

Advances in Analysis

On Neumann and Poincare Problems in A-harmonic Analysis

Download PDF (495.7 KB) PP. 114 - 120 Pub. Date: October 25, 2016

DOI: 10.22606/aan.2016.12006


  • Artyem Yefimushkin
    Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine


The existence of nonclassical solutions is proved for the Neumann and Poincare problems for generalizations of the Laplace equation in anisotropic and nonhomogeneous media in almost smooth domains with arbitrary boundary data that are measurable with respect to logarithmic capacity. Moreover, it is shown that the spaces of such solutions have the infinite dimension.


Neumann problem, poincare problem, A-harmonic functions, logarithmic capacity, anisotropic and nonhomogeneous media.


[1] K. Astala, T. Iwaniec, and G. Martin, Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton University Press, 2009, vol. 48.

[2] V. Gutlyanskii, V. Ryazanov, and A. Yefimushkin, “On the boundary-value problems for quasiconformal functions in the plane,” Journal of Mathematical Sciences, vol. 214, no. 2, pp. 200–219, 2016.

[3] V. Ryazanov, “On the Riemann-Hilbert problem without Index,” Ann. Univ. Bucharest, Ser. Math, vol. 5, pp. 169–178, 2014.

[4] ——, “Infinite dimension of solutions of the Dirichlet problem,” Open Math.(the former Central European J. Math.), vol. 13, pp. 348–350, 2015.

[5] ——, “On Neumann and Poincare problems for Laplace equation,” Analysis and Mathematical Physics, pp. 1–5, 2016. [Online]. Available:

[6] A. Yefimushkin and V. Ryazanov, “On the Riemann-Hilbert problem for the Beltrami Equations,” Contemporary Mathematics, vol. 667, pp. 299–316, 2016.

[7] S. G. Mikhlin, Partielle Differentialgleichungen in der mathematischen Physik. Akademie-Verlag, Berlin, 1978.

[8] L. V. Ahlfors, Lectures on quasiconformal mappings. Van Nostrand, 1966.

[9] O. Lehto and K. Virtanen, Quasiconformal mappings in the plane. Springer-Verlag, 1973, vol. 126.

[10] G. M. Goluzin, Geometric theory of functions of a complex variable. Amer. Math. Soc., Providence, 1969.

[11] P. Koosis, Introduction to Hp spaces. Cambridge University Press, 1998, vol. 115.

[12] M. Fekete, “über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten,” Mathematische Zeitschrift, vol. 17, no. 1, pp. 228–249, 1923.

[13] R. Nevanlinna, Eindeutige analytische Funktionen. Ann Arbor, Michigan, 1944.

[14] T. Iwaniec, “Regularity of solutions of certain degenerate elliptic systems of equations that realize quasiconformal mappings in n-dimensional space,” Differential and Integral Equations. Boundary-Value Problems, pp. 97–111, 1979.

[15] ——, “Regularity theorems for solutions of partial differential equations for quasiconformal mappings in several dimensions,” Dissertationes Math. (Rozprawy Mat.), 1982.

[16] M. Krasnoselskii, P. Zabreiko, E. Pustylii, and P. Sobolevskii, Integral operators in spaces of summable functions. Noordhoff International Publishing, Leiden, 1976.

[17] S. Agard, “Angles and quasiconformal mappings in space,” Journal d’Analyse Mathématique, vol. 22, no. 1, pp. 177–200, 1969.

[18] S. Agard and F. Gehring, “Angles and quasiconformal mappings,” Proceedings of the London Mathematical Society, vol. 3, no. 1, pp. 1–21, 1965.

[19] O. Taari, “Charakterisierung der Quasikonformitat mit Hilfe der Winkelverzerrung,” Annales Academiae Scientiarum Fennicae, Ser. A, vol. 390, pp. 1–43, 1966.