On a new approach to the study of plane boundary-value problems

1Gutlyanskii, VYa., 1Ryazanov, VI, 2Yefimushkin, AS
1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk
2Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2017, 4:12-18
Section: Mathematics
Language: English

We give a short description of our recent results obtained by a new approach to the boundary-value problems, such as the Dirichlet, Hilbert, Neumann, Poincaré and Riemann problems, for the Beltrami equations and for analogs of the Laplace equation in anisotropic and inhomogeneous media. We show that the approach makes it possible to study many problems of mathematical physics with arbitrary boundary data which are measurable with respect to logarithmic capacity.

Keywords: anisotropic media, Beltrami equation, boundary-value problems, inhomogeneous media
  1. Ahlfors, L. V. (1966). Lectures on Quasiconformal Mappings. Princeton, N.J.: Van Nostrand. Reprinted by Wadsworth Inc., Belmont, 1987.
  2. Beurling, A. & Ahlfors, L. (1956). The boundary correspondence under quasiconformal mappings. Acta Math, 96, Iss. 1, pp. 125-142. doi: https://doi.org/10.1007/BF02392360/
  3. Astala, K., Iwaniec, T., & Martin, G. (2009). Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Math. Ser. Vol. 48. Princeton, N.J.: Princeton Univ. Press.
  4. Bagemihl, F., & Seidel, W. (1955). Regular functions with prescribed measurable boundary values almost everywhere. Proc. Nat. Acad. Sci. U. S. A., 41, pp. 740-743.
  5. Bojarski, B., Gutlyanskii, V., Martio, O. & Ryazanov, V. (2013). Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane. Zürich: EMS.
  6. Gutlyanskii, V., Ryazanov, V., Srebro, U. & Yakubov, E. (2012). The Beltrami Equation. A Geometric Approach. Developments in Mathematics. Vol. 26. New York: Springer.
  7. Gutlyanskii, V., Ryazanov, V. & Yakubov, E. (2015). The Beltrami equations and prime ends. Ukr. Mat. Visn., 12, No 1, pp. 27-66; transl. in (2015), J. Math. Sci., 210, Iss. 1, pp. 22-51. doi: https://doi.org/10.1007/s10958-015-2546-7.
  8. Gutlyanskii, V., Ryazanov, V. & Yefimushkin, A. (2015). On the boundary-value problems for quasiconformal functions in the plane. Ukr. Mat. Visn., 12, No 3, pp. 363-389; transl. in (2016), J. Math. Sci., 214, Iss. 2, pp. 200-219. doi: https://doi.org/10.1007/s10958-016-2769-2.
  9. Lehto, O. & Virtanen, K. (1973). Quasiconformal Mappings in the Plane. New York: Springer.
  10. Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2009). Moduli in Modern Mapping Theory. New York: Springer.
  11. Noshiro K. (1960). Cluster sets. Berlin: Springer.
  12. Priwalow, I. I. (1956). Randeigenschaften analytischer Funktionen. Berlin: Wissenschaften.
  13. Ryazanov, V. (2015). On Hilbert and Riemann problems. An alternative approach. Ann. Univ. Buchar. Math. Ser. 6 (LXIV), No. 2, pp. 237-244.