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

TitleOn a new approach to the study of plane boundary-value problems
Publication TypeJournal Article
Year of Publication2017
AuthorsGutlyanskii, VYa., Ryazanov, VI, Yefimushkin, AS
Abbreviated Key TitleDopov. Nac. akad. nauk Ukr.
DOI10.15407/dopovidi2017.04.012
Issue4
SectionMathematics
Pagination12-18
Date Published4/2017
LanguageEnglish
Abstract

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.
Keywordsanisotropic media, Beltrami equation, boundary-value problems, inhomogeneous media
References: 
  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. 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. https://doi.org/10.1073/pnas.41.10.740
  5. Bojarski, B., Gutlyanskii, V., Martio, O. & Ryazanov, V. (2013). Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane. Zürich: EMS. https://doi.org/10.4171/122
  6. Gutlyanskii, V., Ryazanov, V., Srebro, U. & Yakubov, E. (2012). The Beltrami Equation. A Geometric Approach. Developments in Mathematics. Vol. 26. New York: Springer. https://doi.org/10.1007/978-1-4614-3191-6
  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.
    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. https://doi.org/10.1007/978-3-642-65513-5
  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. https://doi.org/10.1007/978-3-642-85928-1
  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.