Boundary extension of mappings that satisfy the Poletsky inverse inequality in terms of prime ends

  • E. A. Sevost’yanov Inst. of Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Slovyansk
Keywords: mappings with a bounded and finite distortion, boundary behavior, prime ends

Abstract

УДК 517.5

For mappings with branching points that satisfy the Poletsky inverse inequality, we obtain some results related to their continuous boundary extension in terms of prime ends. Under certain conditions, the specified classes of mappings are also equicontinuous in the closure of a given domain.

References

A. K. Bakhtin, I. V. Denega, Inequalities for the inner radii of nonoverlapping domains, Ukr. Math. J., 71, № 7, 1138 – 1145 (2019). DOI: https://doi.org/10.1007/s11253-019-01703-x

A. K. Bakhtin, I. V. Denega, Estimation of the maximum product of inner radii of mutually disjoint domains, Ukr. Math. J., 72, № 2, 191 – 202 (2020). DOI: https://doi.org/10.1007/s11253-020-01775-0

O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Sci. + Business Media, LLC, New York (2009).

R. Salimov, B. Klishchuk, An extremal problem for the volume functional, Mat. Stud., 50, № 1, 36 – 43 (2018), https://doi.org/10.15330/ms.50.1.36-43 DOI: https://doi.org/10.15330/ms.50.1.36-43

B. A. Klishhuk, R. R. Salimov, Nizhnie oczenki ob`ema obraza shara, Ukr. mat. zhurn. 71, № 6, 774 – 785 (2019).

R. R. Salimov, E. A. Sevost’yanov, On the equicontinuity of one family of inverse mappings in terms of prime ends, Ukr. Math. J., 70, № 9, 1456 – 1466 (2019). DOI: https://doi.org/10.1007/s11253-019-01579-x

Ye. O. Sevost`yanov, S. O. Skvorczov, N. S. Il`kevich, Pro povedinku obernenikh gomeomorfizmiv v terminakh prostikh kincziv, Praczi In-tu prikl. matematiki i mekhaniki NAN Ukrayini, 33, 188 – 203 (2019).

E. A. Sevost’yanov, S. O. Skvortsov, N. S. Ilkevych, On the global behavior of inverse mappings in terms of prime ends, Ann. Acad. Sci. Fenn. Math (accepted for print), https://doi.org/10.30970/ms.52.1.24-31 DOI: https://doi.org/10.30970/ms.52.1.24-31

V. Gutlyanskii, V. Ryazanov, E. Yakubov, The Beltrami equations and prime ends, Укр. мат. вiсн., 12, № 1, 27—66 (2015), https://doi.org/10.1007/s10958-015-2546-7 DOI: https://doi.org/10.1007/s10958-015-2546-7

D. A. Kovtonyuk, V. I. Ryazanov, K teorii prosty`kh konczov dlya prostranstvenny`kh oblastej, Ukr. mat. zhurn.,67, № 4, 467 – 479 (2015).

D. A. Kovtonyuk, V. I. Ryazanov, Prime ends and Orlicz – Sobolev classes, St. Petersburg Math. J., 27, № 5, 765 – 788 (2016), https://doi.org/10.1090/spmj/1416 DOI: https://doi.org/10.1090/spmj/1416

R. N¨akki, Prime ends and quasiconformal mappings, J. Anal. Math., 35, 13 – 40 (1979), https://doi.org/10.1007/BF02791061 DOI: https://doi.org/10.1007/BF02791061

O. Martio, S. Rickman, and J. V¨ais¨al¨a, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 448, 1 – 40 (1969). DOI: https://doi.org/10.5186/aasfm.1969.448

S. Rickman, Quasiregular mappings, Springer-Verlag, Berlin (1993), https://doi.org/10.1007/978-3-642-78201-5 DOI: https://doi.org/10.1007/978-3-642-78201-5

M. Vuorinen, Exceptional sets and boundary behavior of quasiregular mappings in $n$-space, Ann. Acad. Sci. Fenn. Math. Diss., 11, 1 – 44 (1976).

J. V¨ais¨al¨a, Lectures on n-dimensional quasiconformal mappings, Lect. Notes Math., 229, Springer–Verlag, Berlin etc. (1971). DOI: https://doi.org/10.1007/BFb0061218

D. P. Ilyutko, E. A. Sevost’yanov, On prime ends on Riemannian manifolds, J. Math. Sci., 241, № 1, 47 – 63 (2019). DOI: https://doi.org/10.1007/s10958-019-04406-7

E. O. Sevost’yanov, S. O. Skvortsov, O. P. Dovhopiatyi, On nonhomeomorphic mappings with the inverse Poletsky inequality, J. Math. Sci., 252, № 4, 541 – 557 (2021). DOI: https://doi.org/10.1007/s10958-020-05179-0

K. Kuratovskij, Topologiya, T. 2, Mir, Moskva (1969).

B. Fuglede, Extremal length and functional completion, Acta Math., 98, 171 – 219 (1957), https://doi.org/10.1007/BF02404474 DOI: https://doi.org/10.1007/BF02404474

E`. Kollingvud, A. Lovater, Teoriya predel`ny`kh mnozhestv, Mir, Moskva (1971).

Published
20.07.2021
How to Cite
Sevost’yanov , E. A. “Boundary Extension of Mappings That Satisfy the Poletsky Inverse Inequality in Terms of Prime Ends”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 7, July 2021, pp. 951 -63, doi:10.37863/umzh.v73i7.6507.
Section
Research articles