Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 
 

 

On spatial mappings with integral restrictions on the characteristic


Author: E. A. Sevost′yanov
Translated by: E. Dubtsov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 1.
Journal: St. Petersburg Math. J. 24 (2013), 99-115
MSC (2010): Primary 30C65
DOI: https://doi.org/10.1090/S1061-0022-2012-01233-1
Published electronically: November 15, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For a given domain $ D\subset {\mathbb{R}}^n$, some families $ {\mathfrak{F}}$ of mappings $ f\,:\,D\rightarrow \overline {{\mathbb{R}}^n}$, $ n\ge 2$ are studied; such families are more general than the mappings with bounded distortion. It is proved that a family is equicontinuous if $ \int _{\delta _0}^{\infty } \frac {d\tau }{\tau [\Phi ^{-1}(\tau )]^{\frac {1}{n-1}}}= \infty $, where the integral depends on each mapping $ f\in {\mathfrak{F}}$, $ \Phi $ is a special function, and $ \delta _0>0$ is fixed. Under similar restrictions, removability results are obtained for isolated singularities of $ f$. Also, analogs of the well-known Sokhotsky-Weierstrass and Liouville theorems are proved.


References [Enhancements On Off] (What's this?)

  • 1. L. V. Ahlfors, Lectures on quasiconformal mappings, Math. Stud., No. 10D, Van Nostrand Co., Inc., Toronto etc., 1966. MR 0200442 (34:336)
  • 2. P. A. Biluta, Certain extremal problems for mappings which are quasiconformal in the mean, Sibirsk. Mat. Zh. 6 (1965), no. 4, 717-726. (Russian) MR 0190328 (32:7741)
  • 3. C. J. Bishop, V. Ya. Gutlyanskii, O. Martio, and M. Vuorinen, On conformal dilatation in space, Int. J. Math. Math. Sci. 2003, no. 22, 1397-1420. MR 1980177 (2004c:30038)
  • 4. N. Bourbaki, Functions of a real variable, Springer, Berlin, 2004. MR 2013000
  • 5. F. W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353-393. MR 0139735 (25:3166)
  • 6. A. Golberg, Homeomorphisms with finite mean dilatations, Complex Analysis and Dynamical Systems II, Contemp. Math., vol. 382, Amer. Math. Soc., Providence, RI, 2005, pp. 177-186. MR 2175886 (2007b:30020)
  • 7. V. M. Gol'dshteĭn and Yu. G. Reshetnyak, Introduction to the theory of functions with generalized derivatives, and quasiconformal mappings, Nauka, Moscow, 1983; English transl., Quasiconformal mappings and Sobolev spaces, Math. Appl. (Soviet Ser.), vol. 54, Kluwer Acad. Publ. Group, Dordrecht, 1990. MR 0738784 (85m:46031a); MR 1136035 (92h:46040)
  • 8. V. I. Kruglikov, Capacities of condensors and quasiconformal in the mean mappings in space, Mat. Sb. (N.S.) 130 (172) (1986), no. 2, 185-206; English transl., Math. USSR-Sb. 58 (1987), no. 1, 185-205. MR 0854971 (88d:30028)
  • 9. S. L. Krushkal', On mean quasiconformal mappings, Dokl. Akad. Nauk SSSR 157 (1964), no. 3, 517-519; English transl., Soviet Math. Dokl. 5 (1964), 966-969. MR 0176067 (31:342)
  • 10. O. Lehto and K. Virtanen, Quasiconformal mappings in the plane, Grundlehren Math. Wiss., Bd. 126, Springer, New York etc., 1973. MR 0344463 (49:9202)
  • 11. O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in modern mapping theory, Springer, New York, 2009. MR 2466579 (2012g:30004)
  • 12. V. M. Miklyukov, The conformal mapping of nonregular surface and its applications, Volgograd. Univ., Volgograd, 2005. (Russian)
  • 13. M. Perović, Isolated singularity of the mean quasiconformal mappings, Romanian-Finnish Seminar on Complex Analysis (Bucharest, 1976), Lecture Notes in Math., vol. 743, Springer, Berlin, 1979, pp. 212-214. MR 0552884 (81i:30040)
  • 14. I. N. Pesin, Mappings which are quasiconformal in the mean, Dokl. Akad. Nauk SSSR 187 (1969), no. 4, 740-742; English transl., Soviet Math. Dokl. 10 (1969), 939-941. MR 0249613 (40:2856)
  • 15. E. A. Poletskiĭ, The method of moduli for nonhomeomorphic quasiconformal mappings, Mat. Sb. (N.S.) 83 (125) (1970), no. 2, 261-272; English transl., Math. USSR-Sb. 12 (1970), 260-270. MR 0274753 (43:513)
  • 16. Yu. G. Reshetnyak, Space mappings with bounded distortion, Nauka, Novosibirsk, 1982; English transl., Transl. Math. Monogr., vol. 73, Amer. Math. Soc., Providence, RI, 1989. MR 0665590 (84d:30033); MR 0994644 (90d:30067)
  • 17. S. Rickman, Quasiregular mappings, Ergeb. Math. Grenzgeb. (3), Bd. 26, Springer-Verlag, Berlin, 1993. MR 1238941 (95g:30026)
  • 18. V. I. Ryazanov, On mappings that are quasiconformal in the mean, Sibirsk. Mat. Zh. 37 (1996), no. 2, 378-388; English transl., Siberian Math. J. 37 (1996), no. 2, 325-334. MR 1425344 (98a:30022)
  • 19. V. Ryazanov, U. Srebro, and E. Yakubov, On integral conditions in the mapping theory, Ukr. Mat. Visn. 7 (2010), no. 1, 73-87. (English) MR 2798548 (2012b:30051)
  • 20. Yu. F. Strugov, The compactness of families of mappings, quasiconformal in the mean, Dokl. Akad. Nauk SSSR 243 (1978), no. 4, 859-861; English transl., Soviet Math. Dokl. 19 (1978), no. 6, 1443-1446. MR 0514486 (80g:30013)
  • 21. G. D. Suvorov, On the art of mathematical investigation, Donets. Firma Naukoemkikh Tekhnologiĭ Akad. Nauk Ukrainy (Firma TEAN), Donetsk, 1999. (Russian)
  • 22. J. Väisälä, Lectures on $ n$-dimensional quasiconformal mappings, Lecture Notes in Math., vol. 229, Springer-Verlag, Berlin etc., 1971. MR 0454009 (56:12260)
  • 23. A. Ukhlov and S. K. Vodopyanov, Mappings associated with weighted Sobolev spaces, Complex Analysis and Dynamical Systems III, Contemp. Math., vol. 455, Amer. Math. Soc., Providence, RI, 2008, pp. 369-382. MR 2408182 (2009f:30061)
  • 24. V. A. Zorich, Admissible order of growth of the characteristic of quasiconformality in M. A. Lavrent'ev's theorem, Dokl. Akad. Nauk SSSR 181 (1968), no. 3, 530-533; English transl., Soviet Math. Dokl. 9 (1968), 866-869. MR 0229816 (37:5382)
  • 25. E. A. Sevost'yanov, Integral description of certain generalization of quasiregular mappings, and a divergence condition for an integral in geometric function theory, Ukrain. Mat. Zh. 61 (2009), no. 10, 1367-1380; English transl. in Ukrainian Math. J. 61 (2009), no. 10. MR 2888547
  • 26. -, On the branch points of mappings with an unbounded characteristic of quasiconformality, Sibirsk. Mat. Zh. 51 (2010), no. 5, 1129-1146; English transl., Siberian Math. J. 51 (2010), no. 5, 899-912. MR 2768509 (2011k:30027)
  • 27. -, A generalization of a lemma of E. A. Poletskiĭ on classes of space mappings, Ukrain. Mat. Zh. 61 (2009), no. 7, 1151-1157; English transl., Ukrainian Math. J. 61 (2009), no. 7, 969-975. MR 2768898 (2012a:30063)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 30C65

Retrieve articles in all journals with MSC (2010): 30C65


Additional Information

E. A. Sevost′yanov
Affiliation: Institute of Applied Mathematics and Mechanics, Rozy Luxembourg str. 74, Donetsk 83114, Ukraine
Email: brusin2006@rambler.ru, esevostyanov2009@mail.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01233-1
Keywords: Spatial mappings, capacity, integral restrictions
Received by editor(s): November 25, 2010
Published electronically: November 15, 2012
Article copyright: © Copyright 2012 American Mathematical Society

American Mathematical Society