Address: Vatutina st. 53, Vladikavkaz,
362025, RNO-A, Russia
Phone: (8672)23-00-54
E-mail: rio@smath.ru
DOI: 10.23671/VNC.2017.2.6507
On the Power Order of Growth of Lower \(Q\)-Homeomorphisms
Salimov R. R.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 2.
Abstract: In the present paper we investigate the asymptotic behavior of \(Q\)-homeomorphisms with respect to a \(p\)-modulus at a point. The sufficient conditions on \(Q\) under which a mapping has a certain order of growth are obtained. We also give some applications of these results to Orlicz-Sobolev classes \(W^{1,\varphi}_{\rm loc}\) in \(\mathbb{R}^n\), \(n\geqslant 3\), under conditions of the Calderon type on \(\varphi\) and, in particular, to Sobolev classes \(W_{\rm loc}^{1,p},\) \(p>n-1\). We give also an example of a homeomorphism demonstrating that the established order of growth is precise.
For citation: Salimov R. R. On the Power Order of Growth of Lower \(Q\)-Homeomorphisms. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 36-48. DOI 10.23671/VNC.2017.2.6507
1. Martio O., Ryazanov V., Srebro U., Yakubov E. Moduli in Modern
Mapping Theory, Springer Monographs in Mathematics. Springer, New
York etc., 2009. 367 p.
2. Ryazanov V., Salimov R., Srebro U., Yakubov E. On Boundary
Value Problems for the Beltrami Equations. Contemp. Math. 2013,
vol. 591, pp. 211-242.
3. Kovtonjuk D. A., Petkov I. V., Rjazanov V. I., Salimov R. R.
Boundary Behavior and the Dirichlet Problem for the Beltrami
Equations. St. Petersburg Math. J. [Algebra i Analiz, 2013, vol. 25,
no. 4, pp. 101-124], 2014, vol. 25, no. 4, pp. 587-603.
4. Afanas'eva E. S., Rjazanov V. I., Salimov R. R. About Mappings
in Orlicz-Sobolev Classes on Riemannian Manifolds. Ukrainskii
Matematichnij Visnik [Ukrainian Mathematical Bulletin], 2011, vol.
8, no. 3, pp. 319-342 (Russian).
5. Kovtonyuk D., Salimov R. and Sevost'yanov E. To the theory of
the mappings of Sobolev and Orlicz-Sobolev classes. Kiev: Naukova
Dumka, 2013, 304 p. (Russian).
6. Kovtonyuk D.A., Ryazanov V.I., Salimov R.R. and Sevost'yanov
E.A., St. Petersburg Mathematical Journal [Algebra i Analiz, 2013,
vol. 25, no. 6, pp. 50-102], 2014, vol. 25, no. 6, pp. 929-963.
7. Kovtonjuk D. A., Rjazanov V. I., Salimov R. R., Sevost'janov E.
A. Boundary behavior of Orlicz–Sobolev ñlasses. Math. Notes. [Mat.
Zametki, 2014, vol. 95, no. 4, pp. 564–576], 2014, vol. 95, no. 3-4,
pp. 509-519.
8. Salimov R. R. Lower estimates of p-modulus and mappings of
Sobolev's class. St. Petersburg Math. J. [Algebra i Analiz, 26:6
(2014), 143-171], 2015, vol. 26, no. 6, pp. 965-984.
9. Salimov R. R. Metricheskie svojstva klassov Orlicha-Soboleva.
Ukr. mat. visnyk. 2016, vol. 13, no. 1, pp. 129-141.
10. Salimov R. R. On finite lipschitz Orlicz-Sobolev classes.
Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.],
2015, vol. 17, no. 1, pp. 64–77 (Russian).
11. Salimov R. R. On new condition of finite Lipschitz of
Orlicz-Sobolev class. Mat. Studii. 2015, vol. 44, no. 1, pp. 27–35
(in Russian).
12. Ikoma K. On the distortion and correspondence under
quasiconformal mappings in space. Nagoya Math. J. 1965, vol. 25, pp.
175-203.
13. Kovtonyuk D., Ryazanov V. K teorii nizhnih Q-gomeomorfizmov.
Ukr. mat. visnyk. 2008, vol. 5, no. 2, pp. 157-181.
14. Saks S. Theory of the integral [Teoriya integrala]. Izdat.
inostr. lit. Moscow, 1949 (in Russian)
15. Martio O., Rickman S., Vaisala J. Definitions for quasiregular
mappings. Ann. Acad. Sci. Fenn. Ser. A1. Math. 1969, Vol. 448, pp.
1-40.
16. Shlyk V. A. On the Equality Between p-capacity and p-modulus.
Siberian Math. J. [Sibirsk. Mat. Zh., 1993, vol. 34, no. 6, pp.
216-221], 1993, vol. 34, no. 6, pp. 1196-1200.
17. Maz'ya V. Lectures on Isoperimetric and Isocapacitary
Inequalities in the Theory of Sobolev Spaces. Contemp. Math., 2003,
vol. 338, pp. 307-340.
18. Iwaniec T., Sverak V. On Mappings with Integrable Dilatation.
Proc. Amer. Math. Soc., 1993, vol. 118, p. 181-188.
19. Iwaniec T., Martin G. Geometrical Function Theory and Non-Linear
Analysis, Oxford, Clarendon Press, 2001.
20. Krasnosel'skij M. A., Rutickij Ja. B. Vypuklye Funkcii i
Prostranstva Orlicha, Moskva, Gos. Izd-vo Fiz.-Mat. Lit-ry, 1958
(Russian).
21. Maz'ja V. G. Prostranstva S. L. Soboleva, Leningrad, LGU, 1985,
416 p. (Russian).
22. Reshetnjak Ju. G. Prostranstvennye Otobrazhenija s Ogranichennym
Iskazheniem [Space Mappings with Bounded Distortion], Novosibirsk,
Nauka, 1982 (Ruassian).
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses.
Сайт использует файлы cookie, необходимые для корректной работы сайта, и сервисы Яндекс-метрики, используемые для анализа статистики посещаемости, которые не содержат сведений, на основании которых можно идентифицировать личность пользователя. Продолжение пользования сайтом является согласием на применение данных технологий.
