We study open discrete maps of two-dimensional Riemannian manifolds from the Sobolev class. For these mappings, we establish the lower estimates of distortions of the moduli of families of the curves. As a consequence, we establish the equicontinuity of Sobolev classes at the interior points of the domain.
This is a preview of subscription content, log in via an institution to check access.
References
E. A. Sevost’yanov and S. A. Skvortsov, “On the local behavior of Orlicz–Sobolev classes,” Ukr. Mat. Vestn., 13, No. 4, 543–569 (2016).
E. S. Afanas’eva, V. I. Ryazanov, and R. R. Salimov, “On mappings in the Orlicz–Sobolev classes on Riemannian manifolds,” Ukr. Mat. Vestn., 8, No. 3, 319–342 (2011).
D. A. Kovtonyuk, R. R. Salimov, and E. A. Sevost’yanov, On the Theory of Mappings of Sobolev and Orlicz–Sobolev Classes [in Russian], Naukova Dumka, Kiev (2013).
R. R. Salimov, “Lower estimates for the p-module and mappings of the Sobolev classes,” Algebra Analiz, 26, No. 6, 143–171 (2014).
V. Ryazanov and S. Volkov, “On the boundary behavior of mappings in the class \( {W}_{\mathrm{loc}}^{1,1} \) on Riemann surfaces,” Complex Anal. Oper. Theory, 11, 1503–1520 (2017).
V. M. Miklyukov, Conformal Mapping of an Irregular Surface and Its Applications [in Russian], Volgograd University, Volgograd (2005).
T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer, Berlin (1955).
O. Lehto and O. Virtanen, Quasiconformal Mappings in the Plane, Springer, New York (1973).
A. Ignat’ev and V. Ryazanov, “Finite mean oscillation in the mapping theory,” Ukr. Mat. Vestn., 2, No. 3, 395–417 (2005).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer Science + Business Media, New York (2009).
H. Federer, Geometric Measure Theory, Springer, New York (1969).
J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin (1971).
J. M. Lee, Riemannian Manifolds: an Introduction to Curvature, Springer, New York (1997).
D. P. Il’yutko and E. A. Sevost’yanov, “On the open discrete mappings with unbounded characteristic on Riemannian manifolds,” Mat. Sb., 207, No. 4, 65–112 (2016).
V. G. Maz’ya, Sobolev Spaces [in Russian], Leningrad Univ., Leningrad (1985).
T. Lomako, R. Salimov, and E. Sevost’yanov, “On equicontinuity of solutions to the Beltrami equations,” Ann. Univ. Bucharest. Math. Ser., 59, No. 2, 263–274 (2010).
E. A. Sevost’yanov, “On the local behavior of open discrete mappings from the Orlicz–Sobolev classes,” Ukr. Math. Zh., 68, No. 9, 1259–1272 (2016); English translation:Ukr. Math. J., 68, No. 9, 1447–1465 (2017).
K. Kuratowski, Topology, Vol. 2, Academic Press, New York (1968).
B. Fuglede, “Extremal length and functional completion,” Acta Math., 98, 171–219 (1957).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 5, pp. 663–676, May, 2019.
Rights and permissions
About this article
Cite this article
Sevost’yanov, E.A. On the Local Behavior of Sobolev Classes on Two-Dimensional Riemannian Manifolds. Ukr Math J 71, 758–773 (2019). https://doi.org/10.1007/s11253-019-01672-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-019-01672-1