Abstract
This article is devoted to the study of mappings with bounded and finite distortion, as well as the Sobolev classes that have been actively studied recently. We study the homeomorphisms of the Sobolev classes \(W^{1,p}_{\textrm{loc}}\) for the case when \(p=n-1\), where \(n\) is the corresponding dimension of space. For these classes, we prove the estimate of the distortion of the modulus of families of curves under mapping. As an application, we obtain results on the boundary extension of homeomorphisms of the indicated class in terms of prime ends.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
C. Caratheodory, ‘‘Über die Begrenzung der einfachzusammenhängender Gebiete,’’ Math. Ann. 73, 323–370 (1913).
E. S. Afanas’eva, V. I. Ryazanov, and R. R. Salimov, ‘‘To the theory of mappings of the Sobolev class with the critical index,’’ J. Math. Sci. 239, 1–16 (2019).
B. Bojarski, V. Gutlyanski, and V. Ryazanov, ‘‘On the Beltrami equations with two characteristics,’’ Complex Var. Ellipt. Equat. 54, 935–950 (2009).
V. Ryazanov, U. Srebro, and E. Yakubov, ‘‘On strong solutions of the Beltrami equations,’’ Complex Var. Ellipt. Equat. 55, 219–236 (2010).
V. Ryazanov, U. Srebro, and E. Yakubov, ‘‘Integral conditions in the theory of the Beltrami equations,’’ Complex Var. Ellipt. Equat. 57, 1247–1270 (2012).
R. Näkki, ‘‘Prime ends and quasiconformal mappings,’’ J. Anal. Math. 35, 13–40 (1979).
V. Gol’dshtein and A. Ukhlov, ‘‘Traces of functions of \(L^{1}_{2}\) Dirichlet spaces on the Caratheéodory boundary,’’ Studia Math. 235, 209–224 (2016).
D. A. Kovtonyuk and V. I. Ryazanov, ‘‘Prime ends and the Orlicz-Sobolev classes,’’ SPb. Math. J. 27, 765–788 (2016).
D. Ilyutko and E. Sevost’yanov, ‘‘On prime ends on Riemannian manifolds,’’ J. Math. Sci. 241, 47–63 (2019).
J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Vol. 229 of Lecture Notes in Mathematics (Springer, Berlin, 1971).
B. Fuglede, ‘‘Extremal length and functional completion,’’ Acta Math. 98, 171–219 (1957).
G. T. Whyburn, Analytic Topology (Am. Math. Soc., Providence, RI, 1942).
O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov, Moduli in Modern Mapping Theory, Springer Monographs in Mathematics (Springer, New York, 2009).
A. Ignat’ev and V. Ryazanov, ‘‘Finite mean oscillation in mapping theory,’’ Ukr. Math. Bull. 2, 403–424 (2005).
T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis (Springer, Berlin, 1955).
S. Saks, Theory of the Integral (Dover, New York, 1964).
V. Ryazanov, U. Srebro, and E. Yakubov, ‘‘On ring solutions of Beltrami equation,’’ J. Anal. Math. 96, 117–150 (2005).
V. I. Ryazanov and E. A. Sevost’yanov, ‘‘Equicontinuous classes of ring \(Q\)-homeomorphisms,’’ Sib. Math. J. 48, 1093–1105 (2007).
V. Gutlyanskii, V. Ryazanov, U. Srebro, and E. Yakubov, The Beltrami Equation: A Geometric Approach (Springer, New York, 2012).
D. A. Kovtonyuk and V. I. Ryazanov, ‘‘On the theory of lower \(Q\)-homeomorphisms,’’ Ukr. Math. Bull. 5, 157–181 (2008).
D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, and E. A. Sevost’yanov, ‘‘To the theory of Orlicz–Sobolev classes,’’ SPb. Math. J. 25, 929–963 (2014).
H. Federer, Geometric Measure Theory (Springer, Berlin, 1996).
D. Kovtonyuk and V. Ryazanov, ‘‘On the theory of mappings with finite area distortion,’’ J. Anal. Math. 104, 291–306 (2008).
D. Kovtonyuk and V. Ryazanov, ‘‘On the boundary behavior of generalized quasi-isometries,’’ J. Anal. Math. 115, 103–119 (2011).
E. Sevost’yanov, ‘‘On boundary extension of mappings in metric spaces in terms of prime ends,’’ Ann. Acad. Sci. Fenn. Math. 44, 65–90 (2019).
R. Näkki and B. Palka, ‘‘Uniform equicontinuity of quasiconformal mappings,’’ Proc. Am. Math. Soc. 37, 427–433 (1973).
O. Martio and J. Sarvas, ‘‘Injectivity theorems in plane and space,’’ Ann. Acad. Sci. Fenn., Ser. A1: Math. 4, 384–401 (1978–1979).
E. Sevost’yanov and S. Skvortsov, ‘‘On equicontinuous families of mappings in the case of variable domains,’’ Ukr. Math. Zh. 71, 938–951 (2019).
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by F. G. Avkhadiev)
Rights and permissions
About this article
Cite this article
Afanas’eva, E., Ryazanov, V., Salimov, R. et al. On Boundary Extension of Sobolev Classes with Critical Exponent by Prime Ends. Lobachevskii J Math 41, 2091–2102 (2020). https://doi.org/10.1134/S1995080220110025
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080220110025