Boundary behaviour of open discrete mappings on Riemannian manifolds. II

D. P. Ilyutko   ; E. A. Sevost'yanov

Sbornik. Mathematics, Tome 211 (2020) no. 4, pp. 539-582

Résumé

The boundary behaviour of classes of ring mappings, which generalize quasiconformal mappings in the sense of Gehring, is under investigation. Theorems proving that they have continuous boundary extensions are established in terms of prime ends of regular domains. Results on the equicontinuity of mappings in these classes in the closure of a fixed domain are also established in these terms. Bibliography: 45 titles.

Détail

Keywords: Riemannian manifold, moduli of families of curves and surfaces, end, mapping with bounded distortion, mapping with finite distortion, Orlicz-Sobolev class.

Bibliographie

[1] T. Adamowicz, A. Björn, J. Björn, N. Shanmugalingam, “Prime ends for domains in metric spaces”, Adv. Math., 238 (2013), 459–505 | DOI | MR | Zbl

[2] T. Adamowicz, Prime ends in metric spaces and boundary extensions of mappings, arXiv: 1608.02393

[3] C. J. Bishop, V. Ya. Gutlyanskiĭ, O. Martio, M. Vuorinen, “On conformal dilatation in space”, Int. J. Math. Math. Sci., 2003, no. 22, 1397–1420 | DOI | MR | Zbl

[4] M. Cristea, “Open discrete mappings having local $ACL^n$ inverses”, Complex Var. Elliptic Equ., 55:1-3 (2010), 61–90 | DOI | MR | Zbl

[5] A. Golberg, “Differential properties of $(\alpha, Q)$-homeomorphisms”, Further progress in analysis, World Sci. Publ., Hackensack, NJ, 2009, 218–228 | DOI | MR | Zbl

[6] V. Gol'dshtein, A. Ukhlov, “Traces of functions of $L^1_2$ Dirichlet spaces on the Carathéodory boundary”, Studia Math., 235:3 (2016), 209–224 | DOI | MR | Zbl

[7] V. Ya. Gutlyanskiĭ, O. Martio, V. I. Ryazanov, M. Vuorinen, “On convergence theorems for space qusiregular mappings”, Forum Math., 10:3 (1998), 353–375 | DOI | MR | Zbl

[8] V. Ya. Gutlyanskiĭ, O. Martio, V. I. Ryazanov, M. Vuorinen, “On local injectivity and asymptotic linearity of quasiregular mappings”, Studia Math., 128:3 (1998), 243–271 | MR | Zbl

[9] V. Gutlyanskiĭ, V. Ryazanov, U. Srebro, E. Yakubov, The Beltrami equation. A geometric approach, Dev. Math., 26, Springer, New York, 2012, xiv+301 pp. | DOI | MR | Zbl

[10] O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Monogr. Math., Springer, New York, 2009, xii+367 pp. | DOI | MR | Zbl

[11] O. Martio, S. Rickman, J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, 448, Suomalainen Tiedeakatemia, Helsinki, 1969, 40 pp. | DOI | MR | Zbl

[12] O. Martio, S. Rickman, J. Väisälä, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I, 465, Suomalainen Tiedeakatemia, Helsinki, 1970, 13 pp. | MR | Zbl

[13] Yu. G. Reshetnyak, Space mappings with bounded distortion, Transl. Math. Monogr., 73, Amer. Math. Soc., Providence, RI, 1989, xvi+362 pp. | MR | MR | Zbl | Zbl

[14] S. Rickman, Quasiregular mappings, Ergeb. Math. Grenzgeb. (3), 26, Springer-Verlag, Berlin, 1993, x+213 pp. | DOI | MR | Zbl

[15] M. Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Math., 1319, Springer-Verlag, Berlin, 1988, xx+209 pp. | DOI | MR | Zbl

[16] D. A. Kovtonyuk, V. I. Ryazanov, “Prime ends and Orlicz–Sobolev classes”, St. Petersburg Math. J., 27:5 (2016), 765–788 | DOI | MR | Zbl

[17] V. Gutlyanskii, V. Ryazanov, E. Yakubov, “The Beltrami equations and prime ends”, Ukr. matem. visn., 12:1 (2015), 27–66 | MR | Zbl

[18] D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov, E. A. Sevost'yanov, “Toward the theory of Orlicz–Sobolev classes”, St. Petersburg Math. J., 25:6 (2014), 929–963 | DOI | MR | Zbl

[19] D. A. Kovtonyuk, R. R. Salimov, E. A. Sevostyanov, K teorii otobrazhenii klassov Soboleva i Orlicha–Soboleva, Naukova dumka, Kiev, 2013, 303 pp.

[20] E. A. Sevostyanov, “Equicontinuity of homeomorphisms with unbounded characteristic”, Siberian Adv. Math., 23:2 (2013), 106–122 | DOI | MR | Zbl

[21] E. S. Afanasieva, V. I. Ryazanov, “On mappings in the Orlicz–Sobolev classes on Riemannian manifolds”, J. Math. Sci. (N.Y.), 181:1 (2012), 1–17 | DOI | MR | Zbl

[22] E. S. Afanaseva, “O granichnom povedenii koltsevykh $Q$-gomeomorfizmov na rimanovykh mnogoobraziyakh”, Dopov. NAN Ukraini, 2011, no. 8, 7–12 | MR | Zbl

[23] E. S. Afanas'eva, “On the boundary behavior of one class of mappings in metric spaces”, Ukrainian Math. J., 66:1 (2014), 16–29 | DOI | MR | Zbl

[24] E. S. Afanas'eva, “Generalized quasi-isometries on smooth Riemannian manifolds”, Math. Notes, 102:1 (2017), 12–21 | DOI | DOI | MR | Zbl

[25] E. S. Afanas'eva, “Boundary behavior of ring $Q$-homeomorphisms on Riemannian manifolds”, Ukrainian Math. J., 63:10 (2012), 1479–1493 | DOI | MR | Zbl

[26] D. P. Il'yutko, E. A. Sevost'yanov, “Open discrete mappings with unbounded coefficient of quasi-conformality on Riemannian manifolds”, Sb. Math., 207:4 (2016), 537–580 | DOI | DOI | MR | Zbl

[27] R. Näkki, “Prime ends and quasiconformal mappings”, J. Analyse Math., 35 (1979), 13–40 | DOI | MR | Zbl

[28] B. Fuglede, “Extremal length and functional completion”, Acta Math., 98 (1957), 171–219 | DOI | MR | Zbl

[29] G. T. Whyburn, Analytic topology, Amer. Math. Soc. Colloq. Publ., 28, Amer. Math. Soc., New York, 1942, x+278 pp. | MR | Zbl

[30] D. P. Ilyutko, E. A. Sevost'yanov, “Boundary behaviour of open discrete mappings on Riemannian manifolds”, Sb. Math., 209:5 (2018), 605–651 | DOI | DOI | MR | Zbl

[31] K. Kuratowski, Topology, v. II, New ed., rev. and augm., Academic Press, New York–London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968, xiv+608 pp. | MR | MR

[32] J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Math., 229, Springer–Verlag, Berlin, 1971, xiv+144 pp. | DOI | MR | Zbl

[33] E. A. Sevost'yanov, “Boundary behavior and equicontinuity for families of mappings in terms of prime ends”, St. Petersburg Math. J., 30:6 (2019), 973–1005 | DOI | MR | Zbl

[34] T. Adamowicz, N. Shanmugalingam, “Non-conformal Loewner type estimates for modulus of curve families”, Ann. Acad. Sci. Fenn. Math., 35:2 (2010), 609–626 | DOI | MR | Zbl

[35] J. M. Lee, Riemannian manifolds. An introduction to curvature, Grad. Texts in Math., 176, Springer-Verlag, New York, 1997, xvi+224 pp. | DOI | MR | Zbl

[36] M. Vuorinen, Exceptional sets and boundary behavior of quasiregular mappings in $n$-space, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 11, Helsinki, Suomalainen tiedeakatemia, 1976, 44 pp. | MR | Zbl

[37] W. P. Ziemer, “Extremal length and conformal capacity”, Trans. Amer. Math. Soc., 126:3 (1967), 460–473 | DOI | MR | Zbl

[38] W. P. Ziemer, “Extremal length and $p$-capacity”, Michigan Math. J., 16 (1969), 43–51 | DOI | MR | Zbl

[39] V. A. Shlyk, “The equality between $p$-capacity and $p$-modulus”, Siberian Math. J., 34:6 (1993), 1196–1200 | DOI | MR | Zbl

[40] D. Kovtonyuk, V. Ryazanov, “New modulus estimates in Orlicz–Sobolev classes”, Ann. Univ. Buchar. Math. Ser., 5(LXIII):1 (2014), 131–135 | MR | Zbl

[41] J. Väisälä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I, 392, Suomalainen Tiedeakatemia, Helsinki, 1966, 10 pp. | MR | Zbl

[42] E. A. Poleckiĭ, “The modulus method for nonhomeomorphic quasiconformal mappings”, Math. USSR-Sb., 12:2 (1970), 260–270 | DOI | MR | Zbl

[43] E. S. Smolovaya, “Boundary behavior of ring $Q$-homeomorphisms in metric spaces”, Ukrainian Math. J., 62:5 (2010), 785–793 | DOI | MR | Zbl

[44] E. A. Sevost'yanov, E. A. Petrov, “On the equicontinuity of homeomorphisms of Orlicz and Orlicz–Sobolev classes in the closure of a domain”, Ukrainian Math. J., 69:11 (2018), 1821–1834 | DOI | MR | Zbl

[45] E. A. Sevostyanov, S. A. Skvortsov, “O ravnostepennoi nepreryvnosti obobschennykh kvaziizometrii na rimanovykh mnogoobraziyakh”, Mat. Stud., 45:2 (2016), 159–169 | DOI | MR | Zbl