Boundary behaviour of open discrete mappings on Riemannian manifolds

D. P. Il'yutko   ; E. A. Sevost'yanov

Sbornik. Mathematics, Tome 209 (2018) no. 5, pp. 605-651

Résumé

The paper is concerned with problems of continuous extension of certain classes of mappings on Riemannian manifolds to boundary points of a given domain. In particular, the so-called ring mappings are shown to be continuously extendable to an isolated boundary point. Analogous theorems are also derived under more general conditions on the boundaries of the given and the target domains. As an application of the machinery thus developed, an arbitrary open discrete boundary-preserving mapping from the Orlicz-Sobolev class is shown to extend continuously to an isolated boundary point. Bibliography: 40 titles.

Détail

Keywords: Riemannian manifold, moduli of families of paths and surfaces, mappings with bounded or finite distortion, local and boundary behaviour of mappings, Sobolev class, Orlicz-Sobolev class.

Bibliographie

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

[2] O. Martio, S. Rickman, J. Väisälä, “Distortion and singularities of quasiregular mappings”, Ann. Acad. Sci. Fenn. Ser. A I, 465 (1970), 1–13 | MR | Zbl

[3] 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

[4] V. M. Mikljukov, “Removable singularities of quasi-conformal mappings in space”, Soviet Math. Dokl., 10 (1969), 1153–1155 | MR | Zbl

[5] R. Näkki, “Boundary behavior of quasiconformal mappings in $n$-space”, Ann. Acad. Sci. Fenn. Ser. A I, 484 (1970), 1–50 | MR | Zbl

[6] E. A. Sevost'yanov, R. R. Salimov, E. A. Petrov, “On the removal of singularities of the Orlicz–Sobolev classes”, J. Math. Sci. (N. Y.), 222:6 (2017), 723–740 | DOI | MR | Zbl

[7] E. A. Poleckii, “On the removal of singularities of quasiconformal mappings”, Math. USSR-Sb., 21:2 (1973), 240–254 | DOI | MR | Zbl

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

[9] B. V. Shabat, “On the theory of quasiconformal mappings in space”, Soviet Math. Dokl., 1 (1960), 730–733 | MR | Zbl

[10] U. Srebro, “Conformal capacity and quasiregular mappings”, Ann. Acad. Sci. Fenn. Ser. A I, 529 (1973), 1–8 | MR | Zbl

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

[12] M. Vuorinen, “Exceptional sets and boundary behavior of quasiregular mappings in $n$-space”, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 1976, no. 11, 1–44 | MR | Zbl

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

[14] I. Holopainen, P. Pankka, “Mappings of finite distortion: global homeomorphism theorem”, Ann. Acad. Sci. Fenn. Math., 29:1 (2004), 59–80 | MR | Zbl

[15] T. Iwaniec, G. Martin, Geometric function theory and non-linear analysis, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press, New York, 2001, xvi+552 pp. | MR | Zbl

[16] P. Koskela, K. Rajala, “Mappings of finite distortion: removable singularities”, Israel J. Math., 136 (2003), 269–283 | DOI | MR | Zbl

[17] K. Rajala, “Mappings of finite distortion: removable singularities for locally homeomorphic mappings”, Proc. Amer. Math. Soc., 132:11 (2004), 3251–3258 | DOI | MR | Zbl

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

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

[20] A. A. Ignat'ev, V. I. Ryazanov, “On the theory of boundary behavior of spatial mappings”, Ukr. Math. Bull., 3:2 (2006), 189–201 | MR | Zbl

[21] 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

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

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

[24] R. R. Salimov, E. A. Sevost'yanov, “The theory of shell-based $Q$-mappings in geometric function theory”, Sb. Math., 201:6 (2010), 909–934 | DOI | DOI | MR | Zbl

[25] 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

[26] E. A. Sevost'yanov, “Local and boundary behavior of maps in metric spaces”, St. Petersburg Math. J., 28:6 (2017), 807–824 | DOI | MR | Zbl

[27] E. A. Sevost'yanov, “Generalization of one Poletskii lemma to classes of space mappings”, Ukrainian Math. J., 61:7 (2009), 1151–1157 | DOI | MR | Zbl

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

[29] F. W. Gehring, “Lipschitz mappings and $p$-capacity of rings in $n$-space”, Advances in the theory of Riemann surfaces (Stony Brook, N.Y., 1969), Ann. of Math. Studies, 66, Princeton Univ. Press, Princeton, N.J., 1971, 175–193 | MR | Zbl

[30] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001, x+140 pp. | DOI | MR | Zbl

[31] 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

[32] 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

[33] T. Rado, P. V. Reichelderfer, Continuous transformations in analysis. With an introduction to algebraic topology, Grundlehren Math. Wiss., LXXV, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1955, vii+442 pp. | MR | Zbl

[34] S. Saks, Theory of the integral, Monogr. Mat., 7, 2nd ed., G. E. Stechert, Warszawa–New York, 1937, vi+347 pp. | MR | Zbl

[35] H. Federer, Geometric measure theory, Grundlehren Math. Wiss., 153, Springer-Verlag, New York, 1969, xiv+676 pp. | MR | MR | Zbl | Zbl

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

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

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

[39] W. Hurewicz, H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, NJ, 1948, vii+165 pp. | MR | Zbl

[40] C. J. Titus, G. S. Young, “The extension of interiority, with some applications”, Trans. Amer. Math. Soc., 103:2 (1962), 329–340 | DOI | MR | Zbl