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Tytuł artykułu

On local injectivity and asymptotic linearity of quasiregular mappings

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EN
Abstrakty
EN
It is shown that the approximate continuity of the dilatation matrix of a quasiregular mapping f at $x_0$ implies the local injectivity and the asymptotic linearity of f at $x_0$. Sufficient conditions for $log|f(x) - f(x_0)|$ to behave asymptotically as $log|x - x_0|$ are given. Some global injectivity results are derived.
Słowa kluczowe
Twórcy
  • Institute of Applied Mathematics and Mechanics, NAS of Ukraine, ul. Roze Luxemburg 74, 340114 Donetsk, Ukraine, math@iamm.ac.donetsk.ua
autor
  • Department of Mathematics, P.O. Box 4 (Yliopistonk. 5), FIN-00014 University of Helsinki, Finland, martio@cc.helsinki.fi
  • Institute of Applied Mathematics and Mechanics, NAS of Ukraine, ul. Roze Luxemburg 74, 340114 Donetsk, Ukraine, math@iamm.ac.donetsk.ua
autor
  • Department of Mathematics, P.O. Box 4 (Yliopistonk. 5), FIN-00014 University of Helsinki, Finland, vuorinen@csc.fi
Bibliografia
  • [A$_1$] L. Ahlfors, On a class of quasiconformal mappings, Österreich. Akad. Wiss. Math. Naturwiss. Kl. S. B. II 185 (1976), 5-10.
  • [A$_2$] L. Ahlfors, Quasiconformal deformations and mappings in $R^n$, J. Analyse Math. 30 (1976), 74-97.
  • [B] P. P. Belinskiĭ, General Properties of Quasiconformal Mappings, Nauka, Novosibirsk, 1974 (in Russian).
  • [Bell] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1970.
  • [BI] B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $ℝ^n$, Ann. Acad. Sci. Fenn. Ser. A I 8 (1983), 257-324.
  • [BIK] B. Bojarski, T. Iwaniec and R. Kopiecki, Riemannian manifolds with non-smooth metric tensors and QC-maps, in: Monge-Ampère Equations and Related Topics (Firenze, 1980), Ist. Naz. Alta Mat., Roma, 1982, 123-167.
  • [D] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
  • [Fer] J. Ferrand, Regularity of conformal mappings of Riemannian manifolds, in: Lecture Notes in Math. 743, Springer, 1979, 191-203.
  • [Fi] G. M. Fichtengolz, Differential- und Integralrechnung, Band 1, Deutscher Verlag Wiss., Berlin, 1964.
  • [FG] I. Fonseca and W. Gangbo, Degree Theory in Analysis and Applications, Clarendon Press, Oxford, 1995.
  • [Gol] V. M. Gol'dshteĭn, On behavior of mappings with bounded distortion under distortion coefficient close to 1, Sibirsk. Mat. Zh. 12 (1971), 1250-1258 (in Russian).
  • [GMRV$_1$] V. Gutlyanskiĭ, O, Martio, V. Ryazanov and M. Vuorinen, On asymptotical behavior of quasiconformal mappings in the space, to appear.
  • [GMRV$_2$] V. Gutlyanskiĭ, O, Martio, V. Ryazanov and M. Vuorinen, Convergence theorems for quasiregular mappings in the space, Forum Math., to appear.
  • [H] P. R. Halmos, Finite-Dimensional Vector Spaces, 2nd ed., Van Nostrand Reinhold, New York, 1958.
  • [Iw] T. Iwaniec, Regularity theorems for solutions of partial differential equations related to quasiregular mappings in several variables, Dissertationes Math. 198 (1982).
  • [LV] O. Lehto and K. Virtanen, Quasikonforme Abbildungen, Springer, Berlin, 1965.
  • [MRV$_1$] O. Martio, S. Rickman and J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I 448 (1969), 1-40.
  • [MRV$_2$] O. Martio, S. Rickman and J. Väisälä, Distortion and singularities of quasiregular mappings, ibid. 465 (1970), 1-13.
  • [MRV$_3$] O. Martio, S. Rickman and J. Väisälä, Topological and metric properties of quasiregular mappings, ibid. 488 (1971), 1-31.
  • [MSA] O. Martio and J. Sarvas, Injectivity theorems in plane and space, ibid. 4 (1978/1979), 383-401.
  • [MSR] O. Martio and U. Srebro, Universal radius of injectivity for locally quasiconformal mappings, Israel J. Math. 29 (1978), 17-23.
  • [O] A. Ostrowski, Solution of Equations and Systems of Equations, 2nd ed., Academic Press, New York, 1966.
  • [RR] T. Rado and P. V. Reichelderfer, Continuous Transformations in Analysis, Springer, Berlin, 1955.
  • [Re$_1$] Yu. G. Reshetnyak, Space Mappings with Bounded Distortion, Transl. Math. Monographs 73, Amer. Math. Soc., 1989.
  • [Re$_2$] Yu. G. Reshetnyak, Stability Theorems in Geometry and Analysis, Nauka, Novosibirsk, 1982 (in Russian).
  • [RS] S. Rickman and U. Srebro, Remarks on the local index of quasiregular mappings, J. Analyse Math. 46 (1986), 246-250.
  • [S] S. Saks, Theory of the Integral, Dover Publ., New York, 1964.
  • [Sar] J. Sarvas, Coefficient of injectivity for quasiregular mappings, Duke Math. J. 43 (1976), 147-158.
  • [V$_1$] J. Väisälä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I 392 (1966), 1-10.
  • [V$_2$] J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math. 229, Springer, Berlin, 1971.
  • [Vu] M. Vuorinen, Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math. 1319, Springer, Berlin, 1988.
  • [Z] V. A. Zorich, M. A. Lavrent'ev's theorem on quasiconformal space mappings, Mat. Sb. 74 (1967), 417-433 (in Russian).
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Bibliografia
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