On modulus inequality of the order $p$ for the inner dilatation
Abstract
The article is devoted to mappings with bounded
and finite distortion of planar domains. Our investigations are
devoted to the connection between mappings of the Sobolev class and
upper bounds for the distortion of the modulus of families of paths.
For this class, we have proved the Poletsky-type inequality with
respect to the so-called inner dilatation of the order~$p.$ We
separately considered the situations of homeomorphisms and mappings
with branch points. In particular, we have established that
homeomorphisms of the Sobolev class satisfy the upper estimate of
the distortion of the modulus at the inner and boundary points of
the domain. In addition, we have proved that similar estimates of
capacity distortion occur at the inner points of the domain for open
discrete mappings. Also, we have shown that open discrete and closed
mappings satisfy some estimates of the distortion of the modulus of
families of paths at the boundary points. The results of the
manuscript are obtained mainly under the condition that the
so-called inner dilatation of mappings is locally integrable. The
main approach used in the proofs is the choice of admissible
functions, using the relations between the modulus and capacity, and
connections between different modulus of families of paths (similar
to Hesse, Ziemer and Shlyk equalities). In this context, we have
obtained some lower estimate of the modulus of families of paths in
Sobolev classes. The manuscript contains some examples related to
applications of obtained results to specific mappings.
References
L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, New York, 1966.
M. Cristea, Open discrete mappings having local ACLn inverses, Complex Variables and Elliptic Equations, 55 (2010), №1–3, 61–90.
M. Cristea, The limit mapping of generalized ring homeomorphisms, Complex Variables and Elliptic Equations, 61 (2016), 608–622.
M. Cristea, Eliminability results for mappings satisfying generalized modular inequalities, Complex Variables and Elliptic Equations, 64 (2019), 676–622.
O.P. Dovhopiatyi, E.A. Sevost’yanov, On compact classes of Beltrami solutions and Dirichlet problem, Complex Variables and Elliptic Equations, (2022), https://www.tandfonline.com/doi/abs/10.1080/17476933.2022.2040020.
H. Federer, Geometric Measure Theory, Springer, Berlin etc., 1969.
F.W. Gehring, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103 (1962), 353–393.
V.Ya. Gutlyanskii, V.I. Ryazanov, U. Srebro, E. Yakubov, The Beltrami equation: a geometric approach, Springer, New York etc., 2012.
J. Hesse, A p−extremal length and p−capacity equality, Ark. Mat., 13 (1975), 131–144.
D. Ilyutko, E. Sevost’yanov, Boundary behaviour of open discrete mappings on Riemannian manifolds, Sb. Math., 209 (2018), №5, 605–651.
T. Iwaniec, G. Martin, Geometrical function theory and non-linear analysis, Clarendon Press, Oxford, 2001.
P. Koskela, J. Onninen, Mappings of finite distortion: capacity and modulus inequalities, J. Reine Angew. Math., 599 (2006), 1–26.
D. Kovtonuyk, V. Ryazanov, New modulus estimates in Orlicz-Sobolev classes, Annals of the University of Bucharest (mathematical series), 5 (2014), №LXIII, 131–135.
O. Lehto, K. Virtanen, Quasiconformal mappings in the plane, Springer, New York etc., 1973.
T. Lomako, R. Salimov, E. Sevost’yanov, On equicontinuity of solutions to the Beltrami equations, Ann. Univ. Bucharest (math. series), LIX (2010), №2, 263–274.
O. Martio, S. Rickman, J. Vaisala, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I. Math., 448 (1969), 1–40.
O. Martio, S. Rickman, J. Vaisala, Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 465 (1970), 1–13.
O. Martio, S. Rickman, J. Vaisala, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 488 (1971), 1–31.
O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Monographs in Mathematics, Springer, New York etc., 2009.
V. Mazya, Sobolev Spaces, Springer-Verlag, Berlin, 1985.
R. Nakki, Boundary behavior of quasiconformal mappings in n-space, Ann. Acad. Sci. Fenn. Ser. A., 484 (1970), 1–50.
S. Rickman, Quasiregular mappings, Springer-Verlag, Berlin, 1993.
Yu.G. Reshetnyak, Space mappings with bounded distortion, Transl. of Math. Monographs, V.73, AMS, 1989.
V. Ryazanov, U. Srebro, E. Yakubov, On ring solutions of Beltrami equations, J. d’Anal. Math., 96 (2005), 117–150.
R.R. Salimov, E.A. Sevost’yanov, Analogs of the Ikoma-Schwartz lemma and Liouville theorem formappings with unbounded characteristics, Ukr. Math. J., 63 (2012), №10, 1551–1565.
S. Saks, Theory of the integral, Dover Publ. Inc., New York, 1964.
E. Sevost’yanov, R. Salimov, E. Petrov, On the removable of singularities of the Orlicz-Sobolev classes, J. Math. Sci., 222 (2017), №6, 723–740.
E.A. Sevost’yanov, On the local behavior of open discrete mappings from the Orlicz-Sobolev classes, Ukr. Math. J., 68 (2017), №9, 1447–1465.
E.A. Sevost’yanov, Boundary behavior and equicontinuity for families of mappings in terms of prime ends, St. Petersburg Math. J., 30 (2019), №6, 973–1005.
E.A. Sevost’yanov, On boundary behavior of some classes of mappings, J. Math. Sci., 243 (2019), №6, 934–948.
E.A. Sevost’yanov, S.A. Skvortsov, On the local behavior of the Orlicz-Sobolev classes, Journ. of Math. Sciences, 224 (2017), №4, 563–581.
E.A. Sevost’yanov, S.A. Skvortsov, Logarithmic H¨older continuous mappings and Beltrami equation, Analysis and Mathematical Physics, 11 (2021), №3, 138.
V.A. Shlyk, The equality between p-capacity and p-modulus, Siberian Mathematical Journal, 34 (1993), №6, 1196-–1200.
S. Stoїlow, L¸econs sur les principes topologiques de la th´eorie des fonctions analytiques, Gauthier-Villars,
Paris, 1956.
J. Vaisala, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math., V.229, Springer–Verlag, Berlin etc., 1971.
M. Vuorinen, Exceptional sets and boundary behavior of quasiregular mappings in n−space, Ann. Acad.Sci. Fenn. Ser. AI Math. Dissertattiones, 11 (1976), 1–44.
W.P. Ziemer, Extremal length and conformal capacity, Trans. Amer. Math. Soc., 126 (1967), №3, 460–473.
W.P. Ziemer, Extremal length and p−capacity, Michigan Math. J., 16 (1969), 43–51.
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