On modulus  inequality of the order $p$ for the inner dilatation

R. R. Salimov; E. O.  Sevost'yanov; V. A.  Targonskii

doi:10.30970/ms.59.2.141-155

R. R. Salimov Institute of Mathematics of NAS of Ukraine, 3 Tereschenkivska Str., 01 024 Kiev-4
E. O. Sevost'yanov Zhytomyr Ivan Franko State University, Bol'shaya Berdichevskaya Str., 40, Zhytomyr, 10 008, UKRAINE; Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Dobrovo'skogo Str., 1, Slavyansk, 84 100,  UKRAINE  https://orcid.org/0000-0001-7892-6186
V. A. Targonskii Zhytomyr Ivan Franko State University, 40 Bol'shaya Berdichevskaya Str., 10 008 Zhytomyr

DOI: https://doi.org/10.30970/ms.59.2.141-155

Keywords: mappings, mappings with bounded and finite distortion, equicontinuity, moduli of families of paths

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.

Author Biographies

R. R. Salimov, Institute of Mathematics of NAS of Ukraine, 3 Tereschenkivska Str., 01 024 Kiev-4

Institute of Mathematics of NAS of Ukraine, 3 Tereschenkivska Str., 01 024 Kiev-4

E. O. Sevost'yanov, Zhytomyr Ivan Franko State University, Bol'shaya Berdichevskaya Str., 40, Zhytomyr, 10 008, UKRAINE; Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Dobrovo'skogo Str., 1, Slavyansk, 84 100,  UKRAINE

Profeesor of Math. Analysis Dept. of Zhytomyr Ivan Franko State University; Leading Researcher of Function Theory Dept., Institute of Applied Mathematics and Mechanics of NAS of Ukraine

V. A. Targonskii, Zhytomyr Ivan Franko State University, 40 Bol'shaya Berdichevskaya Str., 10 008 Zhytomyr

Zhytomyr Ivan Franko State University, 40 Bol'shaya Berdichevskaya Str., 10 008 Zhytomyr

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