On modulus inequality of the order $p$ for the inner dilatation
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 in...
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| Format: | Article |
| Language: | German |
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Ivan Franko National University of Lviv
2023-06-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/376 |
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| author | R. R. Salimov E. O. Sevost'yanov V. A. Targonskii |
| author_facet | R. R. Salimov E. O. Sevost'yanov V. A. Targonskii |
| author_sort | R. R. Salimov |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-f68a9f81c49e4fab9731cd4c1aa9abd0 |
| institution | Matheson Library |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2023-06-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-f68a9f81c49e4fab9731cd4c1aa9abd02025-07-08T09:05:54ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202023-06-0159214115510.30970/ms.59.2.141-155376On modulus inequality of the order $p$ for the inner dilatationR. R. Salimov0E. O. Sevost'yanov1V. A. Targonskii2Institute of Mathematics of NAS of Ukraine, 3 Tereschenkivska Str., 01 024 Kiev-4<p>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</p><p> </p>Zhytomyr Ivan Franko State University, 40 Bol'shaya Berdichevskaya Str., 10 008 ZhytomyrThe 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.http://matstud.org.ua/ojs/index.php/matstud/article/view/376mappings, mappings with bounded and finite distortion, equicontinuity, moduli of families of paths |
| spellingShingle | R. R. Salimov E. O. Sevost'yanov V. A. Targonskii On modulus inequality of the order $p$ for the inner dilatation Математичні Студії mappings, mappings with bounded and finite distortion, equicontinuity, moduli of families of paths |
| title | On modulus inequality of the order $p$ for the inner dilatation |
| title_full | On modulus inequality of the order $p$ for the inner dilatation |
| title_fullStr | On modulus inequality of the order $p$ for the inner dilatation |
| title_full_unstemmed | On modulus inequality of the order $p$ for the inner dilatation |
| title_short | On modulus inequality of the order $p$ for the inner dilatation |
| title_sort | on modulus inequality of the order p for the inner dilatation |
| topic | mappings, mappings with bounded and finite distortion, equicontinuity, moduli of families of paths |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/376 |
| work_keys_str_mv | AT rrsalimov onmodulusinequalityoftheorderpfortheinnerdilatation AT eosevostyanov onmodulusinequalityoftheorderpfortheinnerdilatation AT vatargonskii onmodulusinequalityoftheorderpfortheinnerdilatation |