| Autor: |
SALIMOV, R. R., SEVOST'YANOV, E. O., TARGONSKII, V. A. |
| Předmět: |
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| Zdroj: |
Matematychni Studii; 2023, Vol. 59 Issue 2, p141-155, 15p |
| Abstrakt: |
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. [ABSTRACT FROM AUTHOR] |
| Databáze: |
Complementary Index |
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