On Poletsky-type Modulus Inequalities for Some Classes of Mappings

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/w5793-5981-8894-o On Poletsky-type Modulus Inequalities for Some Classes of Mappings Vodopyanov, S. K. Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 4. Abstract: It is well-known that the theory of mappings with bounded distortion was laid by Yu. G. Reshetnyak in 60-th of the last century [1]. In papers [2, 3], there was introduced the two-index scale of mappings with weighted bounded \((q, p)\)-distortion. This scale of mappings includes, in particular, mappings with bounded distortion mentioned above (under \(q=p=n\) and the trivial weight function). In paper [4], for the two-index scale of mappings with weighted bounded \((q, p)\)-distortion, the Poletsky-type modulus inequality was proved under minimal regularity; many examples of mappings were given to which the results of [4] can be applied. In this paper we show how to apply results of [4] to one such class. Another goal of this paper is to exhibit a new class of mappings in which Poletsky-type modulus inequalities is valid. To this end, for \(n=2\), we extend the validity of the assertions in [4] to the limiting exponents of summability: \(1 < q\leq p \leq \infty\). This generalization contains, as a special case, the results of recently published papers. As a consequence of our results, we also obtain estimates for the change in capacitу of condensers. Keywords: quasiconformal analysis, Sobolev space, modulus of a family of curves, modulus estimate Language: English Download the full text For citation: Vodopyanov, S. K. On Poletsky-type Modulus Inequalities for Some Classes of Mappings, Vladikavkaz Math. J., 2022, vol. 24, no. 4, pp. 58-69. DOI 10.46698/w5793-5981-8894-o + References 1. Reshetnyak Yu. G. Space Mappings with Bounded Distortion, Providence, Amer. Math. Soc., 1989. 2. Vodopyanov, S. K. Basics of the Quasiconformal Analysis of a Two-Index Scale of Space Mappings, Siberian Mathematical Journal, 2018, vol. 59, no. 5, pp. 805-834. DOI: 10.1134/S0037446618050075. 3. Vodopyanov, S. K. Differentiability of Mappings of the Sobolev Space \(W^1_{n-1}\) with Conditions on the Distortion Function, Siberian Mathematical Journal, 2018, vol. 59, no. 6, pp. 983-1005. DOI: 10.1134/S0037446618060034. 4. Vodopyanov, S. K. Moduli Inequalities for \(W^1_{n-1, loc}\)-Mappings with Weighted Bounded \((q, p)\)-Distortion, Complex Variables and Elliptic Equations, 2021, vol. 66, no. 6-7, pp. 1037-1072. DOI: 10.1080/17476933.2020.1825396. 5. Vaisala J. Lectures on \(n\)-Dimensional Quasiconformal Mappings, Lecture Notes in Mathematics, vol. 229, Berlin-Heidelberg-New York, Springer, 1971. 6. Fuglede, B. Extremal Length and Functional Completion, Acta Mathematica, 1957, vol. 98, pp. 171-219. DOI: 10.1007/BF02404474. 7. Vodopyanov, S. K. The Regularity of Inverses to Sobolev Mappings and the Theory of \(Q_{q,p}\)-Homeomorphisms, Siberian Mathematical Journal, 2020, vol. 61, no. 6, pp. 1002-1038. DOI: 10.1134/S0037446620060051. 8. Vodopyanov, S. K. and Tomilov, A. O. Functional and Analytic Properties of a Class of Mappings in Quasi-Conformal Analysis, Izvestiya: Mathematics, 2021, vol. 85, no. 5, pp. 883-931. DOI: 10.1070/IM9082. 9. Vodopyanov, S. K. Regularity of Mappings Inverse to Sobolev Mappings, Sbornik: Mathematics, 2012, vol. 203, no. 10, pp. 1383-1410. DOI: 10.1070/SM2012v203n10ABEH004269. 10. Hencl, S. and Koskela, P. Regularity of the Inverse of a Planar Sobolev Homeomorphism, Archive for Rational Mechanics and Analysis, 2006, vol. 180, pp. 75-95. DOI: 10.1007/s00205-005-0394-1. 11. Rickman S. Quasiregular mappings, Berlin, Springer-Verlag, 1993, 213 p. 12. Poletsky, E. A. The Modulus Method for Nonhomeomorphic Quasiconformal Mappings, Mathematics of the USSR-Sbornik, 1970, vol. 12, no. 2, pp. 260-270. DOI: 10.1070/SM1970v012n02ABEH000921. 13. Salimov, R. R., Sevost'yanov, E. A. and Targonskii, V. A. On Modulus Inequality of the Order \(p\) for the Inner Dilatation. arXiv - MATH - Complex Variables. 2022. DOI:arxiv-2204.07870. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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