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	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. 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. ← Contents of issue


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