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 Sibirsk. Mat. Zh., 2015, Volume 56, Number 2, Pages 290–321 (Mi smj2639)

Capacity estimates, Liouville's theorem, and singularity removal for mappings with bounded $(p,q)$-distortion

A. N. Baykinab, S. K. Vodop'yanovab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Novosibirsk State University, Novosibirsk, Russia

Abstract: The mappings with bounded weighted $(p,q)$-distortion are natural generalizations of the class of mappings with bounded distortion which appears as a doubly indexed scale for $p=q=n$ in the absence of weight functions. In case $n-1<q\le p=n$, the mappings with bounded $(p,q)$-distortion were studied previously in a series of articles under the additional assumption that the mapping enjoys Luzin's $\mathscr N$-property. In this article we present the first facts of the theory of mappings with bounded $(p,q)$-distortion which are obtained without additional analytical assumptions. The core of the theory consists of the new analytical properties of pushforward functions; in particular, we prove that the gradient of the pushforward function vanishes almost everywhere on the image of the branch set. Some estimates are given on the capacity of the images of condensers under mappings with bounded $(p,q)$-distortion. We obtain Liouville-type theorems and the singularity removal theorems for the mappings of this class, and we apply these theorems to classifying manifolds.

Keywords: mappings with bounded weighted $(p,q)$-distortion, capacity estimate, Liouville-type theorem, singularity removal.

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English version:
Siberian Mathematical Journal, 2015, 56:2, 237–261

Bibliographic databases:

UDC: 517.54

Citation: A. N. Baykin, S. K. Vodop'yanov, “Capacity estimates, Liouville's theorem, and singularity removal for mappings with bounded $(p,q)$-distortion”, Sibirsk. Mat. Zh., 56:2 (2015), 290–321; Siberian Math. J., 56:2 (2015), 237–261

Citation in format AMSBIB
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\by A.~N.~Baykin, S.~K.~Vodop'yanov
\paper Capacity estimates, Liouville's theorem, and singularity removal for mappings with bounded $(p,q)$-distortion
\jour Sibirsk. Mat. Zh.
\yr 2015
\vol 56
\issue 2
\pages 290--321
\mathnet{http://mi.mathnet.ru/smj2639}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3381241}
\elib{https://elibrary.ru/item.asp?id=23112840}
\transl
\jour Siberian Math. J.
\yr 2015
\vol 56
\issue 2
\pages 237--261
\crossref{https://doi.org/10.1134/S0037446615020056}
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. M. V. Tryamkin, “On asymptotic curves and values in the theory of mappings with weighted bounded distortion”, Sib. elektron. matem. izv., 12 (2015), 688–697
2. M. V. Tryamkin, “Modulus inequalities for mappings with weighted bounded $(p,q)$-distortion”, Siberian Math. J., 56:6 (2015), 1114–1132
3. M. V. Tryamkin, “Otsenki na moduli semeistv krivykh dlya otobrazhenii s vesovym ogranichennym $(p,q)$-iskazheniem”, Vladikavk. matem. zhurn., 17:3 (2015), 65–74
4. M. V. Tryamkin, “Asymptotic curves and asymptotic values for mappings with weighted bounded $(p,q)$-distortion”, Russian Math. (Iz. VUZ), 60:1 (2016), 76–80
5. S. K. Vodop'yanov, A. O. Molchanova, “Lower semicontinuity of mappings with bounded $(\theta,1)$-weighted $(p,q)$-distortion”, Siberian Math. J., 57:5 (2016), 778–787
6. M. V. Tryamkin, “Boundary Correspondence for Homeomorphisms with Weighted Bounded $(p,q)$-Distortion”, Math. Notes, 102:4 (2017), 591–595
7. S. K. Vodop'yanov, N. A. Kudryavtseva, “On the Convergence of Mappings with $k$-Finite Distortion”, Math. Notes, 102:6 (2017), 878–883
8. N. A. Kudryavtseva, S. K. Vodopyanov, “On the convergence of mappings with $k$-finite distortion”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 88–100
9. S. K. Vodopyanov, “Basics of the quasiconformal analysis of a two-index scale of spatial mappings”, Siberian Math. J., 59:5 (2018), 805–834
10. S. K. Vodopyanov, “Differentiability of mappings of the Sobolev space $W^1_{n-1}$ with conditions on the distortion function”, Siberian Math. J., 59:6 (2018), 983–1005
11. A. Molchanova, S. Vodopyanov, “Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity”, Calc. Var. Partial Differ. Equ., 59:1 (2019), 17
12. S. K. Vodopyanov, “Foundations of quasiconformal analysis of a two-index scale of spatial mappings”, Dokl. Math., 99:1 (2019), 23–27
13. S. K. Vodopyanov, “O regulyarnosti otobrazhenii, obratnykh k sobolevskim, i teoriya $\mathscr{Q}_{q,p}$-gomeomorfizmov”, Sib. matem. zhurn., 61:6 (2020), 1257–1299
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