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This article is cited in 13 scientific papers (total in 13 papers)
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
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UDC:
517.54
Received: 06.10.2014
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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\jour Siberian Math. J.
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\vol 56
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http://mi.mathnet.ru/eng/smj2639 http://mi.mathnet.ru/eng/smj/v56/i2/p290
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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
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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
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M. V. Tryamkin, “Boundary Correspondence for Homeomorphisms with Weighted Bounded $(p,q)$-Distortion”, Math. Notes, 102:4 (2017), 591–595
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S. K. Vodop'yanov, N. A. Kudryavtseva, “On the Convergence of Mappings with $k$-Finite Distortion”, Math. Notes, 102:6 (2017), 878–883
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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
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S. K. Vodopyanov, “Basics of the quasiconformal analysis of a two-index scale of spatial mappings”, Siberian Math. J., 59:5 (2018), 805–834
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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
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A. Molchanova, S. Vodopyanov, “Injectivity almost everywhere and mappings with finite distortion in nonlinear elasticity”, Calc. Var. Partial Differ. Equ., 59:1 (2019), 17
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S. K. Vodopyanov, “Foundations of quasiconformal analysis of a two-index scale of spatial mappings”, Dokl. Math., 99:1 (2019), 23–27
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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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