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This article is cited in 25 scientific papers (total in 25 papers)
Regularity of mappings inverse to Sobolev mappings
S. K. Vodopyanov
Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
Abstract:
For homeomorphisms $\varphi\colon\Omega\to \Omega'$ on Euclidean domains in $\mathbb R^n$, $n\geq2$, necessary and sufficient conditions ensuring that the inverse mapping belongs to a Sobolev class are investigated. The result obtained is used to describe a new two-index scale of homeomorphisms in some Sobolev class such that their inverses also form a two-index scale of mappings, in another Sobolev class.
This scale involves quasiconformal mappings and also homeomorphisms in the Sobolev class $W^1_{n-1}$ such that $\operatorname{rank}D\varphi(x)\leq n-2$ almost everywhere on the zero set of the Jacobian
$\det D\varphi(x)$.
Bibliography: 65 titles.
Keywords:
Sobolev class of mappings, approximate differentiability, distortion and codistortion of mappings, generalized quasiconformal mapping, composition operator.
DOI:
https://doi.org/10.4213/sm7792
 Full text:
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English version:
Sbornik: Mathematics, 2012, 203:10, 1383–1410
Bibliographic databases:
    
UDC:
517.518.23+517.548.2
MSC: 30C65, 46E35
Received: 29.09.2010 and 05.08.2012
Citation:
S. K. Vodopyanov, “Regularity of mappings inverse to Sobolev mappings”, Mat. Sb., 203:10 (2012), 3–32; Sb. Math., 203:10 (2012), 1383–1410
Citation in format AMSBIB
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http://mi.mathnet.ru/eng/msb7792https://doi.org/10.4213/sm7792 http://mi.mathnet.ru/eng/msb/v203/i10/p3
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N. A. Evseev, “Change of variables operators in weighted Sobolev spaces on Carnot groups”, J. Math. Sci., 221:6 (2017), 826–832
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M. V. Tryamkin, “Otsenki na moduli semeistv krivykh dlya otobrazhenii s vesovym ogranichennym $(p,q)$-iskazheniem”, Vladikavk. matem. zhurn., 17:3 (2015), 65–74
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A. V. Menovshchikov, “Composition operators in Orlicz–Sobolev spaces”, Siberian Math. J., 57:5 (2016), 849–859
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N. A. Evseev, “Bounded Composition Operator on Lorentz Spaces”, Math. Notes, 102:6 (2017), 763–769
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M. Brakalova, I. Markina, A. Vasil'ev, “Extremal functions for modules of systems of measures”, J. Anal. Math., 133:1 (2017), 335–359
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A. V. Menovshchikov, “Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space”, Siberian Math. J., 58:4 (2017), 649–662
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A. V. Menovshchikov, “The lower semicontinuity of distortion coefficients of the homeomorphisms inducing bounded composition operators of Sobolev–Orlicz spaces”, Siberian Math. J., 59:2 (2018), 332–340
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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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S. K. Vodopyanov, “Admissible changes of variables for Sobolev functions on (sub-)Riemannian manifolds”, Sb. Math., 210:1 (2019), 59–104
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L. Kleprlík, A. O. Molchanova, T. Roskovec, “Example of a smooth homeomorphism violating the Luzin $N^{-1}$ property”, Siberian Math. J., 60:5 (2019), 886–895
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Molchanova A., Vodopyanov S., “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, “On the Analytic and Geometric Properties of Mappings in the Theory of $\mathscr Q_{q,p}$-Homeomorphisms”, Math. Notes, 108:6 (2020), 889–894
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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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