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Mat. Sb., 2012, Volume 203, Number 10, Pages 3–32 (Mi msb7792)  

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.


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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
\by S.~K.~Vodopyanov
\paper Regularity of mappings inverse to Sobolev mappings
\jour Mat. Sb.
\yr 2012
\vol 203
\issue 10
\pages 3--32
\jour Sb. Math.
\yr 2012
\vol 203
\issue 10
\pages 1383--1410

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    This publication is cited in the following articles:
    1. S. K. Vodop'yanov, N. A. Evseev, “Isomorphisms of Sobolev spaces on Carnot groups and quasi-isometric mappings”, Siberian Math. J., 55:5 (2014), 817–848  mathnet  crossref  mathscinet  isi
    2. S. K. Vodop'yanov, “On the regularity of the Poletskii function under weak analytic assumptions on the given mapping”, Dokl. Math., 89:2 (2014), 157–161  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    3. A. N. Baykin, S. K. Vodop'yanov, “Capacity estimates, Liouville's theorem, and singularity removal for mappings with bounded $(p,q)$-distortion”, Siberian Math. J., 56:2 (2015), 237–261  mathnet  crossref  mathscinet  isi  elib  elib
    4. D. Henao, C. Mora-Corral, “Regularity of inverses of Sobolev deformations with finite surface energy”, J. Funct. Anal., 268:8 (2015), 2356–2378  crossref  mathscinet  zmath  scopus
    5. N. A. Evseev, “Change of variables operators in weighted Sobolev spaces on Carnot groups”, J. Math. Sci., 221:6 (2017), 826–832  mathnet  crossref  crossref
    6. N. A. Evseev, “Composition operators in weighted Sobolev spaces on the Carnot group”, Siberian Math. J., 56:6 (2015), 1042–1059  mathnet  crossref  crossref  mathscinet  isi  elib
    7. M. V. Tryamkin, “Modulus inequalities for mappings with weighted bounded $(p,q)$-distortion”, Siberian Math. J., 56:6 (2015), 1114–1132  mathnet  crossref  crossref  mathscinet  isi  elib
    8. M. V. Tryamkin, “Otsenki na moduli semeistv krivykh dlya otobrazhenii s vesovym ogranichennym $(p,q)$-iskazheniem”, Vladikavk. matem. zhurn., 17:3 (2015), 65–74  mathnet
    9. Vodop'yanov S.K., Molchanova A.O., “Variational problems of nonlinear elasticity in certain classes of mappings with finite distortion”, Dokl. Math., 92:3 (2015), 739–742  crossref  mathscinet  zmath  isi  scopus
    10. 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  mathnet  crossref  crossref  isi  elib  elib
    11. A. V. Menovshchikov, “Composition operators in Orlicz–Sobolev spaces”, Siberian Math. J., 57:5 (2016), 849–859  mathnet  crossref  crossref  isi  elib  elib
    12. N. A. Evseev, “Bounded Composition Operator on Lorentz Spaces”, Math. Notes, 102:6 (2017), 763–769  mathnet  crossref  crossref  isi  elib
    13. S. K. Vodop'yanov, N. A. Kudryavtseva, “On the Convergence of Mappings with $k$-Finite Distortion”, Math. Notes, 102:6 (2017), 878–883  mathnet  crossref  crossref  isi  elib
    14. M. Brakalova, I. Markina, A. Vasil'ev, “Extremal functions for modules of systems of measures”, J. Anal. Math., 133:1 (2017), 335–359  crossref  mathscinet  zmath  isi  scopus
    15. A. V. Menovshchikov, “Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space”, Siberian Math. J., 58:4 (2017), 649–662  mathnet  crossref  crossref  isi  elib  elib
    16. 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  mathnet  crossref  crossref  isi  elib
    17. 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  mathnet  crossref  elib
    18. S. K. Vodopyanov, “Basics of the quasiconformal analysis of a two-index scale of spatial mappings”, Siberian Math. J., 59:5 (2018), 805–834  mathnet  crossref  crossref  isi  elib
    19. 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  mathnet  crossref  crossref  isi  elib
    20. S. K. Vodopyanov, “Admissible changes of variables for Sobolev functions on (sub-)Riemannian manifolds”, Sb. Math., 210:1 (2019), 59–104  mathnet  crossref  crossref  adsnasa  isi  elib
    21. S. K. Vodopyanov, “Foundations of quasiconformal analysis of a two-index scale of spatial mappings”, Dokl. Math., 99:1 (2019), 23–27  crossref  crossref  zmath  isi  elib  scopus
    22. 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  mathnet  crossref  crossref  isi  elib
    23. Molchanova A., Vodopyanov S., “Injectivity Almost Everywhere and Mappings With Finite Distortion in Nonlinear Elasticity”, Calc. Var. Partial Differ. Equ., 59:1 (2019), 17  crossref  mathscinet  isi
    24. 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  mathnet  crossref  crossref
    25. S. K. Vodopyanov, “O regulyarnosti otobrazhenii, obratnykh k sobolevskim, i teoriya $\mathscr{Q}_{q,p}$-gomeomorfizmov”, Sib. matem. zhurn., 61:6 (2020), 1257–1299  mathnet  crossref
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