Toward the theory of Orlicz–Sobolev classes
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D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov and E. A. Sevost′yanov
Translated by: V. I. Ryazanov - St. Petersburg Math. J. 25 (2014), 929-963
- DOI: https://doi.org/10.1090/S1061-0022-2014-01324-6
- Published electronically: September 8, 2014
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Abstract:
It is shown that, under a Calderón type condition on the function $\varphi$, the continuous open mappings that belong to the Orlicz–Sobolev classes $W^{1,\varphi }_{\mathrm {loc}}$ have total differential almost everywhere; this generalizes the well-known theorems of Gehring–Lehto–Menchoff in the case of ${\mathbb R}^2$ and of Väisälä in ${\mathbb R}^n$, $n\geq 3$. Appropriate examples show that the Calderón type condition is not only sufficient but also necessary. Moreover, under the same condition on $\varphi$, it is also proved that the continuous mappings of class $W^{1,\varphi }_{\mathrm {loc}}$ and, in particular, of class $W^{1,p}_{\mathrm {loc}}$ for $p>n-1$ have Lusin’s $(N)$-property on a.e. hyperplane. On that basis, it is shown that, under the same condition on $\varphi$, the homeomorphisms $f$ with finite distortion of class $W^{1,\varphi }_{\mathrm {loc}}$ and, in particular, those belonging to $W^{1,p}_{\mathrm {loc}}$ for $p>n-1$, are what is called lower $Q$-homeomorphisms, where $Q$ is equal to their outer dilatation $K_f$; also, they are so-called ring $Q_*$-homeomorphisms with $Q_*=K_{f}^{n-1}$. The latter fact makes it possible to fully apply the theory of the boundary and local behavior of the ring and lower $Q$-homeomorphisms, as developed earlier by the authors, to the study of mappings in the Orlicz–Sobolev classes.References
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Bibliographic Information
- D. A. Kovtonyuk
- Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Roze Luxemburg str. 74, Donetsk 83114, Ukraine
- Email: denis_kovtonyuk@bk.ru
- V. I. Ryazanov
- Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Roze Luxemburg str. 74, Donetsk 83114, Ukraine
- Email: vlryazanov1@rambler.ru
- R. R. Salimov
- Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Roze Luxemburg str. 74, Donetsk 83114, Ukraine
- MR Author ID: 824987
- Email: salimov@rambler.ru
- E. A. Sevost′yanov
- Affiliation: Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Roze Luxemburg str. 74, Donetsk 83114, Ukraine
- Email: brusin2006@rambler.ru
- Received by editor(s): May 26, 2013
- Published electronically: September 8, 2014
- © Copyright 2014 American Mathematical Society
- Journal: St. Petersburg Math. J. 25 (2014), 929-963
- MSC (2010): Primary 46E35
- DOI: https://doi.org/10.1090/S1061-0022-2014-01324-6
- MathSciNet review: 3234840