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Toward the theory of Orlicz-Sobolev classes


Authors: D. A. Kovtonyuk, V. I. Ryazanov, R. R. Salimov and E. A. Sevost′yanov
Translated by: V. I. Ryazanov
Original publication: Algebra i Analiz, tom 25 (2013), nomer 6.
Journal: St. Petersburg Math. J. 25 (2014), 929-963
MSC (2010): Primary 46E35
DOI: https://doi.org/10.1090/S1061-0022-2014-01324-6
Published electronically: September 8, 2014
MathSciNet review: 3234840
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Abstract | References | Similar Articles | Additional Information

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.


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Additional 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
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

DOI: https://doi.org/10.1090/S1061-0022-2014-01324-6
Keywords: Moduli of families of curves and surfaces, mappings with bounded and finite distortion, differentiability, Lusin and Sard properties, Sobolev classes, Orlicz--Sobolev classes, boundary and local behavior.
Received by editor(s): May 26, 2013
Published electronically: September 8, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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