| Afanas’eva, OS 1Ryazanov, VI 2Salimov, RR 1Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Sloviansk 2Institute of Mathematics of the NAS of Ukraine, Kyiv |
| Dopov. Nac. akad. nauk Ukr. 2019, 8:3-8 |
| https://doi.org/10.15407/dopovidi2019.08.003 |
| Section: Mathematics |
| Language: Russian |
Abstract: It is established that an arbitrary homeomorphism f in the Sobolev class \[W^{1,n-1}_{loc}\] with the outer dilatation \[K_{0}(x,f) \in L^{n-1}_{loc}\] is the socalled
lower Q - homeomorphism with \[Q=K_{0} (x,f)\] and the ring Q* homeomorphism with \[Q*=K^{n-1}_{0} (x,f)\]. These results make it possible to research the local and boundary behaviors of the mappings \[W^{1,n-1}_{loc}\]
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| Keywords: critical exponent, lower and ring Q homeomorphisms, outer and inner dilatations, Sobolev’s classes |
1. Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2009). Moduli in modern mapping theory. New York: Springer.
2. Rado, T. & Reichelderfer, P. V. (1955). Continuous transformations in analysis. Berlin: Springer. doi: https://doi.org/10.1007/978-3-642-85989-2
3. Saks, S. (1949). Integral teory. Ìoscow: Izdvo Inostr. lit. (in Russian).
4. Ryazanov, V., Srebro, U. & Yakubov, E. (2005). On ring solutions of Beltrami equation. J. Anal. Math., 96, pp. 117150. doi: https://doi.org/10.1007/BF02787826
5. Ryazanov, V. I. & Sevostyanov, E. A. (2007). Equicontinuous classes of ring homeomorphisms. Sib. matem. zhurn., 48, No. 6, pp. 13611376. doi: https://doi.org/10.1007/s11202-007-0111-4
6. Gutlyanskii, V., Ryazanov, V., Srebro, U. & Yakubov, E. (2012). The Beltrami equation: A geometric approach. Developments of Mathematics, Vol. 26. New York etc.: Springer. doi: https://doi.org/10.1007/978-1-4614-3191-6
7. Kovtonyuk, D. A. & Ryazanov, V. I. (2008). On the theory of lower Q homeomorphisms. Ukr. mat. vestnik, 5, No. 2, pp. 159184 (in Russian).
8. Kovtonyuk, D. A., Petkov, I. V., Ryazanov, V. I. & Salimov, R. R. (2013). Boundary behavior and the Dirichlet problem for the Beltrami equations. Algebra i analiz, 25, No. 4, pp. 101124 (in Russian). doi: https://doi.org/10.1090/S1061-0022-2014-01308-8
9. Kovtonyuk, D. A., Ryazanov, V. I., Salimov, R. R. & Sevostyanov, E. A. (2013). On the theory of OrlichSobolev classes. Algebra i analiz, 25, No. 6, pp. 50102 (in Russian). doi: https://doi.org/10.1090/S1061-0022-2014-01324-6
10. Kovtonyuk, D. A., Salimov, R. R. & Sevostyanov, E. A. (2013). On the theory of mappings of classes Sobolev and OrlichSobolev. Kyiv: Naukova Dumka (in Russian).
11. Tengvall, V. (2014). Differentiability in the Sobolev space 1, n 1 W − . Calc. Var. Part. Differ. Equat., 51, No. 12, pp. 381399. doi: https://doi.org/10.1007/s00526-013-0679-4
12. Csörnyei, M., Hencl, S. & Maly, J. (2010). Homeomorphisms in the Sobolev space 1, n 1 W − . J. Reine Angew. Math., No. 644, pp. 221235. doi: https://doi.org/10.1515/crelle.2010.057
13. Ponomarev, S. P. (1987). On N properties of classes 1 Wp homeomorphisms. Sib. matem. zhurn., 28, No. 2, pp. 140148 (in Russian). doi: https://doi.org/10.1007/BF00970876
14. Mazya, V. G. (1985). Spaces S.L. Sobolev. Leningrad: Izdvo Leningr. unta (in Russian).
15. Kuratovskij, Ê. (1966). Òopology. Vol. 1. Ìoscow: Ìir (in Russian).
