Boundary behavior of the Sobolev classes with critical exponent

Authors

  • O.S. Afanas’eva Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yansk
  • V.I. Ryazanov Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slov’yansk
  • R.R. Salimov Institute of Mathematics of the NAS of Ukraine, Kyiv

DOI:

https://doi.org/10.15407/dopovidi2019.10.003

Keywords:

boundary behavior., critical exponent, outer dilation, Sobolev’s classes

Abstract

The conditions for outer dilation KO(x, f ) and the boundaries of domains under which the homeomorphisms of the Sobolev classes W 1,1 loc  admit a continuous or homeomorphic extension to the boundary are founded.

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References

Martio, O., Ryazanov, V., Srebro, U. & Yakubov, E. (2009). Moduli in modern mapping theory. New York: Springer.

Kovtonyuk, D., Petkov, I. & Ryazanov, V. (2013). On the boundary behaviour of solutions to the Beltrami equations. Complex Var. Elliptic Equ., 58, No. 5, pp. 647-663. doi: https://doi.org/10.1080/17476933.2011.603494

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. 101-124 (in Russian). doi: https://doi.org/10.1090/S1061-0022-2014-01308-8

Kovtonyuk, D., Ryazanov, V., Salimov, R. & Sevost’yanov, E. (2012). On mappings in the Orlicz—Sobolev classes. Ann. Univ. Buchar. (Math. Ser.), 3 (LXI), pp. 67-78.

Tengvall, V. (2014). Differentiability in the Sobolev space W1, n−1 . Calc. Var. Part. Differ. Equat., 51, No. 1-2, pp. 381-399. doi: https://doi.org/10.1007/s00526-013-0679-4

Afanas’eva, O. S., Ryazanov, V. I. & Salimov, R. R. (2019). Toward the theory of the Sobolev classes with critical exponent. Dopov. Nac. akad. nauk Ukr., No. 8, pp. 3-8. doi: https://doi.org/10.15407/dopovidi2019.08.003

Kovtonyuk, D. A. & Ryazanov, V. I. (2008). On the theory of lower Q-homeomorphisms. Ukr. mat. visnyk, 5, No. 2, pp. 159-184 (in Russian).

Lomako, T. V. (2009). On the extension of some generalizations of quasiconformal mappings to the boundary. Ukr. mat. zhurn., 61, No. 10, pp. 1329-1337 (in Russian). doi: https://doi.org/10.1007/s11253-010-0298-6

Ryazanov, V. I. & Salimov, R. R. (2007). Weakly flat spaces and boundaries in the theory of mappings. Ukr. mat. visnyk, 4, No. 2,pp. 199-234 (in Russian).

Kovtonyuk, D. & Ryazanov, V. (2011). On the boundary behavior of generalized quasi-isometries. J. Anal. Math., 115, pp. 103-119. doi: https://doi.org/10.1007/s11854-011-0025-8

Gehring, F. W. & Martio, O. (1985). Quasiextremal distance domains and extension of quasiconformal mappings. J. Anal. Math., 45, pp. 181-206. doi: https://doi.org/10.1007/BF02792549

Afanas’eva, E. S., Ryazanov, V. I. & Salimov, R. R. (2018). On the theory of mappings of the Sobolev class with a critical exponent. Ukr. mat. visnyk, 15, No. 2, pp. 154-176 (in Russian).

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

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Published

24.04.2024

How to Cite

Afanas’eva, O., Ryazanov, V., & Salimov, R. (2024). Boundary behavior of the Sobolev classes with critical exponent . Reports of the National Academy of Sciences of Ukraine, (10), 3–10. https://doi.org/10.15407/dopovidi2019.10.003

Issue

No. 10 (2019): Reports of the National Academy of Sciences of Ukraine

Section

Mathematics

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