On the existence of solutions of quasilinear Beltrami equations with two characteristics
Abstract
UDC 517.5
We study Beltrami-type equations with two given complex characteristics. Under certain conditions imposed on the complex coefficients, we prove theorems on the existence of homeomorphic ACL-solutions of this equation. In addition, under some relatively weak conditions, we establish theorems on the existence of the corresponding continuous ACL-solutions of this equation that are logarithmic Hölder continuous in a given domain. ¨
References
L. Al'fors, Lekcii po kvazikonformnym otobrazheniyam, Mir, Moskva (1969).
K. Astala, T. Iwaniec, G. Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Univ. Press, Princeton, NY (2009). DOI: https://doi.org/10.1515/9781400830114
B. Bojarski, V. Gutlyanskii, V. Ryazanov, General Beltrami equations with two characteristics, Ukr. Math. Bull., 5, № 3, 305 – 326 (2008).
B. Bojarski, V. Gutlyanskii, V. Ryazanov, On the Beltrami equations with two characteristics, Complex Var. and Elliptic Equat., 54, № 10, 935 – 950 (2009), https://doi.org/10.1080/17476930903030069 DOI: https://doi.org/10.1080/17476930903030069
B. Bojarski, V. Gutlyanskii, V. Ryazanov, On existence and representation of solutions for general degenerate Beltrami equations, Complex Var. and Elliptic Equat., 59, № 1, 67 – 75 (2013), https://doi.org/10.1080/17476933.2013.795955 DOI: https://doi.org/10.1080/17476933.2013.795955
A. Golberg, R. Salimov, Nonlinear Beltrami equation, Complex Var. and Elliptic Equat., 65, № 1, 6 – 21 (2020), https://doi.org/10.1080/17476933.2019.1631292 DOI: https://doi.org/10.1080/17476933.2019.1631292
V. Ya. Gutlyanskii, V. I. Ryazanov, U. Srebro, E. Yakubov, The Beltrami equation: a geometric approach, Springer, New York etc. (2012), https://doi.org/10.1007/978-1-4614-3191-6 DOI: https://doi.org/10.1007/978-1-4614-3191-6
T. Lomako, R. Salimov, E. Sevost’yanov, On equicontinuity of solutions to the Beltrami equations, Ann. Univ. Bucharest. Math. Ser., 59, № 2, 263 – 274 (2010).
R. Salimov, M. Stefanchuk, On the local properties of solutions of the nonlinear Beltrami equation, J. Math. Sci., 248, 203 – 216 (2020), https://doi.org/10.1007/s10958-020-04870-6 DOI: https://doi.org/10.1007/s10958-020-04870-6
R. Salimov, M. Stefanchuk, Logarithmic asymptotics of the nonlinear Cauchy – Riemann – Beltrami equation, Ukr. Math. J., 73, № 4, 463 – 478 (2021), https://doi.org/10.37863/umzh.v73i3.6403 DOI: https://doi.org/10.1007/s11253-021-01936-9
E. A. Sevost'yanov, O kvazilinejnyh uravneniyah tipa Bel'trami s vyrozhdeniem, Mat. zametki, 90, vyp. 3, 445 – 453 (2011). DOI: https://doi.org/10.4213/mzm8406
Є. О. Севостьянов, Про iснування розв’язкiв рiвнянь Бельтрамi з умовами на оберненi дилатацiї, Укр. мат. вiсн., 18, № 2, 243 – 254 (2021).
E. A. Sevost’yanov, S. A. Skvortsov, Logarithmic Holder continuous mappings and Beltrami equation, Anal. and Math. Phys., Article 138 (2021), https://doi.org/10.1007/s13324-021-00573-6 DOI: https://doi.org/10.1007/s13324-021-00573-6
O. Martio, S. Rickman, J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1, 448, 1 – 40 (1969). DOI: https://doi.org/10.5186/aasfm.1969.448
V. Ryazanov, U. Srebro, E. Yakubov, Finite mean oscillation and the Beltrami equation, Israel J. Math., 153, 247 – 266 (2006), https://doi.org/10.1007/BF02771785 DOI: https://doi.org/10.1007/BF02771785
B. Bojarski, Generalized solutions of a system of differential equations of the first order of the elliptic type with discontinuous coefficients, Mat. Sb., 43(85), 451 – 503 (1957).
O. Lehto, K. Virtanen, Quasiconformal mappings in the plane, Springer, New York etc. (1973). DOI: https://doi.org/10.1007/978-3-642-65513-5
J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lect. Notes Math., 229, Springer-Verlag, Berlin etc. (1971). DOI: https://doi.org/10.1007/BFb0061216
V. I. Ryazanov, R. R. Salimov, E. A. Sevost’yanov, On convergence analysis of space homeomorphisms, Siberian Adv. Math., 23, № 4, 263 – 293 (2013), https://doi.org/10.3103/s1055134413040044 DOI: https://doi.org/10.3103/S1055134413040044
J. Maly, O. Martio, Lusin’s condition $N$ and mappings of the class $W^{1,n}$, J. reine und angew. Math., 458, 19 – 36 (1995), https://doi.org/10.1515/crll.1995.458.19 DOI: https://doi.org/10.1515/crll.1995.458.19
S. P. Ponomarev, $N - 1$-svojstvo otobrazhenij i uslovie $(N)$ Luzina, Mat. zametki, 58, 411 – 418 (1995).
E. A. Sevost’yanov, Equicontinuity of homeomorphisms with unbounded characteristic, Siberian Adv. Math., 23, № 2, 106 – 122 (2013). DOI: https://doi.org/10.3103/S1055134413020053
G. Federer, Geometricheskaya teoriya mery, Nauka, Moskva (1987).
YU. G. Reshetnyak, Prostranstvennye otobrazheniya s ogranichennym iskazheniem, Nauka, Novosibirsk (1982).
V. Ryazanov, U. Srebro, E. Yakubov, On convergence theory for Beltrami equations, Ukr. Mat. Visn., 5, № 4, 524 – 535 (2008).
S. Saks, Teoriya integrala, Izd-vo inostr. lit., Moskva (1949).
O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Mappings with finite length distortion, J. Anal. Math., 93, 215 – 236 (2004), https://doi.org/10.1007/BF02789308 DOI: https://doi.org/10.1007/BF02789308
Є. O. Sevost'yanov, S. O. Skvorcov, O. P. Dovgopyatij, Pro negomeomorfni vidobrazhennya z obernenoyu nerivnistyu Polec'kogo, Ukr. mat. visn., 17, № 3, 414 – 436 (2020).
V. Ryazanov, E. Sevost’yanov, Toward the theory of ring $Q$-homeomorphisms, Israel J. Math., 168, 101 – 118 (2008), https://doi.org/10.1007/s11856-008-1058-2 DOI: https://doi.org/10.1007/s11856-008-1058-2
O. Martio, V. Ryazanov, U. Srebro, E. Yakubov, Moduli in modern mapping theory, Springer Sci. + Business Media, LLC, New York (2009).
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