Logarithmic asymptotics of the nonlinear Cauchy – Riemann – Beltrami equation

  • R. R. Salimov Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv
  • M. V. Stefanchuk Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv

Abstract

UDC 517.54; 517.12

We investigate regular solutions of the nonlinear Cauchy – Riemann – Beltrami equation in terms of lower limits and solve an extreme problem for the disk image area functional on a certain class of solutions to the nonlinear Cauchy – Riemann – Beltrami system.

References

V. Gutlyanskii, V. Ryazanov, U. Srebro, E. Yakubov, The Beltrami equations: a geometric approach , Dev. Math., 26, 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

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

V. Gutlyanskii, V. Ryazanov, U. Srebro, E. Yakubov, On recent advances in the degenerate Beltrami equations , Ukr. Mat. Visn., 4, № 7, 467 – 515 (2010), https://doi.org/10.1007/s10958-011-0355-1 DOI: https://doi.org/10.1007/s10958-011-0355-1

U. Srebro, E. Yakubov, The Beltrami equation , Handbook in Complex Analysis: Geometric Function Theory, 2, 555 – 597 (2005), https://doi.org/10.1016/S1874-5709(05)80016-2 DOI: https://doi.org/10.1016/S1874-5709(05)80016-2

E. A. Sevost’yanov, On quasilinear Beltrami-type equations with degeneration , Math. Notes, 90, № 3-4, 431 – 438 (2011). DOI: https://doi.org/10.1134/S0001434611090112

E. A. Sevost’yanov, Generalization of one Poletskii lemma to classes of space mappings , Ukr. Math. J., 61, № 7, 1151 – 1157 (2009).

D. A. Kovtonyuk, R. R. Salimov, E. A. Sevost`yanov, K teorii otobrazhenij klassov Soboleva i Orlicha – Soboleva, Nauk. dumka, Kiev (2013).

M. Cristea, Local homeomorphisms having local ${ ACL}^n$ inverses, Complex Var. and Elliptic Equat., 53 , № 1, 77 – 99 (2008), https://doi.org/10.1080/17476930701666924 DOI: https://doi.org/10.1080/17476930701666924

M. Cristea, Open, discrete mappings having local ${ ACL}^n$ inverses, Complex Var. and Elliptic Equat., 55 , № 1-3, 61 – 90 (2010), https://doi.org/10.1080/17476930902998985 DOI: https://doi.org/10.1080/17476930902998985

M. Cristea, Local homeomorphisms satisfying generalized modular inequalities, Complex Var. and Elliptic Equat., 59 , № 10, 1363 – 1387 (2014), https://doi.org/10.1080/17476933.2013.845176 DOI: https://doi.org/10.1080/17476933.2013.845176

M. Cristea, Some properties of open, discrete generalized ring mappings, Complex Var. and Elliptic Equat., 61, № 5, 623 – 643 (2016), https://doi.org/10.1080/17476933.2015.1108311 DOI: https://doi.org/10.1080/17476933.2015.1108311

K. Astala, T. Iwaniec, G. Martin, Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Math. Ser., 48 (2009). DOI: https://doi.org/10.1515/9781400830114

C.-Y. Guo, M. Kar, Quantitative uniqueness estimates for p-Laplace type equations in the plane, Nonlinear Analysis: Theory, Methods and Appl., 143, 19 – 44 (2016), https://doi.org/10.1016/j.na.2016.04.015 DOI: https://doi.org/10.1016/j.na.2016.04.015

M. A. Lavrent`ev, B. V. Shabat, Geometricheskie svojstva reshenij nelinejny`kh sistem uravnenij s chastny`mi proizvodny`mi, Dokl. AN SSSR, 112, № 5, 810 – 811 (1957).

M. A. Lavrent`ev, Obshhaya zadacha teorii kvazikonformny`kh otobrazhenij ploskikh oblastej, Mat. sb., 21(63), № 2, 285 – 320 (1947).

M. A. Lavrent`ev, Variaczionny`j metod v kraevy`kh zadachakh dlya sistem uravnenij e`llipticheskogo tipa, Izd-vo AN SSSR, Moskva (1962).

B. V. Shabat, Geometricheskij smy`sl ponyatiya e`lliptichnosti, Uspekhi mat. nauk, 12, № 6 (78), 181 – 188 (1957).

B. V. Shabat, K ponyatiyu proizvodnoj sistemy` v smy`sle M. A. Lavrent`eva , Dokl. AN SSSR, 136, № 6, 1298 – 1301 (1961).

R. Kuhnau, Minimal surfaces and quasiconformal mappings in the mean, Zb. pracz` In-tu matematiki NAN Ukrayini, 7, № 2, 104 – 131 (2010).

S. L. Krushkalʹ, R. Kyunau, Квазиконформные отображения—новые методы и приложения. (Russian) [[Quasiconformal mappings—new methods and applications]] Nauka Sibirsk. Otdel., Novosibirsk, (1984)

T. Adamowicz, On $p$-harmonic mappings in the plane, Nonlinear Anal., 71, № 1-2, 502 – 511 (2009), https://doi.org/10.1016/j.na.2008.10.088 DOI: https://doi.org/10.1016/j.na.2008.10.088

G. Aronsson, On certain $p$-harmonic functions in the plane, Manuscripta Math., 61, № 1, 79 – 101 (1988), https://doi.org/10.1007/BF01153584 DOI: https://doi.org/10.1007/BF01153584

A. S. Romanov, Emkostny`e sootnosheniya v ploskom chety`rekhstoronnike, Sib. mat. zhurn., 49, № 4, 886 – 897 (2008).

B. Bojarski, T. Iwaniec, $p$-Harmonic equation and quasiregular mappings, Banach Center Publ., 19, № 1, 25 – 38 (1987).

K. Astala, A. Clop, D. Faraco, J. J¨a¨askel¨ainen, A. Koski, Nonlinear Beltrami operators. Schauder estimates and bounds for the Jacobian, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 34, № 6, 1543 – 1559 (2017), https://doi.org/10.1016/j.anihpc.2016.10.008 DOI: https://doi.org/10.1016/j.anihpc.2016.10.008

M. Carozza, F. Giannetti, A. Passarelli di Napoli, C. Sbordone, R. Schiattarella, Bi-Sobolev mappings and $K_p$ distortions in the plane, J. Math. Anal. and Appl, 457, № 2, 1232 – 1246 (2018), https://doi.org/10.1016/j.jmaa.2017.02.050 DOI: https://doi.org/10.1016/j.jmaa.2017.02.050

A. Golberg, R. Salimov, M. Stefanchuk, Asymptotic dilation of regular homeomorphisms, Complex Anal. and Oper. Theory, 13, № 6, 2813 – 2827 (2019), https://doi.org/10.1007/s11785-018-0833-2 DOI: https://doi.org/10.1007/s11785-018-0833-2

R. R. Salimov, M. V. Stefanchuk, On the local properties of solutions of the nonlinear Beltrami equation, J. Math. Sci., 248, 203 – 216 (2020). DOI: https://doi.org/10.1007/s10958-020-04870-6

E. A. Sevost`yanov, R. R. Salimov, O neravenstve tipa Vyajsyalya dlya uglovoj dilataczii otobrazhenij i nekotory`kh ego prilozheniyakh, Ukr. mat. visn., 12, № 4, 511 – 538 (2015).

M. Cristea, On Poleckii-type modular inequalities, Complex Var. and Elliptic Equat., https://doi.org/10.1080/17476933.2020.1783660 DOI: https://doi.org/10.1080/17476933.2020.1783660

A. Golberg, R. Salimov, Nonlinear Beltrami equation, Complex Var. and Elliptic Equat., 65, № 1, 6 – 21 (2019), https://doi.org/10.1080/17476933.2019.1631292 DOI: https://doi.org/10.1080/17476933.2019.1631292

O. Lehto, K. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag, New York (1973). DOI: https://doi.org/10.1007/978-3-642-65513-5

B. Bojarski, V. Gutlyanskii, O. Martio, V. Ryazanov, Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane, Tracts Math., 19, Warsaw etc. (2013), https://doi.org/10.4171/122 DOI: https://doi.org/10.4171/122

E. Reich, H. Walczak, On the behavior of quasiconformal mappings at a point, Trans. Amer. Math. Soc., 117, 338 – 351 (1965), https://doi.org/10.2307/1994211 DOI: https://doi.org/10.2307/1994211

A. Schatz, On the local behavior of homeomorphic solutions of Beltrami equation, Duke Math. J., 35, 289 – 306 (1968). DOI: https://doi.org/10.1215/S0012-7094-68-03528-X

C. Andreian Cazacu, Influence of the orientation of the characteristic ellipses on the properties of the quasiconformal mappings, Proc. Rom. Finn. Sem., Romania (1969), Publ. House Acad. Soc. Rep. Rom., Bucharest (1971), p. 65 – 85.

M. A. Brakalova, J. A. Jenkins, On solutions of the Beltrami equation, J. Anal. Math., 76, 67 – 92 (1998), https://doi.org/10.1007/BF02786930 DOI: https://doi.org/10.1007/BF02786930

V. Gutlyanskii, T. Sugawa, On Lipschitz continuity of quasiconformal mappings, Rep. Univ. Jyv¨askyl¨a Dep. Math. Stat., 83, 91 – 108 (2001).

V. Gutlyanskii, A. Golberg, On Lipschitz continuity of quasiconformal mappings in space, J. Anal. Math., 109, 233 – 251 (2009), https://doi.org/10.1007/s11854-009-0032-1 DOI: https://doi.org/10.1007/s11854-009-0032-1

V. Gutlyanskii, A. Golberg, Rings and Lipschitz continuity of quasiconformal mappings, Analysis and Math. phys. Trends Math., Birkh¨auser, Basel , p. 187 – 192 (2009), https://doi.org/10.1007/978-3-7643-9906-1_10 DOI: https://doi.org/10.1007/978-3-7643-9906-1_10

V. Gutlyanskii, O. Martio, T. Sugawa, M. Vuorinen, On the degenerate Beltrami equation, Trans. Amer. Math. Soc., 357, 875 – 900 (2005), https://doi.org/10.1090/S0002-9947-04-03708-0 DOI: https://doi.org/10.1090/S0002-9947-04-03708-0

V. Ryazanov, R. Salimov, U. Srebro, E. Yakubov, On boundary value problems for the Beltrami equations, Contemp. Math., 591, 211 – 242 (2013), https://doi.org/10.1090/conm/591/11839 DOI: https://doi.org/10.1090/conm/591/11839

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

K. Ikoma, On the distortion and correspondence under quasiconformal mappings in space, Nagoya Math. J., 25, 175 – 203 (1965). DOI: https://doi.org/10.1017/S0027763000011521

S. Saks, Teoriya integrala, Izd-vo inostr. lit., Moskva (1949).

Published
19.03.2021
How to Cite
Salimov , R. R., and M. V. Stefanchuk. “Logarithmic Asymptotics of the Nonlinear Cauchy – Riemann – Beltrami Equation”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 3, Mar. 2021, pp. 395 -07, doi:10.37863/umzh.v73i3.6403.
Section
Research articles