Geodesic
On the construction of regular solutions for a class of generalized Cauchy–Riemann systems with coefficients bounded on the entire plane
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 166 (2024) no. 3, pp. 297-305 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

This article explores the generalized Cauchy–Riemann system on the entire complex plane. The coefficient for the conjugation of the desired function belongs to the Hölder space and, for $|z|>1$, equals $e^{im\varphi}$, where $m$ is an integer. For $m\le 0$, the system was shown to have no nonzero solutions that grow no faster than a polynomial. For $m\ge 0$, the complete set of regular solutions, i.e., those without singularities in the finite part of the plane, was constructed. The obtained solutions were expressed as series of Bessel functions of an imaginary argument. From the resulting set, the solutions bounded on the entire plane were distinguished, and the dimension of the real linear space of these solutions, which equals $m$, was determined.
Keywords: generalized Cauchy–Riemann system, bounded coefficients, bounded solutions.
Mots-clés : Hölder spaces 
@article{UZKU_2024_166_3_a1,
     author = {S. Baizaev and R. N. Barotov},
     title = {On the construction of regular solutions for a class of generalized {Cauchy{\textendash}Riemann} systems with coefficients bounded on the entire plane},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {297--305},
     year = {2024},
     volume = {166},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2024_166_3_a1/}
}
TY  - JOUR
AU  - S. Baizaev
AU  - R. N. Barotov
TI  - On the construction of regular solutions for a class of generalized Cauchy–Riemann systems with coefficients bounded on the entire plane
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2024
SP  - 297
EP  - 305
VL  - 166
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UZKU_2024_166_3_a1/
LA  - ru
ID  - UZKU_2024_166_3_a1
ER  - 
%0 Journal Article
%A S. Baizaev
%A R. N. Barotov
%T On the construction of regular solutions for a class of generalized Cauchy–Riemann systems with coefficients bounded on the entire plane
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2024
%P 297-305
%V 166
%N 3
%U http://geodesic.mathdoc.fr/item/UZKU_2024_166_3_a1/
%G ru
%F UZKU_2024_166_3_a1
S. Baizaev; R. N. Barotov. On the construction of regular solutions for a class of generalized Cauchy–Riemann systems with coefficients bounded on the entire plane. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 166 (2024) no. 3, pp. 297-305. http://geodesic.mathdoc.fr/item/UZKU_2024_166_3_a1/

[1] Vekua I.N., Generalized Analytic Functions, Nauka, Gl. Red. Fiz.-Mat. Lit., M., 1988, 512 pp. (In Russian)

[2] Bers L., Theory of Pseudo-Analytic Functions, Courant Institute of Mathematical Sciences, New York Univ., Inst. Math. Mech., New York, NY, 1953, iii+187 pp.

[3] Mikhailov L.G., A New Class of Singular Integral Equations and Its Application to Differential Equations with Singular Coefficients, Tr. Akad. Nauk Tadzh. SSR, Dushanbe, 1963, 183 pp. (In Russian)

[4] Vinogradov V.S., “Liouville theorems for an equation of generalized analytic functions”, Differ. Uravn., 16:1 (1980), 42–46 (In Russian)

[5] Muhamadiev E., Baizaev S., “On the theory of bounded solutions of a generalized Cauchy–Riemann system”, Dokl. Akad. Nauk SSSR, 287:2 (1986), 280–283 (In Russian)

[6] Baizaev S., Muhamadiev E., “On the index of first-order elliptic operators in the plane”, Differ. Equations, 28:5 (1992), 663–672

[7] Baizaev S., Muhamadiev E., “On the normal solvability of elliptic equations in the Holder space functions on plane”, Vestn. NGU. Ser.: Mat., 6:1 (2006), 3–13 (In Russian)

[8] Baizaev S., Selected Works, Khujand, 2024, 350 pp. (In Russian)

[9] Baizaev S., Barotov R.N., “Some estimates for elliptic systems generalizing the Bitsadze system of equations”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 166, no. 1, 2024, 22–35 (In Russian) | DOI

[10] Tikhonov A.N., Samarskii A.A., Equations of Mathematical Physics. A Study Guide, 6th ed., revis. enlarged, Izd. MGU, M., 1999, 799 pp. (In Russian)

[11] Watson G.N., A Treatise on the Theory of Bessel Functions, Berman V.S. (Trans.), ed. Shilov G., Izd. Inostr. Lit., M., 1949, 799 pp. (In Russian)