k holomorphic functions are a type of generation of holomorphic functions. In this paper, a nonlinear boundary value problem for k holomorphic functions is primarily discussed on generalized polycylinders in $\mathbb{C}^2$. The existence of the solution for the problem is studied in detail with the help of the boundary properties of Cauchy type singular integral operators with a k holomorphic kernel. Furthermore, the integral representation for the solution is obtained.
Yanyan CUI
,
Zunfeng LI
,
Yonghong XIE
,
Yuying QIAO
. THE NONLINEAR BOUNDARY VALUE PROBLEM FOR k HOLOMORPHIC FUNCTIONS IN $\mathbb{C}^2$∗[J]. Acta mathematica scientia, Series B, 2023
, 43(4)
: 1571
-1586
.
DOI: 10.1007/s10473-023-0408-9
[1] Yang P. On Riemann-Hilbert boundary value problems for analytic functions of several complex variables. Journal of Sichuan Normal University (Natural Science), 1992, 15(1): 26-31
[2] Lin L Y, Qiu C. The cauchy boundary value problems on closed piecewise smooth manifolds in $\mathbb{C}^n$. Acta Mathematica Sinica, 2004, 20(6): 989-998
[3] Bezrodnykh S I, Vlasov V I. Singular Riemann-Hilbert problem in complex-shaped domains. Computational Mathematics and Mathematical Physics, 2014, 54(12): 1826-1875
[4] Polunin V A, Soldatov A P. Riemann-Hilbert problem for the Moisil-Teodorescu system in multiple connected domains. Electronic Journal of Differential Equations, 2016, 310: 1-5
[5] Kokilashvili V, Paatashvili V. Riemann-Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue spaces. Complex Variables and Elliptic Equations, 2018, 63(9): 1233-1257
[6] Laurent-Thiébaut C, Shaw M C. Solving $\overline{\partial}$ with prescribed support on Hartogs triangles in $\mathbb{C}^2$ and $\mathbb{C}\mathbb{P}^2$. Transactions of the American Mathematical Society,2019, 371(9): 6531-6546
[7] Fassina M, Pinton S. Existence and interior regularity theorems for $\overline{\partial}$ on $q$-convex domains. Complex Analysis and Operator Theory, 2019, 13: 2487-2494
[8] Chakrabarti D, Harrington P S. Exact sequences and estimates for the $\overline{\partial}$-problem. Mathematische Zeitschrift, 2021, 299: 1837-1873
[9] Wu X. Some oscillation criteria for a class of higher order nonlinear dynamic equations with a delay argument on time scales. Acta Mathematica Scientia, 2021, 41B(5): 1474-1492
[10] Chen Z, Zhang R, Lan S, Chen C. The growth of difference equations and differential equations. Acta Mathematica Scientia, 2021, 41B(6): 1911-1920
[11] Yang J S, Li T X. Oscillation for a class of second-order damped Emden-Fowler dynamic equations on time scales. Acta Mathematica Scientia, 2018, 38A(1): 134-155
[12] Balk M B. Polyanalytic Functions. Berlin: Akademie Verlay, 1991
[13] Hayrapetyan H M, Hayrapetyan A R. Boundary value problems in weighted spaces of polyanalytic functions in half-plane. Journal of Contemporary Mathematical Analysis, 2012, 47(1): 1-15
[14] Soldatov A P, Vuong T Q. The linear conjugation problem for bi-analytic functions. Russian Mathematics.2016, 60: 62-66
[15] Ku M, He F, Wang Y. Riemann-Hilbert problems for Hardy space of meta-analytic functions on the unit disc. Complex Analysis and Operator Theory, 2018, 12(2): 457-474
[16] Aksoy Ü, Begehr H, Çelebi A O. A.V. Bitsadze's observation on bianalytic functions and the Schwarz problem. Complex Variables and Elliptic Equations,2019, 64(8): 1257-1274
[17] Han H, Liu H, Wang Y. Riemann boundary-value problem for doubly-periodic bianalytic functions. Boundary Value Problems, 2018, 1: 1-20
[18] Aksoy Ü, Begehr H, Çelebi A O. A.V. Bitsadze's observation on bianalytic functions and the Schwarz problem. Complex Variables and Elliptic Equations,2019, 64(8): 1257-1274
[19] Aksoy Ü, Begehr H, Çelebi A O. A.V. Bitsadze's observation on bianalytic functions and the Schwarz problem revisited. Complex Variables and Elliptic Equations,2021, 66(4): 583-585
[20] Begehr H, Shupeyeva B. Polyanalytic boundary value problems for planar domains with harmonic Green function. Analysis and Mathematical Physics, 2021, 11(137): 1-22
[21] Qiao Y, Cui Y, Li Z, Wang L. $k$-holomorphic functions in spaces of several complex variables. Complex Variables and Elliptic Equations, 2020, 65(11): 1826-1845
[22] Huang S. A nonlinear boundary value problem for analytic functions of several complex variables. Acta Mathematica Scientia, 1997, 17A(4): 382-388
