Abstract
In the paper, an elementary solution of polyharmonic equation is determined and, with the help of it, an integral representation of solutions of polyharmonic equation in a bounded domain is presented.
REFERENCES
H. Begehr, ‘‘Biharmonic Green functions,’’ Matematiche LXI, 395–405 (2006).
H. Begehr and T. Vaitekhovich, ‘‘Modified harmonic Robin function,’’ Complex Variab. Ellipt. Equat. 58, 483–496 (2013).
M. A. Sadybekov, B. T. Torebek, and B. Kh. Turmetov, ‘‘On an explicit form of the Green function of the Robin problem for the Laplace operator in a circle,’’ Adv. Pure Appl. Math. 6, 163–172 (2015).
Wang Ying and Ye. Liuqing, ‘‘Biharmonic Green function and biharmonic Neumann function in a sector,’’ Complex Variab. Ellipt. Equat. 58, 7–22 (2013).
Wang Ying, ‘‘Tri-harmonic boundary value problems in a sector,’’ Complex Variab. Ellipt. Equat. 59, 732–749 (2014).
V. V. Karachik, ‘‘The Green function of the Dirichlet problem for the triharmonic equation in the ball,’’ Math. Notes 107 (1), 105–120 (2020).
T. Boggio, ‘‘Sulle funzioni di Green d’ordine \(m\),’’ Palermo Rend. 20, 97–135 (1905).
T. Sh. Kalmenov, B. D. Koshanov. and M. Y. Nemchenko, ‘‘Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere,’’ Complex Variab. Ellipt. Equat. 53, 177–183 (2008).
H. Begehr, J. Du, and Y. Wang, ‘‘A Dirichlet problem for polyharmonic functions,’’ Ann. Mat. Pura Appl. 187, 435–457 (2008).
V. V. Karachik and B. Kh. Turmetov, ‘‘On Green’s function of the Robin problem for the Poisson equation,’’ Adv. Pure Appl. Math. 10, 203–214 (2019).
M. A. Sadybekov, B. T. Torebek, and B. K. Turmetov, ‘‘Representation of Green’s function of the Neumann problem for a multi-dimensional ball,’’ Complex Variab. Ellipt. Equat. 61, 104–123 (2016).
V. V. Karachik and B. T. Torebek, ‘‘On the Dirichlet–Riquier problem for biharmonic equations,’’ Math. Notes 102 (1), 31–42 (2017).
A. P. Soldatov, ‘‘On the Fredholm property and index of the generalized Neumann problem,’’ Differ. Equat. 56, 212–220 (2020).
V. V. Karachik, ‘‘Green’s functions of the Navier and Riquier–Neumann problems for the biharmonic equation in the ball,’’ Differ. Equat. 57, 654–668 (2021).
G. Sweers, ‘‘A survey on boundary conditions for the biharmonic,’’ Complex Variab. Ellipt. Equat. 54, 79–93 (2009).
V. V. Karachik, B. Turmetov, and H. Yuan, ‘‘Four boundary value problems for a nonlocal biharmonic equation in the unit ball,’’ Mathematics 10 (7), 1–21 (2022).
A. Cabada and F. A. F. Tojo, ‘‘On linear differential equations and systems with reflection,’’ Appl. Math. Comput. 305, 84–102 (2017).
F. A. F. Tojo, ‘‘Computation of Green’s functions through algebraic decomposition of operators,’’ Boundary Value Probl. 167, 1–15 (2016).
A. V. Bitsadze, Equations of Mathematical Physics (Nauka, Moscow, 1982) [in Russian].
V. V. Karachik, ‘‘Green’s function of Dirichlet problem for biharmonic equation in the ball,’’ Complex Variab. Ellipt. Equat. 64, 1500–1521 (2019).
Author information
Authors and Affiliations
Corresponding author
Additional information
(Submitted by T. K. Yuldashev)
Rights and permissions
About this article
Cite this article
Karachik, V.V. On One Integral Representation of Solutions of Polyharmonic Equation. Lobachevskii J Math 44, 2749–2756 (2023). https://doi.org/10.1134/S1995080223070247
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080223070247