Robin Boundary Value Problem Depending on Parameters in a Ring Domain

Abstract

This study is devoted to give solvability conditions and solutions of the Robin boundary problem with constant coefficients for the homogeneous and the inhomogeneous Cauchy-Riemann equation in an annular domain. In order to get results, known representations and theorems in the literature are used. The representations for the solutions and solvability conditions are given in explicit form and here only a special Robin problem is considered. At the end of the paper, it is concluded that with some choices, boundary value problems for the Cauchy-Riemann equation reduce to some basic boundary problems in the ring domain.

Keywords

• Cauchy-Riemann operator
• Ring domain
• Robin problem

References

1. [1] J. Diaz, V. Peron, Equivalent Robin boundary conditions for acoustic and elastic media, Math. Models Methods Appl. Sci., 26(08) (2016), 1531-1566.
2. [2] B. Gustafsson, A. Vasilev, Conformal and Potential Analysis in Hele-Shaw Cells, Springer Science Business Media, 2006.
3. [3] K. Gustafson, T. Abe, The third boundary condition—was it Robin’s?, Math. Intelligencer, 20 (1998), 63–71.
4. [4] Y. R. Linares, C. J. Vanegas, A Robin boundary value problem in the upper half plane for the Bitsadze equation, J. Math. Anal. Appl., 419(1) (2014), 200-217.
5. [5] G. Lozada-Cruz, C. E. Rubio-Mercedes, J. Rodrigues-Ribeiro, Numerical solution of heat equation with singular Robin boundary condition, TEMA Tend. Mat. Apl. Comput., 19(2) (2018), 209-220.
6. [6] D. Medkova, P. Krutitskii, Neumann and Robin problems in a cracked domain with jump conditions on cracks, J. Math. Anal. Appl., 301(1) (2005), 99–114.
7. [7] R. Novak, Bound states in waveguides with complex Robin boundary conditions, Asymptot. Anal., 96(3-4) (2016), 251–281.
8. [8] H. Begehr, E. A. Gaertner, Dirichlet problem for the inhomogeneous polyharmonic equation in the upper half plane, Georgian Math. J., 14(1) (2007), 33-52.

9. [9] H. Begehr, G. Harutyunyan, Robin boundary value problem for the Cauchy-Riemann operator, Complex Var. Elliptic Equ., 50(15) (2005), 1125-1136.
10. [10] İ. Gençtürk, K. Koca, Dirichlet boundary value problem for an nth order complex partial differential equation, Gen. Math., 23(1-2) (2015), 39-48.
11. [11] T. Ünver, Homogen ve homogen olmayan Cauchy-Riemann denklemleri ic¸in parametreye bag˘lıRobin sınır deg˘er problemi, Kırıkkale Uni. J. Adv. Sci., 1 (2012), 58-63.
12. [12] İ. Gençtürk, The Dirichlet-Neumann boundary value problem for the inhomogeneous Bitsadze equation in a ring domain, Thai J. Math., (in press).
13. [13] İ. Gençtürk, K. Koca, Neumann boundary value problem for Bitsadze equation in a ring domain, J. Anal., 28 (2020), 799–815.
14. [14] T. Vaitekhovich, Boundary value problems for complex partial differential equations in a ring domain, Ph.D. Thesis, FU Berlin, 2008.
15. [15] H. Begehr, Boundary value problems in complex analysis I, Bol. Asoc. Mat. Venezolana, 12(1) (2005), 65-85.
16. [16] H. Begehr, Boundary value problems in complex analysis II, Bol. Asoc. Mat. Venezolana, 12(2) (2005), 217-250.
17. [17] H. Begehr, Complex Analytic Methods for Partial Differential Equations. An Introductory Text, World Scientific, Singapore, 1994.