Higher-Order Equations with Robin Boundary Conditions in the Upper Half Complex Plane

Abstract

This study examines the conditions for solvability and derives solution of a Robin problem linked to a higher-order differential equation in the upper half plane. The focus lies on extending classical boundary value problem techniques to higher-order equations, leveraging advanced tools from complex analysis. By formulating the problem within the framework of the higher-order Cauchy-Riemann operator, we address challenges arising from the coupling of boundary conditions and the operator’s intricate structure. The study begins by investigating necessary conditions for solutions to the Robin problem, identifying critical constraints on the data and parameters. These conditions are derived using an analysis of the kernel and range of the associated boundary operator. The interplay between the boundary terms and the underlying differential operator is systematically studied to establish the problem’s well-posedness. To construct explicit solutions, we introduce an integral approach combining two linked Robin boundary problems, reducing the higherorder equation to a tractable form. Utilizing integral transforms, such as the Cauchy transform and its higher-order extensions, we develop representations that encapsulate both the interior and boundary behaviors of the solution. The derived integral formulae highlight the role of analytic continuation and specific kernel functions tailored to the geometry of the upper half-plane.

Keywords

• Robin Boundary value problem
• upper half plane
• higher order equation

References

1. M. Akel, M. Hidan, M. Abdalla, Complex boundary value problems for the Cauchy– Riemann operator on a triangle, Fractals, 30(10) (2022) 1–15.
2. H. Begehr, S. Burgumbayeva, A. Dauletkulova, H. Lin, Harmonic Green functions for the Almaty apple, Complex Variables and Elliptic Equations 65(11) (2020) 1814–1825.
3. E. Gaertner, Basic Complex Boundary Value Problems in the Upper Half Plane, Ph.D. dissertation, Free University, Berlin, (2006).
4. B. Karaca, Dirichlet Problem for complex model partial differential equations, Complex Variables and Elliptic Equations 65(10) (2020) 1748–1762.
5. R.Y. Linares, C. J. Vanegas, A Robin boundary value problem in the upper half plane for