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

Year 2025, Volume: 7 Issue: 1, 10 - 15, 30.06.2025

Bahriye Karaca

https://doi.org/10.47086/pims.1589091

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

• M. Akel, M. Hidan, M. Abdalla, Complex boundary value problems for the Cauchy– Riemann operator on a triangle, Fractals, 30(10) (2022) 1–15.
• 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.
• E. Gaertner, Basic Complex Boundary Value Problems in the Upper Half Plane, Ph.D. dissertation, Free University, Berlin, (2006).
• B. Karaca, Dirichlet Problem for complex model partial differential equations, Complex Variables and Elliptic Equations 65(10) (2020) 1748–1762.
• R.Y. Linares, C. J. Vanegas, A Robin boundary value problem in the upper half plane for