Generalized Hyers–Ulam Stability of Laplace Equation With Neumann Boundary Condition in the Upper Half-Space

This paper investigates the generalized Hyers–Ulam stability of the Laplace equation subject to Neumann boundary conditions in the upper half-space. Traditionally, Hyers–Ulam stability problems for differential equations are analyzed by examining the system's error, particularly in relation to a forcing term. Hyers–Ulam stability problems are traditionally studied by examining whether approximate solutions satisfy the equation sufficiently well and identifying the conditions under which exact solutions can be guaranteed near such approximations. In this work, we significantly extend the classical framework by establishing a generalized Hyers–Ulam stability that simultaneously accounts for errors within the equation and on the boundary. This approach represents a significant advancement as it addresses stability arising from both the equation and the boundary conditions. Using an integral methodology based on the Fourier transform, we derive explicit stability estimates while effectively managing the singularities of Green's functions in the half-space. The use of a Gaussian kernel in our analysis is essential for controlling these singularities and ensuring the validity of our stability estimates. By applying the Fourier transform to the tangential variables, we bypass the complexities of these singularities and provide precise results for both interior and boundary contributions. This extended framework enhances the understanding of generalized Hyers–Ulam stability and establishes a comprehensive approach to stability analysis in higher dimensional boundary value problems.