Fourier transform and estimates for stability of Laplace equation with Dirichlet boundary condition on half space

This paper presents an extended framework for the generalized Hyers–Ulam stability of boundary value problems with Dirichlet conditions in the half-space (Formula presented.). Unlike traditional approaches, which focus solely on equation errors, our framework simultaneously addresses errors within the equation and at the boundary, marking a significant advancement. Using an integral methodology based on the Fourier transform, we derive explicit stability estimates while managing the singularities of Green’s functions in the half-space. By transforming tangential variables, we circumvent singularity complexities and provide precise results for both interior and boundary contributions. This work enhances the understanding of generalized Hyers–Ulam stability and offers a comprehensive approach to stability analysis in higher-dimensional boundary value problems.