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

Abstract

This paper presents an extended framework for the generalized Hyers-Ulam stability of boundary value problems with Dirichlet conditions in the half-space $ {\mathbb {R}<^>{n+1}_{+}} $ R+n+1. 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.