Estimates for Resonant Frequencies Under Boundary Deformation in Multi-dimensional Space

In multi-dimensional space, we address the integral equation method to investigate the interplay between the geometry, boundary conditions and the properties of the resonant frequencies and their associated eigenfunctions under boundary variations of domain. We provide a rigorous derivation of asymptotic expansions for eigenfunctions and we establish error estimations for both resonant frequencies and eigenfunctions of the Helmholtz eigenvalue problem.