Asymptotic expansions for eigenvalues of the Steklov problem in singularly perturbed domains
С. А. Назаров · St Petersburg Mathematical Journal · 2015
Full asymptotic expansions are constructed and justified for two series of eigenvalues and the corresponding eigenfunctions of the spectral Steklov problem in a domain with a singular boundary perturbation having the form of a small cavity. The terms of those series are of type $\lambda _k+o(1)$ and $\varepsilon ^{-1}(\mu _m+o(1))$, where $\lambda _k$ and $\mu _m$ are the eigenvalues of the Steklov problem in a bounded domain without cavity and the exterior Steklov problem for a cavity of unit size. A similar problem of the surface wave is also treated. The smoothness requirements on the boundary are discussed and unsolved problems are stated.