Simple eigenvalues for the Steklov problem in a domain with a small hole. A functional analytic approach

Let ${\mathbb{I}}^{o}$ be a bounded open domain of ${\mathbb{R}}^{n}$. Let $ \nu_{ {\mathbb{I}}^{o} }$ denote the outward unit normal to $\partial{\mathbb{I}}^{o}$. We assume that the Steklov problem $ \Delta u=0 $ in ${\mathbb{I}}^{o}$, $\frac{\partial u}{\partial \nu_{ {\mathbb{I}}^{o} } }=\lambda u$ on $\partial {\mathbb{I}}^{o}$ has a simple eigenvalue $\tilde{\lambda}$. Then we consider an annular domain ${\mathbb{A}}(\epsilon)$ obtained by removing from ${\mathbb{I}}^{o}$ a small cavity of size $\epsilon>0$, and we show that under proper assumptions there exists a real valued real analytic function $\hat{\lambda} (\cdot,\cdot)$ defined in an open neighborhood of $(0,0)$ in ${\mathbb{R}}^{2}$ and such that $\hat{\lambda} (\epsilon,\delta_{2,n} \epsilon\log \epsilon)$ is a simple eigenvalue for the Steklov problem $ \Delta u=0 $ in ${\mathbb{A}}(\epsilon)$, $\frac{\partial u}{\partial \nu_{ {\mathbb{A}}(\epsilon) } }=\lambda u$ on $\partial {\mathbb{A}}(\epsilon)$ for all $\epsilon>0$ small enough, and such that $\hat{\lambda} (0,0)=\tilde{\lambda}$. Here $ \nu_{{\mathbb{A}}(\epsilon) }$ denotes the outward unit normal to $\partial {\mathbb{A}}(\epsilon)$, and $\delta_{2,2}\equiv 1$ and $\delta_{2,n}\equiv 0$ if $n\geq 3$. Then related statements have been proved for corresponding eigenfunctions.