Splitting a multiple eigenvalue in the boundary value problem for a membrane clamped on a small part of the boundary

We prove that, under a singular perturbation of boundary conditions, a multiple eigenvalue $\lambda_0$ in the Neumann problem in a bounded connected domain $\Omega\subset\mathbb{R}^2$ with boundary $\Gamma_0\in C^\infty$ splits into the simple eigenvalues $\lambda_\varepsilon^{(i)}$ of the boundary value problem
\begin{gather*} (\Delta+\lambda_{\varepsilon})\varphi_{\varepsilon}=0 \quad \text{for } x\in\Omega, \\ \frac{\partial\varphi_{\varepsilon}}{\partial n}=0 \quad \text{on } \Gamma_0\setminus\overline{\omega}_{\varepsilon}, \quad \varphi_{\varepsilon}=0 \quad \text{on } \omega_{\varepsilon}, \end{gather*}
which possess distinct rates of convergence to $\lambda_0$. Here $\omega_{\varepsilon}$, is an open connected part of $\Gamma_0$ with length of order $\varepsilon$, $0<\varepsilon\ll 1$, and $n$ is the outward normal to $\Omega$.