Let $\mu_k(\Omega)$ be the $k$-th Neumann eigenvalue on a piecewise smooth and bounded domain $\Omega \subset \mathbb{R}^2$, $k \geq 2$. Assume also, for simplicity, that $\Omega$ has only a finite number of nonregular points, and each of such points corresponds to a right inward corner, e.g., as on the figure below.
Let $\Omega_\varepsilon$, $\varepsilon>0$, be a sequence of smooth domains which approximate $\Omega$ from the inside. That is, $\Omega_\varepsilon$ rounds the corners of $\Omega$ in some "good" way, see again the figure. (The requirement that $\Omega_\varepsilon \subset \Omega$ is not essential, in principle.)
I'm looking for some results about the convergence $$ \mu_k(\Omega_\varepsilon) \to \mu_k(\Omega) \quad \text{as}~ \varepsilon \to 0, $$ for appropriately chosen approximations $\Omega_\varepsilon$.
The problem is that the smoothing of corners is not a smooth perturbation of $\Omega$, and hence the required convergence is not apriori clear. (Recall that we are in the Neumann case).