Good time of day. Can you help me please?
I try to solve the following task for the Laplace operator for bounded domain $\Omega \subset R^2$ with Dirichlet eigenvalues.
I know that that the disc minimizes the first Dirichlet eigenvalue $\lambda_1(\Omega)$ among all planar domains of the same area. Also I know that the disjoint union of two discs of the same radius minimizes $\lambda_2(\Omega)$ among all planar domains of the same area (Faber-Krahn theorem).
I try to understand why that the disjoint union of $n$ discs of the same radius cannot minimize Dirichlet eigenvalues $\lambda_n(\Omega)$ for all $n$. I have found in this article (https://arxiv.org/abs/0808.2968v1) that this contradicts Weyl's law but I don't understand why it contradicts. Please help me and explain it in more detail.