I am trying to come up with an example to the following.
Construct a domain $\Omega$ in all dimensions $n \in \mathbb{N}$, such that the spectrum of the Dirichlet Laplacian on such a domain (i.e., the Laplacian with Dirichlet boundary conditions on the boundary of $\Omega$) has the following properties:
For any given $m \in \mathbb{N} \cup \{\infty\}$, each eigenvalue of the Dirichlet Laplacian on $\Omega$ has multiplicity $m$
I am not sure if I understand the question correctly. Can anybody help?
Thanks