Consider the problem on the domain
with , and .
(a) Derive the eigenvalues and multi-dimensional eigenfunctions . Assume that and that any eigenvalues of sub-problems are non-negative.
(b) For the case find simple, explicit expressions for the smallest three eigenvalues
Here is what I have for (a)
So we have
are the eigenvalues
Really not sure if that is correct though.
For (b) I'm a little lost. If I evaluate the eigenvectors at I get
I'm really confused about the 3 smallest eigenvalues, which makes me think my answer for (a) is incorrect.