I am currently trying to find the spectrum of the Laplace operator for ellipsoids in $\mathbb{R}^{3}$ with Dirichlet boundary conditions, i.e., I am looking for solutions to the following PDE, $$ \begin{equation} \label{eq1} \begin{split} \Delta u & = -\lambda u, \qquad &\text{in } \Omega \\ u & = 0 \qquad &\text{on } \Omega \end{split} \end{equation} $$ where $\Omega$ is an ellipsoid given by $$ \Omega = \{ (x,y,z) : \frac{x^2}{a_{1}^{2}} + \frac{y^2}{a_{2}^{2}} +\frac{z^2}{a_{3}^{2}} =1 \}$$
If no explicit solutions exist, a solution for oblate spheroids would also be helpful. Or some general information on the solution, for example, does it separate in ellipsoidal coordinates?
In a perfect world, I am looking for a similar result as as Troesch and Troesch showed in (https://www.jstor.org/stable/2005509) but in three dimensions.
I found some literature, including the book "Ellipsoidal Harmonics" by George Dassios (https://www.cambridge.org/core/books/ellipsoidal-harmonics/803CBC9F98DF279CC52F6AC53923ECD1) which has been helpful, but it does not contain exactly what I am looking for. I tried to follow the book (for the Lame's function) as much as possible to show that the solution might separate but I could not do it.