Let $\mathcal{E}$ be an ellipse in the $\mathbb{R}^2$ plane with center in $o=(0,0)$, given focal distance $c\geq 0$ and given area $A>0$.
It is a fact that the eigenvalue problem for the Laplace operator with homogeneous Dirichlet boundary condition, i.e.:
$\begin{cases} u_{xx}+u_{yy}=-\lambda u, &\text{in } \mathcal{E}, \\ u=0, &\text{on } \partial \mathcal{E}, \end{cases} $
has solutions for infinite positive values of $\lambda$: these values are called eigenvalues of the Laplace operator in $\mathcal{E}$; in particular, there exists an eigenvalue $\lambda_1(c)$ which is the smallest one: $\lambda_1(c)$ is called the first eigenvalue of the Laplace operator and it has some nice properties (e.g. it is simple, for the eigenspace associated to $\lambda_1(c)$ is one dimensional).
Now, if $\mathcal{E}$ is a circle (this can happen iff $c=0$) it is well known that $\lambda_1(0)=\frac{\pi}{A}\ j_{0,1}^2$, where $j_{0,1}\approx 2.40483$ is the first zero of the Bessel function $\text{J}_0(x)$.
The questions I'm interested in are the following:
- What happens to $\lambda_1(c)$ if $\mathcal{E}$ is a "true" ellipse (i.e. if c> 0)? Can it be evaluated explicitly (in terms of some special functions)?
- And how different values of $c$ bias the value of $\lambda_1(c)$ around $0$?
It is known that $\lambda_1(c)\geq \lambda_1(0)$, with equality iff $\mathcal{E}$ is a circle (i.e., iff $c=0$; this is the famous Faber-Krahn inequality), but it also seems quite obvious that $\lambda_1(c)$ has to exhibit a sort of continuity in $0$: in fact one expects that $\lim \limits_ {c\to 0^+} \lambda_1(c) = \lambda_1(0)$...
Now, I did some researches on the net. In the case $c>0$, one can introduce the elliptic coordinates $(\mu ,\nu)$:
$\begin{cases} x=c\cosh \mu \cos \nu, \\ y=c\sinh \mu \sin \nu ,\end{cases} $
so that equation $u_{xx}+u_{yy}=-\lambda u$ transforms into:
$u_{\mu\mu} +u_{\nu \nu} =-c^2 \lambda (\sinh^2 \mu +\sin^2 \nu) u $
which is harder to solve with separation of variables than the equation for the circle; neverthless separation of variables applies and yields a couple of so-called Mathieu's differential equations, which are a sort of ugly counterpart of Bessel's differential equation...
But then I cannot figure out how to compute $\lambda_1(c)$ (neither for fixed $c$ nor for varying $c$)!
Do I have to use some tables (like the ones in Abramowitz & Stegun, §20)? And, in the positive case, how they can be used?
If you have any reference it could be worth reading, please feel free to suggest.
Thanks in advance for your help.