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The Steklov problem for a compact planar region $\Omega$ is \begin{cases} \Delta u =0 &\text{in $\Omega$}, \\ \frac{\partial u}{\partial n} = \sigma u &\text{on $\partial \Omega$}, \end{cases} where $n$ is the outward unit normal along $\partial \Omega$.

I am finding Steklov eigenfunctions when $\Omega$ is a region bounded by ellipse. I know that a spectrum of the problem is $0=\sigma_0<\sigma_1\le \cdots\rightarrow \infty$.

Can we obtain the eigenfunctions explicitly by seperation of variables? Is there any calculation about the eigenfunctions or multiplicities of the eigenvalues? Or is there any numerical results about level sets of the eigenfunctions?


My approach is as follows.

Consider the elliptic coordinate system $(\mu, \nu)$ which is given by $x=a \cosh \mu \cos \nu, y=a \sinh \mu \sin \nu$. It is a two dimensional orthogonal coordinate system and the curves of $\mu=const$, $\nu=const$ are ellipse, hyperbola, respectively. Now we assume that $\Omega$ is bounded by the curves of $\mu=\mu_0$.

In this coordinate system, Laplacian is calculated by \begin{align} \Delta u = \frac{1}{a^2 (\cosh^2 \mu -\cos^2 \nu)} (\frac{\partial^2 u}{\partial \mu^2}+\frac{\partial^2 u}{\partial \nu^2}). \end{align} By seperation of variables, $\Delta u=0$ gives $u =\mu, \nu, \mu\nu, \cosh \alpha \mu \cos \alpha \nu, \cosh \alpha \mu \sin \alpha \nu, \sinh \alpha \mu \ \alpha \nu, \sinh \alpha \mu \sin \alpha \nu, \\ \cosh \alpha \nu \cos \alpha \mu, \cosh \alpha \nu \sin \alpha \mu, \sinh \alpha \nu \cos \alpha \mu, \sinh \alpha \nu \sin \alpha \mu$.

In addition, since curves of $\nu=const$ is orthogonal to $\partial \Omega$, we can calculate \begin{align} \frac{\partial u}{\partial n} = \left.\frac{\partial u}{\partial \mu}\right|_{\mu=\mu_0} \times a\sqrt{\cosh^2 \mu_0 - \cos^2 \nu}. \end{align} But it seems hard to deal with $\sqrt{\cosh^2 \mu_0 - \cos^2 \nu}$ and find eigenfunctions explicitly.

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    $\begingroup$ Let me remark that your Steklov eigenvalues are nothing but the eigenvalues of the Dirichlet-to-Neumann operator; and that, obviously, $\mu$ is an eigenvalue for the DtN operator on $\partial \Omega$ iff 0 is an eigenvalue for the Laplacian on $\Omega$ with Robin boundary conditions $\frac{\partial u}{\partial n}=\mu u_{|\partial \Omega}$. So you can sometimes take advantage of the spectral theory for the Robin Laplacian, see e.g. an article by Arendt and Mazzeo (CPAA 2012) where this approach is thoroughly discussed. $\endgroup$ – Delio Mugnolo Dec 19 '17 at 10:04

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