Spectrum of Laplacian with zero Dirichlet and Neumann data

Question

Let $\Omega$ be a bounded domain with sufficiently smooth(say piece-wise smooth) boundary and consider the Laplacian eigenvalue problem with the given boundary condition(Cauchy BV ) \begin{cases} -\Delta u=\lambda u \text{, $\,\,\,\, \Omega$} \\ \frac{\partial u}{\partial n}=u=0 \text{,$\,\,\,\,\, \partial\Omega$.} \end{cases} What are some examples of $\Omega$ with explicitly known spectrum? In particular, what are the spectra for $\Omega$=disk, circular annulus, regular polygon?

Any references would be highly appreciated.

$\textbf{Remark:}$ The above problem is a generalization of this question.

Answer 1

The given BVP has no non-trivial solution. This follows immediately from Rellich's formula which states a non-zero eigenfunction $u$ corresponding to an eigenvalue $\lambda$ for the Dirichlet eigenvalue problem on $\Omega$ satisfies the following

$$\lambda=\frac{\int_{\partial\Omega}(\frac{\partial u}{\partial n})^2\frac{\partial(r^2)}{\partial n}ds}{4\int_\Omega u^2dA},$$ where $r^2=x_1^2+x_2^2,$ $dA$ is the area measure and $ds$ is arc-length measure.