Let $\Delta$ be the Laplacian on a smooth domain $\Omega\subset \mathbb{R}^2$ with Dirichlet boundary conditions. I am interested in whether the implication
\begin{align}
\Omega \text{ is asymmetric } \Rightarrow \Delta \text{ only has single eigenvalues}
\end{align}
holds, where by asymmetric I mean that the only planar rotation $\varphi$ such that $\varphi(\Omega) = \Omega$ is the identity.  

It is well known that the set of Dirichlet eigenvalues $\{\lambda\_i\}\_{i\in \mathbb{N}}$ forms a sequence
\begin{align}
0<\lambda\_1< \lambda\_2 \leq \lambda\_3 \leq \dots \nearrow \infty
\end{align}
with the first eigenvalue being single.  

Examples of symmetric domains with multiple eigenvalues (see [1]) include a square, an equilateral triangle or a circular disk.
An example of a symmetric domain with only simple eigenvalues would be a rectangle with side lengths $l\_1$, $l\_2$ such that $\left(\frac{l\_1}{l\_2}\right)^2$ is irrational [1, 3.1].  

I have carried out some numerical experiments with asymmetric shapes:  

<https://i.stack.imgur.com/uwroB.png>   

<https://i.stack.imgur.com/6VJk6.png>   

<https://i.stack.imgur.com/v1suq.png>   

but up to the 200th eigenvalue there are only single ones.


Does anyone know whether the above claim is true or false, or has ideas for counterexample shapes to check (i.e., asymmetric shapes with multiple eigenvalues)?  

[1] Grebenkov, Denis S., and B-T. Nguyen. "Geometrical structure of Laplacian eigenfunctions." siam REVIEW 55.4 (2013): 601-667.



Multiplicity of Dirichlet Laplacian eigenvalues of asymmetric domains
A counterexample is given by $\Omega = (0,2\pi)^2\setminus [0,\pi]^2$. Then $u\_{mn}(x,y)=\sin mx\sin ny$ is an eigenfunction with eigenvalue $m^2+n^2$, and so is $u\_{nm}$.


You can build many other examples in this way by looking at the zero set of eigenfunctions of the square (in particular, the reflection symmetry of the region can be removed also) and perhaps also starting from other regions with multiple eigenvalues.


