Given $\Sigma_\epsilon = \{(x,y)\in\mathbb{R}^2:|\sqrt{x^2+y^2}-1|\le \epsilon \}$ for some small $\epsilon>0$
I want to determine(analytically) all the functions $u$ and constants $\lambda$ (eigenfunctions and eigenvalues?) which satisfy the following conditions.
$-\Delta u=\lambda u $ for $(x,y) \in \Sigma_\epsilon$
and
$ \vec{n} \cdot \nabla u=0$ for $(x,y) \in \partial\Sigma_\epsilon$ (the boundary of $\Sigma_\epsilon$)
where $\Delta$ is the standard Laplacian, $ \vec{n}$ is the normal vector (normal to the boundary of $\Sigma_\epsilon$), $\nabla$ is the gradient.
I think the region is somehow 'circular' thus i tried to convert it to polar coordinate but wasn't able to solve it. Any help would be appreciated!