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Let $ \Omega$ be an open bounded domain with a boundary $ \partial\Omega$ . Consider the following Neumann eigenvalue problem for Laplacian: find $ (\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$ \begin{align*} -\Delta \phi_n& = \lambda_n \phi_n\quad \mbox{in }\Omega,\ \frac{\partial \phi_n}{\partial \nu} & = 0 \quad \mbox{on }\partial\Omega. \end{align*} Now by spectral theory (see the notes at here https://faculty.math.illinois.edu/~laugesen/595Lectures.pdf),Read more