Let $\Omega$ be a domain with $\partial\Omega$ bounded.
Is it possible to find a smooth basis of $H^1(\Omega)$ and $L^2(\Omega)$ which is orthonormal wrt. the $L^2(\partial\Omega)$ inner product?
Often one would take the eigenfunctions of the Laplacian on $\Omega$ but this doesn't give us any information about the boundary requirement.. is there another way?