i'm currently searching for a proper reference or proof to see that the first eigenvalue $\lambda \in \mathbb{R}$ of \begin{equation*} - \Delta u = \lambda u \text{ in } \Omega, \\ u \in H^1(\Omega), \end{equation*} (in a weak sense in the Sobolev-space) with homogenious Neumann or Dirichlet boundary conditions is simple in the case of a bounded and connected Domain $\Omega \subset \mathbb{R}^d$ (where we assume $\Omega$ to be a Lipschitz-Domain in the Neumann-case, so that the outer normals are well-defined).
The only proofs i've seen to far all use the assumption that the boundary $\partial \Omega$ is smooth. Because than we have enough regularity for the eigenfunctions to be in $C^2(\overline{\Omega})$ and can show by using the strong Maximum-Principle , that we can assume the corresponding eigenfunctions of the first eigenvalue to be strict positive. And because two such strict positive eigenfunction can't be orthogonal in $L_2(\Omega)$, we have that the corresponding eigenspace is indeed one-dimensional.
But what about results without the smooth boundary?
Thank you in advance!