I have been studying partial differential equations on my own, and in many sources, I have encountered this characterization, but I have not found proof of it, and I am inquisitive to see proof as I think this characterization is really strong and is not something too easy to believe without a proof. So, if anyone knows where can I find the proof or how to do it and is willing to share I would really appreciate it. :)
Let $\lambda_{k}$ be the k-th eigenvalue of the laplacian with Dirichlet boundary conditions, so the next is true: $$\lambda_{k}=\max_{S\in\sum_{k-1}}\min_{u\in S^{\bot}}\frac{\int_{\Omega}|\nabla u|^2dx}{\int_{\Omega}|u|^2dx}$$ Where $\sum_{k-1}$ denotes the collection of all the subspaces of $H^{1}_{0}(\Omega)$ with dimension $k-1$.
The theorem you're looking for, which is a generalization of the Rayleigh quotient result, is often called the Courant min-max theorem, although I believe Courant only proved the result for symmetric matrices. The wikipedia page gives the proof for compact operators and for symmetric operators, and you can get the main idea for the proof in your case from these. However, you can also find your case discussed on this site.