In my calculus of variations lecture notes it is claimed that if $\Omega$ is an open bounded non-empty subset of $\mathbb{R}^N$, then the eigenvalues of the laplacian on the Sobolev space $H^1_0(\Omega)$, i.e. the values $\lambda\in\mathbb{R}$ for which there exists $u\in H^1_0(\Omega)\setminus\{0\}$ such that: $$\forall v\in H^1_0(\Omega), \langle u,v\rangle_{H^1_0}=\lambda\langle u,v\rangle_{L^2}$$ can be obtained through a inf-sup procedure, i.e. the Courant-Fischer method: $$\lambda_k=\inf_{V\le H^1_0(\Omega)\\\dim(V)\ge k}\sup_{u\in V\cap S} \|u\|^2_{H^1_0}$$ where
- $k\in\mathbb{N}$;
- $S=\{u\in H^1_0(\Omega)\ |\ \|u\|_{L^2}=1\}$;
- the relation $V\le H^1_0(\Omega)$ means that $V$ is a linear subspace of $H^1_0(\Omega)$;
- $\dim(V)$ is the dimension of the linear space $V$.
In the notes, it is also claimed that the Lusternik-Schnirelmann inf-sup procedure is a generalization of the Courant-Fischer procedure and it is suggested that one can prove that the Courant-Fischer procedure produces critical values through a deformation argument, similar to the one given in the Lusternik-Schnirelmann theory: in the Lusternik-Schnirelmann procedure we get the inf-sup values over subsets of the $L^2$-sphere of genus at least $k$ and we use the deformation lemma to prove that these inf-sup values are actually critical values, while in the Courant-Fischer procedure we get the inf-sup values over subsets of the $L^2$-sphere that are the intersection of the $L^2$-sphere with a linear subspace of dimension at least $k$, and we can use a corresponding deformation lemma to prove that these inf-sup values are actually critical values.
How to build such a deformation to prove that the Courant-Fischer procedure produces critical values?
I think that what it's needed is to build a deformation that actually should be linear, in order to get that the deformation of a linear subspace is another linear subspace of at most the same dimension. Also, I think that the deformation should preserve the $L^2(\Omega)$ norm, otherwise the deformation of $V\cap S$ where $V$ is a linear subspace of $H^1_0(\Omega)$ can't be written in the form $W\cap S$ for some linear subspace of $H^1_0(\Omega)$. Also, the deformation should decrease the $H^1_0(\Omega)$ norm somehow, like in the deformation lemma used in Lusternik-Schnirelmann theory.
Can someone help providing hints to go on? Also, proofs or references where Courant-Fischer procedure is proved through a deformation argument are welcome.