This is an exercise function Evans PDE book, Chapter 6.
The theorem states that for $Lu:=-\text{div}(A\cdot\nabla u)+cu$ where $c\geq 0$, we have the eigenvalue of $L$ can be written in the following way:
$$ \lambda_k=\max_{S^{k-1}}\min_{u\in (S^{k-1})^\bot,\|u\|_{L^2}=1} B[u,u]$$ where $$ B[u,u]=\int_\Omega A\nabla u\nabla u+ cu^2 $$ and $S^{k-1}$ is an $k-1$ dimension subspace of $H_0^1$.
By Rayleigh Quotient Theorem we quickly have $$ \lambda_k\leq\max_{S^{k-1}}\min_{u\in (S^{k-1})^\bot,\|u\|_{L^2}=1} B[u,u] $$
To show the other direction, I am trying to pivot that for arbitrary $S^{k-1}$ we have $$ \min_{u\in (S^{k-1})^\bot,\|u\|_{L^2}=1} B[u,u] $$ However, I can not prove this part. What I am trying so far is to project $u\in (S^{k-1})^\bot$ onto $(S_0^{k-1})^\bot $ where $S_0=\{w_1,\ldots,w_{k-1}\}$, the first $k-1$ eigenvectors... But it is not very useful...
Any help is really welcome!
First of all, chatacterize each $\lambda_k$ as $$\lambda_k=\min_{u\in \langle \varphi_1,\cdots,\varphi_{k-1}\rangle ^\bot,\ \|u\|_2=1}B[u,u],$$
where $\varphi_k$ are the eigenfunctions associated with $\lambda_k$. They are orthonormal and normalized.
Let $$\overline{\lambda_k}=\max_{S^{k-1}}\min_{u\in (S^{k-1})^\bot,\ \|u\|_2=1}B[u,u].$$
As you can see, $\lambda_k\le \overline{\lambda _k}$ is trivial. To prove the reverse inequality, note that for each $S^{k-1}$, we have that $$(S^{k-1})^\bot\cap\langle \varphi_1,\cdots, \varphi_k \rangle\neq \emptyset.$$
Take $u$ in the above intersection with $\|u\|_2=1$. Note that $u=\sum_{i=1}^k a_i\varphi_i$ and thus $$B[u,u]=\sum_{i=1}^ka_i^2\lambda_i\le \lambda_k, $$
whence the result follows.