The Courant-Fischer min-max principle for elliptic eigenvalues

For a finite-dimensional nonzero $S\subseteq H_0^1$, choose an $L^2$-orthonormal basis using Every finite-dimensional real or complex inner product space has an orthonormal basis. In its real coordinates (real and imaginary coordinates when the field is complex), the $L^2$ unit sphere is a nonempty compact Euclidean sphere by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, and $a(u,u)$ is a continuous quadratic polynomial there. It has a maximum by A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value. Homogeneity identifies that maximum with ${\max }_{S\setminus \{0\}}Q$, so every inner maximum in the statement exists. Weighted averages. For $u\mathrm{\ne }0$ the quotient is the weighted average $Q(u):=a(u,u)\mathrm{/}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2={\sum }_j{\lambda }_j\mathrm{\mid }{c}_j{\mathrm{\mid }}^2\mathrm{/}{\sum }_j\mathrm{\mid }{c}_j{\mathrm{\mid }}^2$ with ${\sum }_j\mathrm{\mid }{c}_j{\mathrm{\mid }}^2>0$; if $u$ lies in the span of finitely many eigenvectors $e_{j_1},\dots ,e_{j_m}$ then the quotient is the corresponding finite convex combination of the ${\lambda }_{j_i}$.

The min-max identity. Let $k\ge 1$ and put $S_k:=\mathrm{span}\{e_1,\dots ,e_k\}$, a $k$-dimensional subspace on which $Q(u)$ is a weighted average of ${\lambda }_1,\dots ,{\lambda }_k$, hence at most ${\lambda }_k$, with value ${\lambda }_k$ at $u=e_k$; therefore the min over $k$-dimensional $S$ of ${\max }_{S\setminus \{0\}}Q$ is at most ${\lambda }_k$. Conversely, for any $k$-dimensional $S\subseteq H_0^1(\mathrm{\Omega })$ the $(k-1)$ linear functionals $u\mapsto (u,e_j{)}_{L^2}$, $j<k$, have a nonzero common zero $u\in S\setminus \{0\}$ by [F2]; then $c_j=0$ for $j<k$, so $Q(u)$ is a weighted average of ${\lambda }_k,{\lambda }_{k+1},\dots$ and is at least ${\lambda }_k$. Hence every $k$-dimensional $S$ contains a direction of quotient at least ${\lambda }_k$, so the minimum is exactly ${\lambda }_k$, attained at $S_k$; this proves the first displayed identity.

The max-inf identity. Let $T_{k-1}:=\mathrm{span}\{e_1,\dots ,e_{k-1}\}$ (the zero subspace for $k=1$); on $H_0^1(\mathrm{\Omega })\cap T_{k-1}^{{\perp }_{L^2}}$ the quotient is a weighted average of ${\lambda }_k,{\lambda }_{k+1},\dots$ by [F1], so its infimum equals ${\lambda }_k$, attained at $u=e_k$; hence the outer maximum is at least ${\lambda }_k$. Conversely, let $T$ be any $(k-1)$-dimensional subspace and choose a basis $t_1,\dots ,t_{k-1}$ (the empty basis when $k=1$). The linear map $J:\mathrm{span}\{e_1,\dots ,e_k\}\to {\mathbb{K}}^{k-1}$ given by $J(u)=((u,t_1{)}_{L^2},\dots ,(u,t_{k-1}{)}_{L^2})$ has a nonzero kernel by [F2], since its domain has dimension $k$ and its codomain has dimension $k-1$. A nonzero $u$ in that kernel lies in $H_0^1(\mathrm{\Omega })\cap T^{{\perp }_{L^2}}\cap \mathrm{span}\{e_1,\dots ,e_k\}$; its quotient $Q(u)$ is a weighted average of ${\lambda }_1,\dots ,{\lambda }_k$ and is therefore at most ${\lambda }_k$. Thus the infimum over $H_0^1(\mathrm{\Omega })\cap T^{{\perp }_{L^2}}$ is at most ${\lambda }_k$ for every $T$. Hence the outer maximum is exactly ${\lambda }_k$, attained at $T=T_{k-1}$.

Attainment. The extreme subspaces $S_k$ and $T_{k-1}$ are explicit finite-dimensional spans of the eigenbasis, and the values ${\lambda }_k$ are attained at $e_k$; no smoothness of $\mathrm{\partial }\mathrm{\Omega }$ entered the argument, which uses only the eigenbasis expansion and linear algebra.