The first Dirichlet eigenfunction by constrained minimisation

Given: A nonempty bounded open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, the Dirichlet energy $E(u)={\int }_{\mathrm{\Omega }}\mathrm{\mid }Du{\mathrm{\mid }}^{2}dx$ on $H_0^1(\mathrm{\Omega };\mathbb{R})$, and the $L^2$-unit sphere $S=\{u\in H_0^1(\mathrm{\Omega }):\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}=1\}$.

Zero-boundary Sobolev space as a norm closure, $H^k$ is a Hilbert space under the derivative-sum inner product, A closed subspace of a Banach space is Banach: $H_0^1(\mathrm{\Omega })$ is the norm closure of $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ in $H^1(\mathrm{\Omega })=W^{1,2}(\mathrm{\Omega })$, hence a closed subspace of the Banach space $H^1(\mathrm{\Omega })$ and itself a real Banach space with the $H^1$ norm.

W^{1,p}(Omega) is reflexive for 1<p<infinity, Closed subspaces of reflexive spaces are reflexive, Reflexivity is surjectivity of the canonical map: under the ultrafilter lemma, DC and HB the space $W^{1,2}(\mathrm{\Omega })$ is reflexive, and under HB its closed subspace $H_0^1(\mathrm{\Omega })$ is reflexive, hence a real reflexive Banach space.

Convex and strictly convex functionals on a convex subset of a real vector space, A convex norm-lower-semicontinuous functional is weakly lower semicontinuous: $E$ is convex, being the squared norm of the bounded linear map $u\mapsto Du$ composed with the convex square; it is continuous because $\mathrm{\mid }E(u)-E(v)\mathrm{\mid }\le \mathrm{\parallel }Du-Dv{\mathrm{\parallel }}_{L^2}(\mathrm{\parallel }Du{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }Dv{\mathrm{\parallel }}_{L^2})$. By the convex-lower-semicontinuity lemma (Axiom of Choice) $E$ is weakly sequentially lower semicontinuous on every nonempty convex subset of $H_0^1(\mathrm{\Omega })$.

Test function cutoffs and euclidean localization: since $\mathrm{\Omega }$ is nonempty and open there is a nonzero $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega })$, for instance a cutoff equal to one on a neighbourhood of a chosen point; then $u_1:=\phi \mathrm{/}\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^2}$ lies in $S$, so $S\mathrm{\ne }\mathrm{\varnothing }$ and ${\lambda }_1:={\inf }_{S}E\le E(u_1)<\mathrm{\infty }$.

The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, Coercivity of the principal Dirichlet form, Dependent choice implies countable choice: since the bounded set $\mathrm{\Omega }$ is bounded in every direction, Poincaré at $p=2$ gives a constant $C_P$ with $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}\le C_P\mathrm{\parallel }Dv{\mathrm{\parallel }}_{L^2}$ for all $v\in H_0^1(\mathrm{\Omega })$; by the coercivity lemma (Axiom of Choice and Countable Choice, the latter from DC) the model principal form satisfies $E(v)=\mathrm{\parallel }Dv{\mathrm{\parallel }}_{L^2}^2\ge (1+C_P^2{)}^{-1}\mathrm{\parallel }v{\mathrm{\parallel }}_{H_0^1}^2$.

A bounded sequence in a reflexive Banach space has a weakly convergent subsequence: under the ultrafilter lemma, DC and HB every norm-bounded sequence in a real reflexive Banach space has a weakly convergent subsequence.

Weak H^1 convergence plus Rellich preserves the L^2 unit normalisation: if $\mathrm{\Omega }$ is bounded and open, $(v_j)\subseteq H_0^1(\mathrm{\Omega })$ is norm bounded with $v_j\rightharpoonup v$ in $H_0^1(\mathrm{\Omega })$ and $\mathrm{\parallel }{v}_j{\mathrm{\parallel }}_{L^2}=1$, then $v\in H_0^1(\mathrm{\Omega })$ and $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}=1$.

The Lagrange multiplier rule for finitely many regular constraints, Fréchet derivative between Banach spaces: let $X$ be a real Banach space, $U\subseteq X$ open, $I:U\to \mathbb{R}$ Fréchet differentiable at $u$, and $G:U\to {\mathbb{R}}^m$ of class $C^1$ with $DG(u)$ surjective; if $u$ is a local minimiser or maximiser of $I$ on the level set $\{G=G(u)\}$, then there is a unique $\lambda \in {\mathbb{R}}^m$ with $DI(u)={\sum }_i{\lambda }_{i}D{G}_i(u)$.

The absolute value preserves the L^2 norm and the Dirichlet energy on H^1_0: for every open $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ and real $v\in H_0^1(\mathrm{\Omega })$, one has $\mathrm{\mid }v\mathrm{\mid }\in H_0^1(\mathrm{\Omega })$, $\mathrm{\parallel }\mathrm{\mid }v\mathrm{\mid }{\mathrm{\parallel }}_{L^2}=\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}$ and ${\int }_{\mathrm{\Omega }}\mathrm{\mid }D\mathrm{\mid }v\mathrm{\mid }{\mathrm{\mid }}^2={\int }_{\mathrm{\Omega }}\mathrm{\mid }Dv{\mathrm{\mid }}^2$.

The Rayleigh principle for the first Dirichlet eigenvalue, Discrete spectrum of a symmetric elliptic Dirichlet operator, The $L^2$ operator associated with a symmetric elliptic form, Symmetric elliptic weak eigenpairs: in the symmetric case over a bounded open set the discrete spectral theorem provides the nondecreasing eigenvalue list ${\lambda }_1\le {\lambda }_2\le \cdots$ of the principal Dirichlet form and an orthonormal basis $\{e_j\}$ of $L^2(\mathrm{\Omega })$ of weak eigenfunctions; the Rayleigh principle states that its first eigenvalue equals ${\min }_{v\mathrm{\ne }0}{a}_0(v,v)\mathrm{/}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}^2$ with $a_0(v,w)={\int }_{\mathrm{\Omega }}Dv\cdot Dw$, the minimum being attained exactly on $E_{{\lambda }_1}\setminus \{0\}$, and that it is positive whenever the form is coercive, as it is for the principal form by [F5].