Symmetric elliptic weak eigenpairs

Assume Countable Choice. In the symmetric case of The $L^2$ operator associated with a symmetric elliptic form, a weak eigenpair of the Dirichlet problem for $L$ is a pair $(\lambda ,u)$ with $\lambda \in \mathbb{R}$, $u\in H_0^1(\mathrm{\Omega })\setminus \{0\}$ and $a(u,v)=\lambda (u,v{)}_{L^2}\text{for every }v\in H_0^1(\mathrm{\Omega });$ $\lambda$ is a weak eigenvalue and $u$ a weak eigenfunction. The eigenspace $E_{\lambda }:=\{u\in H_0^1(\mathrm{\Omega }):a(u,v)=\lambda (u,v{)}_{L^2}\mathrm{\forall }v\}$ is a closed linear subspace. Weak eigenpairs are exactly operator eigenpairs: $(\lambda ,u)$ is a weak eigenpair if and only if $u\in D(L)\setminus \{0\}$ and $Lu=\lambda u$ (Eigenvalues, eigenvectors, eigenspaces $E_{\lambda }(T)=\ker (T-\lambda I)$, and the spectrum ${\sigma }_F(T)$ of an endomorphism, The $L^2$ operator associated with a symmetric elliptic form). Eigenfunctions are $L^2$ equivalence classes; no canonical representative or canonical vector in a multiple eigenspace is selected. The multiplicity of $\lambda$ is $\dim E_{\lambda }$, with no finiteness assertion for general open $\mathrm{\Omega }$.

Well-definedness, recorded with the definition. $E_{\lambda }$ is a linear subspace because $a$ and the $L^2$ pairing are linear in the first argument; it is closed in $H_0^1(\mathrm{\Omega })$ because both $u\mapsto a(u,v)$ and $u\mapsto (u,v{)}_{L^2}$ are continuous on $H_0^1(\mathrm{\Omega })$ for each fixed $v$: boundedness of $a$ and the estimate $\mathrm{\mid }(u,v{)}_{L^2}\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{H_0^1}\mathrm{\parallel }v{\mathrm{\parallel }}_{H_0^1}$ show that weak limits of vectors in $E_{\lambda }$ remain in $E_{\lambda }$. The equivalence with operator eigenpairs is the definition of $D(L)$ and $Lu$: the weak identity for $(\lambda ,u)$ says exactly that the datum $f=\lambda u$ represents $a(u,\cdot )$ on $H_0^1(\mathrm{\Omega })$, which, together with $u\mathrm{\ne }0$, is the pair of conditions $u\in D(L)\setminus \{0\}$ and $Lu=\lambda u$. The eigenvalue $\lambda$ is required to be real, as is forced for the symmetric form once $u\mathrm{\ne }0$: taking $v=u$ gives $\lambda \mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}^2=a(u,u)=\overline{a(u,u)}$. All equalities are equalities of $L^2$ and $H_0^1$ classes (The space $L^p(\mu )$ as the quotient by null functions, Zero-boundary Sobolev space as a norm closure, Uniformly elliptic divergence-form operators and their sesquilinear forms, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no regularity of eigenfunctions and no boundary values beyond membership in $H_0^1$ are asserted.