(Theory 9.31, from Haim Brezis functional analysis Sobolev space and partial differential equations, P311, chapter 9)
$\Omega$ is bounded open set.
There exist a Hilbert basis $(en)_{n\geq 1} $ of $L^2(\Omega)$ and a sequence $(\lambda_n)_{n\geq 1} $1 of reals with $\lambda_n > 0$ $\forall n$ and $\lambda_n\rightarrow +\infty $ such that
$$ e_n\in H_0^1(\Omega)\cap C^{\infty}(\Omega) $$ $$ -\Delta e_n = \lambda_ne_n ~~~in~~~ \Omega. $$
We say that the $\lambda_n $ ’s are the eigenvalues of −$\Delta$ (with Dirichlet boundary condition) and that the $e_n$’s are the associated eigenfunctions.
the idea of the proof is Given $f\in L^2(\Omega)$ and let $u=Tf$ be the unique solution $u\in H_0^1(\Omega)$ of the problem
$$ \int_{\Omega}\triangledown u\cdot\triangledown \varphi = \int_{\Omega}f\varphi,~~\forall \varphi\in H_0^1(\Omega) .$$
then we can proof $T$ is a self-adjoint compact operator.
BUT IN $Remark ~29$
Under the hypotheses of Theorem 9.31 (for a general bounded domain $\Omega$) it can be proved that $e_n\in L^{\infty}(\Omega)$.On the other hand, if $\Omega$ is of class $C^{\infty}$ then $e_n\in C^{\infty}(\bar{\Omega})$;)
It is natural that $(e_n) \in C^{\infty}(\Omega)$, but I don't know how to proof $e_n\in L^{\infty}(\Omega)$.