(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 $e_n\in L^{\infty}(\Omega)$.

  • $\begingroup$ You mean by " it is natural that $e_n\in C^{\infty}(\Omega)$ in case $\Omega$ is of class $C^{\infty}$, i.e. with smooth boundary because of Th.9.25, I guess? In the general case it can be proved, he says,but may be not that easily. Unfortunately he does not cite any sources. $\endgroup$ – Peter Melech Apr 12 at 13:23
  • $\begingroup$ @PeterMelech yes, according to th.9.25( indeed, Remark 25.), can prove $(e_n) \in C^{\infty}(\Omega)$, but it's not easy to show $e_n \in L^{\infty}$ for general bounded $\Omega$.But in french version, it is a exercise[EX], Unfortunately, i didn't find any hint. $\endgroup$ – Ben Apr 12 at 14:50

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