# Eigenfunctions and Spectral Decomposition

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

• 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. – Peter Melech Apr 12 at 13:23
• @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. – Ben Apr 12 at 14:50