I'm, working in the following problem.
Consider the problem $$\left\{\begin{array}{c}\begin{aligned}-\Delta u=& \lambda u, \hspace{3mm}in \hspace{2mm} \Omega \\ u=&0, \hspace{3mm}in \hspace{2mm} \Omega \end{aligned} \end{array}\right. .$$ Where $\Omega \subset \mathbb{R}^{n}$ is a bounded and open set. Prove
- The eigenfunctions are orthogonal functions in $L^{2}(\Omega)$.
- Let $k \in \mathbb{N}\cup \{0\}$, consider the set $V_{k}\subset H_{0}^{1}(\Omega)$ the subspace generated by $\{\varphi_{i}\}_{i}^{k}$ prove the following variationals inequalities $$\int_{\Omega}|\nabla u|^{2}\leq \lambda_{k}\int_{\Omega}u^{2},\hspace{4mm}\text{for each}\hspace{3mm}u \in V_{k}$$ and $$\int_{\Omega}|\nabla u|^{2}\geq \lambda_{k+1}\int_{\Omega}u^{2},\hspace{4mm}\text{for each}\hspace{3mm}u \in V_{k}$$ If $k=0$ in the fist equation, then we have the Poincaré inequality.
I'm able to prove the first item, my attempt:We will denote $\langle , \rangle $ as the inner product in $L^{2}(\Omega)$, Consider $\varphi_{i}$ and $\varphi_{j}$ the eigenfunctions with eigenvalues $\lambda_{i}$ and $\lambda_{j}$ respectively, and $\lambda_{j}\neq \lambda_{j}$ we have \begin{equation}\begin{aligned} \lambda_{i}\langle \varphi_{i},\varphi_{j}\rangle -\lambda_{j}\langle \varphi_{i},\varphi_{j}\rangle =& \langle \lambda_{i}\varphi_{i},\varphi_{j}\rangle -\langle \varphi_{i},\lambda_{j}\varphi_{j}\rangle \\ &\langle-\Delta\varphi_{i},\varphi_{j}\rangle - \langle \varphi_{i},-\Delta \varphi_{j}\\ &\langle\varphi_{i},-\Delta\varphi_{j}\rangle - \langle \varphi_{i},-\Delta \varphi_{j}\rangle =0\end{aligned}\end{equation} So, we have $(\lambda_{i}-\lambda_{j})\langle \varphi_{i},\varphi_{j}\rangle=0$, since $\lambda_{i}\neq \lambda_{j}$ we have the desired result. This works?
In the item 2. I'm trying to use $\lambda_{1}\int_{\Omega}u^{2}\leq \int_{\Omega}|\nabla u|^{2}$, but really I don't know how use this fact or if this idea work.
Thanks!