Hilbert space representation in $H_0^1$

Question

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain and let $\partial\Omega$ be of class $C^2$. Let $v_k$ be the $k$-th eigenfunction of $-\Delta v=\lambda v$ and boundary condition $v=0$ on $\partial\Omega$ with eigenvalue $\lambda_k$.
We know that $v_k$ are smooth and $0<\lambda_1<…<\lambda_k\to \infty$. It is also given that $(v_k)_{k\in \mathbb{N}}$ is an orthonormal basis of $L^2(\Omega)$ and $v_k\in H_0^1(\Omega)$.
I want to derive a representation formula in $H_0^1(\Omega)$ with respect to $v_k$. Moreover $v_k$ is smooth and Hölder-continuous up to the boundary. Let $v\in H_0^1(\Omega)$. By the representation in $L^2(\Omega)$ and $-\Delta v_k=\lambda_k v_k$
\begin{align*}
v=\sum\limits_{k=1}^{\infty} (v,v_k)_{L^2(\Omega)} v_k&=\sum\limits_{k=1}^{\infty} \int\limits_{\Omega} v(x) v_k(x) dx v_k\\
&=\sum\limits_{k=1}^{\infty}\frac{1}{\lambda_k} \int\limits_{\Omega} v(x)\left(-\Delta v_k(x)\right)dx v_k\\
&=\sum\limits_{k=1}^{\infty} \frac{1}{\lambda_k} (v,v_k)_{H_0^1(\Omega)}v_k.
\end{align*}
I would like to get the last line by using integration by parts. Using first $(\tilde{v}_l)_{l\in \mathbb{N}}\subset C_0^{\infty}(\Omega)$ with $\tilde{v}_l\to v$ in $H_0^1(\Omega)$, we have $\tilde{v}_l\to v$ in $L^2(\Omega)$ and hence $(\tilde{v}_l)_{l\in \mathbb{N}}$ is bounded in $L^2(\Omega)$. I have to make sure that I'm allowed to swap the limit and the series. Unfortunately I don't know how "fast" $\lambda_k$ converges to infinity so I can't just use Hölder's inequality to get a majorante. Is it possible to conclude like this? I would be grateful for any help!

Answer 1

If I understand the question correctly, the OP wants to know how to obtain the identity $$ \int_\Omega v (-\Delta v_k) dx = \int_\Omega \nabla v \cdot \nabla v_k dx. $$ As indicated by the OP himself, this can be obtained via an approximation argument. Since $v\in H^1_0(\Omega)$ there is a sequence of $\phi_l\in C^\infty_0(\Omega)$ such that $\phi_l\rightarrow v$ in $H^1_0(\Omega)$. Consequently, \begin{align*} \int_\Omega v (-\Delta v_k) dx &= \lim_{l\rightarrow \infty} \int_\Omega \phi_k (-\Delta v_k) dx \\ & = \lim_{l\rightarrow \infty} \int_\Omega \nabla \phi_k \cdot \nabla v_k dx \quad\text{(integration by parts)}\\ & = \int_\Omega \nabla v \cdot \nabla v_k dx. \end{align*}