Very good I am trying to fully understand the concept of a weak solution for Laplace's equation. I have seen the following stated in a book.
In the context of $\mathbb{R}^n$.
$u \in L^2$ is a weak solution to Laplace's equation if and only if $u \in W^{1,2}$ and $(u, \Delta \varphi)=0$ for all $\varphi \in \mathcal {C}_{0}^{\infty}$ (that is, $u$ is a distributional solution for Laplace's equation). (Such a statement appears in the book Problems on partial differential equations by the authors Maciej Borodzik, Paweł Goldstein, Piotr Rybka, Anna Zatorska-Goldstein.
I have a direction:
Let $u\in W^{1,2}$ and $(u, \Delta \varphi) = 0, \quad \varphi \in \mathcal{C}_{0}^{\infty} $. We will use the fact that the operator $ \Delta $ is self-adjoint on $ W^ {1,2}$, that is, $ \Delta = \Delta^{*} $ over $ W^{1,2} $ and also that, for $v\in D(\Delta^{*}) $, there exists $\left\{\varphi_k\right\}_{k}\subset\mathcal{C}_{0}^{\infty}:\varphi_k\to v$ in $L^2.$ Then
\begin{align} |(u,\Delta^{*}v)|&\leq |(u,\Delta^{*}v)-(u,\Delta\varphi_k)|+|(u,\Delta\varphi_k)|\\ &=|(\Delta u,v)-(\Delta^* u, \varphi_k)|\\ &=|(\Delta u,v)-(\Delta u,\varphi_k)|\\ &=|(\Delta u,v-\varphi_k)|\\ &\leq \left\|\Delta u\right\|_{2}\left\|v-\varphi_k\right\|_{2}\\ &\to 0 \end{align}
(Here I can see that $ \Delta u\in L^2 $ is needed and the fact that $ u\in W^{1,2} $ allows us to ensure this) Therefore, $ (u, \Delta^{*}v) = 0 $ for all $ v\in D(\Delta^*). $ By definition, $ u $ is a weak solution.
Conversely, Let $\varphi\in\mathcal{C}_{0}^{\infty}$, this implies that $\Delta\varphi\in L^2$. Then, $(\varphi,\Delta u)=(\Delta^{*}\varphi,u)$ for all $u\in D(\Delta)$. Equivalent, $\varphi\in D(\Delta^{*})$. And, $\Delta \varphi \in L^2$, i.e by definition $u\in W^{1,2}$, then, $\Delta\varphi=\Delta^{*}\varphi$ because $\Delta$ is self adjoint over $W^{1,2}$. Therefore, $(u,\Delta\varphi)=(u,\Delta^{*}\varphi)=0$. With this I obtain the condition that $ u $ is a distributional solution for Laplace's equation. All this is correct?
and, I can't see: Why $ u\in W ^{1,2}$?