I am currently working with the weak form of the Biharmonic equation. I am trying to figure out how to prove that in the space $$ H^2_0(\Omega) = \left\{ u \in H^2(\Omega) : u = \partial_\nu u = 0 \text{ on } \Gamma \right\} $$ where $ \Omega $ is a bounded open set and $\Gamma$ is a sufficiently smooth boundary with outward facing unit normal $ \nu $, that the seminorm $| u |_{2, \Omega}$ and the $L^2$-norm of the laplacian $\| \Delta u \|_{2, \Omega}$ are equivalent.
My attempted proof went like this:
Let $u \in H^2_0 (\Omega)$. By definition, we have that
$$ |u|_{2, \Omega} = \int_\Omega \sum_i (\partial_{ii} u)^2 dx + \int_\Omega \sum_{i \neq j} \partial_{ij} u \partial_{ij} u dx \\ \| \Delta u\|_{0, \Omega} = \int_\Omega \sum_i (\partial_{ii} u)^2 dx + \int_\Omega \sum_{i \neq j} \partial_{ii} u \partial_{jj} u dx $$
By two applications of the fundamental Greens formula, and using that the boundary integrals vanish, the two expressions can be seen to be equal. However, I suddenly realised that by doing these manipulations, I am implicitly assuming that $ u \in H^3(\Omega) \cap H^2_0(\Omega) $, am I not?
I have seen the proof where they do the manipulations above, but for $u$ in $C^\infty_0(\Omega)$, and then concluding with a "density" argument. However, I am not sure how such density arguments work. Could anyone help me by pointing me in the right direction, maybe a reference or so?
Thanks in advance!
After looking at the problem in more detail, with help from Rhys Steele, I have come to the following proof.
Observation 1.
The space $H^2_0(\Omega)$ is defined as the completion of the space $C^\infty_0(\Omega)$ with respect to the $\| \cdot \|_{2, \Omega}$-norm. This means that by definition, $C^\infty_0(\Omega)$ is a dense subset of $H^2_0(\Omega)$.
Observation 2.
Under certain smoothness requirements on the boundary $ \Gamma $ of $ \Omega$, one may characterise the space $H^2_0(\Omega)$ as $$ H^2_0(\Omega)= \left\{u \in H^2(\Omega) : u = \partial_\nu u = 0 \text{ on } Γ \right\}. $$
Observation 3.
The two maps $u \mapsto | u |_{2, \Omega}$ and $u \mapsto \| \Delta u \|_{0, \Omega}$ can be shown to be continuous with respect to the $\| \cdot \|_{2, \Omega}$-norm.
Lemma 1:
Let $A \subset M$ be a dense subset of $M$. If $f, g \colon M \to N$ are two continuous functions that are equal on $A$, then they are equal on $M$. That is,
$$ \forall x \in A \, (f(x) = g(x)) \implies \forall x \in M \, (f(x) = g(x)). $$
In view of Lemma 1, we may show that $|u|_{2, \Omega}$ and $\| \Delta u \|_{0, \Omega}$ are equal on the dense subset $ C^\infty_0(\Omega)$ of $H^2_0(\Omega)$. Since the two maps are continuous, the equality may then be extended to hold for all of $H^2_0(\Omega)$.
Proof of equality:
Let $u \in C^\infty_0(\Omega)$. Since the functions in this space are infinitely continuously differentiable, we may move derivatives around as we please. Furthermore, any boundary integrals vanish, as the functions are compactly supported. Hence, by two applications of Green's lemma, we have that
$$ \int_\Omega \sum_{i \neq j} \partial_{ij} u \partial_{ij} u dx = - \int_\Omega \sum_{i \neq j} \partial_{i} u \partial_{ijj} u dx = \int_\Omega \sum_{i \neq j} \partial_{ii} u \partial_{jj} u dx. $$
Consequently, we conclude that $|u|_{2, \Omega}$ and $\| \Delta u \|_{0, \Omega}$ agree on $C^\infty_0(\Omega)$. This result then extends to $H^2_0(\Omega)$ by Lemma 1.