The following questions concerns a problem I am treating in my Masters dissertation.
Let $\Omega $ be an open, bounded domain in $\mathbb{R}^3$. Then the norm
$$
\Vert u\Vert^2 = \Vert u\Vert_2^2 + \Vert\nabla u\Vert_2^2 + \Vert\Delta u\Vert_2^2
$$
equivalent to the usual norm in $H^2 (\Omega)$. Indeed, I know from this question that $(\Vert u\Vert_2^2 + \Vert\Delta u\Vert_2^2)^{1/2}$ is equivalent to the norm in $H^2(\Omega)$. From this follows that for some $c > 0$
\begin{align*}
c \Vert u\Vert_{H^2}^2 & \leq \Vert u \Vert_2^2 + \Vert\Delta u\Vert_2^2 \\
& \leq \Vert u\Vert_2^2 + \Vert\nabla u\Vert_2^2 + \Vert\Delta u\Vert_2^2 \\
& \leq \Vert u\Vert_{H^2}^2
\end{align*}
Now, let
$$
V = \left\{u \in H^2(\Omega) \ : \ \frac{\partial u}{\partial n} = 0 \text{ on } \partial \Omega\right\}
$$
and
$$
\tilde V = \left\{ u \in V \ : \ \int_\Omega u \ dx = 0 \right\}.
$$
How can I show that
$$
\Vert u\Vert = \Vert\nabla u\Vert_2 + \Vert\Delta u\Vert_2
$$
is an equivalent norm on $\tilde V$? Does it follow from the Poincaré-Wirtinger inequality?
Best Answer
As you already noted, $(\|u\|_2+\|\Delta u\|_2)^{1/2}$ is an equivalent norm on $H^2$, hence there is a constant $c>0$ such that $$c\|u\|_{H^2}^2 \leq \|u\|_2^2+\|\Delta u\|_2^2, \quad \forall u \in H^2.$$ This is also true for $u \in \tilde V \subset H^2$. Since the elements in $\tilde V$ have zero mean, we can make use of the following Poincare inequality, $$\|u\|_2^2 \leq C_P\|\nabla u\|^2, \quad \forall u \in \tilde V. $$
Putting these two inequalities together, we have for all $u \in \tilde V$,
$$c\|u\|_{H^2}^2 \leq \|u\|_2^2+\|\Delta u\|_2^2 \leq C_P\|\nabla u\|_2^2 + \|\Delta u\|^2_2 \leq C (\|\nabla u\|_2^2 + \|\Delta u\|^2_2) \leq C \|u\|_{H^2}^2,$$
where $C:=\max\{1,C_P\}$. Hence, $(\|\nabla u\|_2^2 + \|\Delta u\|_2^2)^{1/2}$ is an equivalent norm on $\tilde V$.