Is it true that $C_1\|\Delta u \|_{0,\Omega} \leq |u|_{2,\Omega} \leq C_2\|\Delta u \|_{0,\Omega}$ for a bounded domain $\Omega\subset\mathbb{R}^2$?

Question

Let's say $\Omega$ is a bounded domain in $\mathbb{R}^2$. Is it true that

$C_1\|\Delta u \|_{0,\Omega} \leq |u|_{2,\Omega} \leq C_2\|\Delta u \|_{0,\Omega}$?

The left inclusion is obvious by definition. But I have no idea about the right.

Any help is much appreciated.

Answer 1

The answer is yes if $u$ has zero-boundary values and $\Omega$ is sufficiently regular. This follows from $H^2$-regularity of \begin{align}-\Delta u &= f \text{ in } \Omega \\ u &= 0 \text{ on }\partial\Omega\end{align} with $f = -\Delta u \in L^2(\Omega)$. Otherwise it may not hold.