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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.

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  • $\begingroup$ I think the answer is no. For example, let $u=xy$ then $|u|_{2,\Omega} = \sqrt{|\Omega|}$, while $\| \Delta u\|_{0,\Omega}=0$. $\endgroup$ – Ana Uspekova Jan 26 '16 at 16:53
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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.

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  • $\begingroup$ Thank you, I got the idea. Function $u$ in the question only belongs to $H^1(\Omega)$, so it's not true in general. $\endgroup$ – Ana Uspekova Jan 27 '16 at 0:17

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