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.

  • $\begingroup$ 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

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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.