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Let $\Omega$ be a domain of $R^n, \; n\ge 1.$

I am wondering about the existence of an open subset $\omega$ of $\Omega$ such that $ \|y\|_{L^\infty(\omega)} \le C \|y\|_{H^2(\Omega)},\; \forall y\in H^2(\Omega) $

for some constant $C>0?$

In the case $n=1,$ the response is yes with $\omega=\Omega.$ For $n\ge 2$, the inequality is not true for $\omega=\Omega.$

Thanks!

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This would imply that there is an open subset $\omega \subset \Omega$ such that all $u \in H^2(\Omega)$ are bounded on $\omega$. That's not correct for $n \ge 2$. You can find $u_0 \in H^2(D)$ where $D$ is the unit ball such that $\inf_{|x| < \epsilon} u_0(x) \to \infty$ as $\epsilon \downarrow 0$, that is, $u_0(0) = \infty$. You can then move this singular behavior to any point in $\Omega$ by shifting and rescaling $u_0$.