Poincaré inequality for $H^2\cap H_0^1$

Question

Let $\Omega$ be an open and bounded subset in $\mathbb{R}^2$. Let $H$ be the space defined as $$H=W^{2,2}(\Omega)\cap W^{1,2}_0(\Omega).$$ It is a Hilbert space and equipped with the following norm $$\|u\|_{H}:=(1/2)\int_{\Omega}|\Delta u|^2dx.$$ I want to know the Poincaré inequality in $H$ hold or no. $$\|u\|_{0,\Omega}\leq C_p\|\Delta u\|_{0,\Omega},\qquad\forall u\in H,$$ where $C_p>0$ is the constant in the Poincaré inequality on $\Omega$.

Answer 1

The space $H$ is not the same as what you wrote in the comments. The condition $u \in W^{1,2}_0(\Omega)$ just means $u=0$ in the trace sense on $\partial \Omega$. You would need $u \in W^{3,2}_0(\Omega)$ to conclude that $\Delta u = 0$ on $\partial \Omega$ in the trace sense. I'm also not sure what your notation $\|\cdot \|_{0,\Omega}$ means.

Let me try to answer the question I think you are asking. Since $\Delta u \in L^2(\Omega)$, you can use elliptic regularity (see e.g, Evans Chapter 6) to show that

$\|u\|_{W^{2,2}(\Omega)} \leq C\|\Delta u\|_{L^2(\Omega)}$

provided the boundary is $C^2$. Perhaps this is the estimate you are looking for?