I meet a problem in homework: Assume $\Omega\subset\mathbb{R}^n$ is bounded and $\partial\Omega$ is $C^1$. Does $H_0^2(\Omega)$ equal to $H_0^1(\Omega)\cap H^2(\Omega)$?
Obviously, $H_0^2(\Omega)\subset H_0^1(\Omega)\cap H^2(\Omega)$. However, I cannot prove $ H_0^1(\Omega)\cap H^2(\Omega) \subset H_0^2(\Omega)$.
Consider the simple case $\Omega=(0,1)$. A function $u$ is in $H^2_0(\Omega)$ if there exists a sequence $u_n$ of $C^\infty(\Omega)$ functions with compact support such that $$ \lim_{n\to\infty}\int_0^1\bigl(|u-u_n|^2+|u'-u_n'|^2+|u''-u_n''|^2\bigr)=0. $$ In particular $u'\in H^1_0(\Omega)$. The function $u(x)=\sin(\pi\,x)$ is in $H^1_0\cap H^2$, but $u'\not\in H^1_0$, so that $u\not\in H^2_0$.