Is it possible to say that $$ H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\hookrightarrow H_{0}^{1}(\Omega). $$
Precisely, I am dealing with this question: Is it possible to have the following estimate if we take $u\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$? $$ \int_{\Omega} \nabla u \cdot \nabla u_{t} dx < \frac {d} {2}\|\Delta u\|_{2}^{2}+\frac {1}{2}\|\nabla u_{t}\|_{2}^{2} $$ where $d$ denotes the above embedding constant and the norms $\Bigr(\int_{\Omega}|\Delta u|^{2}dx\Bigr)^{\frac {1}{2}}$ and $\Bigr(\int_{\Omega}|\nabla u|^{2}dx\Bigr)^{\frac {1}{2}}$ have been used in $H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$ and $H_{0}^{1}(\Omega)$ respectively.
I'm assuming that $\Omega$ is a bounded domain. Note that if $u\in H_0^1(\Omega)\cap H^2(\Omega)$ then
\begin{eqnarray} \int_\Omega |\nabla u|^2 &=& \int_\Omega u\Delta u \nonumber \\ &\leq& \|u\|_2\|\Delta u\|_2 \nonumber \\ \end{eqnarray}
It follows from Poincare's inequality that $$\|\nabla u\|_2\leq C\|\Delta u\|_2$$
for some positive constant $C$. Now I'm assuming that $\nabla u_t\in L^2(\Omega)$, which implies
\begin{eqnarray} \int_\Omega \nabla u\nabla u_t &\leq& \|\nabla u\|_2\|\nabla u_t\|_2 \nonumber \\ &\leq& C\|\Delta u\|_2\|\nabla u_t\| \nonumber \\ &\le& \frac{C^2\|\Delta u\|_2^2}{2}+\frac{\|\nabla u_t\|^2_2}{2} \end{eqnarray}