# An Imbedding

1. Jul 11, 2013

### amirmath

Dear friends. I have a PDE to investigate its behavior at a semi-infinite cylinder. The nontrivial solution is considered to be in $H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$. Is it possible to say
$$H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\hookrightarrow H_{0}^{1}(\Omega)$$
If the answer is positive, then the following estimate is possible:
$$\int_{\Omega} \nabla u \nabla u_{t} dx < \frac {d}{2} \|\Delta u \|_{2}^{2} +\frac {1}{2} \|\nabla u_{t} \|_{2}^{2}$$
where $d$ denotes the imbedding constant.