I have a question about compactness of semigroups and boundary conditions.


Let $\Omega$ be an **unbounded** domain of $\mathbb{R}^d$ with smooth boundary and $m(\Omega)=\infty$. Then we can define two bilinear forms (Dirichlet forms) on $L^{2}(\Omega)$.
\begin{align\*}
&\mathcal{E}^{D}(f,g)=\int\_{\Omega}(\nabla f, \nabla g)\,dx,\quad f,g \in H\_{0}^1(\Omega), \\
&\mathcal{E}^{N}(f,g)=\int\_{\Omega}(\nabla f, \nabla g)\,dx,\quad f,g \in H^1(\Omega), \\
\end{align\*}
where $H\_{0}^1(\Omega)$ and $H^1(\Omega)$ denote $(1,2)$-Sobolev space with Dirichlet boundary condition and $(1,2)$-Sobolev space with Neumann boundary condition, respectively. $\{T\_t^D\}\_{t>0}$ (resp. $\{T\_t^{N}\}\_{t>0}$) denotes the semigroup on $L^{2}(\Omega)$ associated with $(\mathcal{E}^D,H\_0^1(\Omega))$ (resp. $(\mathcal{E}^N,H^1(\Omega))$). We say that $\{T\_t^D\}\_{t>0}$ (resp. $\{T\_t^{N}\}\_{t>0}$)
 is compact if $T\_{t}^{D}$ (resp. $\{T\_t^{N}\}\_{t>0}$) is compact operator for every $t>0$. 


**Question**


Do you know any examples of $\Omega$ such that $\{ T\_{t}^D \}$ is compact and $\{T\_{t}^N\}$ is not compact ?



Compactness of semigroups, boundary conditions
You will find examples in this paper:
<http://math.caltech.edu/SimonPapers/212.pdf>


