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 ?
