denoted $$\label{eq:Dmax}
  \mathcal{D}_{\max} := \mathcal{D}_{\max}(\Delta^{g_{ice}}) = {\left\{  u \in L^2 \mid P u \in
    L^2 \right\} }.$$ This is the largest subset of $L^2$ which is a closed subspace in the graph norm $\| u \|_\Gamma = \| u \|_{L^2} + \| P u \|_{L^2}$. On the other hand, the smallest such closed extension from the domain $C^\infty_c(M)$ is the closure, i.e. the minimal domain $$\label{eq:Dmin}
  \mathcal{D}_{\min} := \mathcal{D}_{\min}(\Delta^{g_{ice}}) = {\left\{  u \in L^2 \mid
    \exists u_k \in C^\infty_c(M) \mbox{ with }\lim_{k \to \infty} \|
    u_k - u \|_\Gamma = 0 \right\} }.$$ The essential self-adjointness statement in Theorem \[thm:essentiallyselfadjoint\] says that the smallest closed extension is equal to the largest, i.e. that $$\label{eq:Dmax=Dmin}
  \mathcal{D}_{\max} = \mathcal{D}_{\min},$$ and therefore there is exactly one closed extension. On the other hand, $\mathcal{D}_{\max}$ is dual to $\mathcal{D}_{\min}$ with respect to $L^2$ and thus if holds then $P$ with core domain $C^\infty_c(M)$ has exactly one closed extension, which we denote by $\mathcal{D} = \mathcal{D}_{\min} = \mathcal{D}_{\max}$ and $(P, \mathcal{D})$ is a self-adjoint, unbounded operator on $L^2$. Equation is a statement about decay in the sense that to prove it we will show that a differential form $\alpha \in \mathcal{D}_{\max}$ decays fast enough near ${\partial}M$ 