Let $D \subset \mathbb{R}^d$ be a bounded $C^1$ domain. We consider a reflected Brownian motion $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in \overline{D}})$ on $\overline{D}$. Let $\{p_t\}_{t>0}$ denote the semigroup of $X$ (in other words, $\{p_t\}_{t>0}$ is the Neumann semigroup). It is known that $\{p_t\}_{t>0}$ is a feller semigroup. That is, we have $p_t f \in C(\overline{D})$ for any $t>0$ and $f \in C(\overline{D})$. Here, $C(\overline{D})$ is the space of continuous functions on $\overline{D}.$ We always equip $C(\overline{D})$ with the sup-norm so that it becomes a real Banach space.
Let $(L,D(L))$ be the Neumann Laplacian on $C(\overline{D})$. That is, \begin{align*} D(L)&=\left\{ f \in C(\overline{D})\mid \lim_{t \to 0}\frac{p_t f-f}{t} \text{ exists in }C(\overline{D})\right\},\\ Lf&=\lim_{t \to 0}\frac{p_t f-f}{t}. \end{align*}
Let $\nu$ denote the inward unit normal vector on $\partial D$, and define \begin{align*} \mathcal{C}=\{f \in C^2(\mathbb{R}^d)|_{\overline{D}} \mid \langle \nabla f(x),\nu(x) \rangle=0\text{ for every $x \in \partial D$}\}. \end{align*} Can we prove that $\mathcal{C}$ is a core for $(L,D(L))$?