I have a question about a Dirichlet form.

Let $D$ be a open subset of $\mathbb{R}^d$. Then, we can define $H^{1}(D)$ by \begin{equation*} H^{1}(D)=\{f \in L^{2}(D,dx):\frac{\partial f}{\partial x_i} \in L^{2}(D,dx),\,1\le i\le d\}. \end{equation*} It is well known that $H^{1}(D)$ becomes a Hilbert space with inner norm \begin{equation*} (f,g)_{H}:=\mathcal{E}(f,g)+\int_{D}fg\,dx, \end{equation*} where $\mathcal{E}(f,g):=\frac{1}{2}\int_{D}\frac{\partial f}{\partial x_i}\frac{\partial g}{\partial x_i}\,dx$. Moreover, $(\mathcal{E}, H^{1}(D))$ becomes a Dirichlet form on $L^{2}(D,dx)$. Hence, from a general theory of Dirichlet form, there exists a unique (non-positive) closed linear operator $(L,\text{Dom}(L))$ such that \begin{equation*} (-Lf,g)=\mathcal{E}(f,g),\quad f \in \text{Dom}(L),\ g\in H^{1}(D). \end{equation*}

**My question**

If $D =\mathbb{R}^d$, I know $\text{Dom}(L)=W^{2,2}(\mathbb{R}^d)$.

- For a general open subset $D$, $\text{Dom}(L)=W^{2,2}(D)$?
- Is there a sufficient condition for $f \in \text{Dom}(L)$?

If you know related results, please let me know.