I have a question about Dirichlet forms.
Let $D$ be a domain of $\mathbb{R}^d$ and $H^{1}(D)$ denotes $(1,2)$-Sobolev space on $D$ with Neumann boundary condition. We define the following a Dirichlet form on $L^{2}(D,dx)$: \begin{align*} \mathcal{E}(f,g)=\frac{1}{2}\int_{D}(\nabla f,\nabla g)\,dx,\quad f,g \in H^{1}(D). \end{align*}
Let $\bar{D}$ denotes the closure of $D$ in $\mathbb{R}^d$. We define a measure $m$ on $(\bar{D},\mathcal{B}(\bar{D}))$ by $$m(A)=\lambda(A \cap D),\quad A \in \mathcal{B}(\bar{D}),$$ where $\lambda$ is the Lebesgue measure on $\mathbb{R}^d$. In the following, we identify $L^{2}(D,dx)$ with $L^{2}(\bar{D},m)$. Then, $(\mathcal{E},H^{1}(D))$ is a Dirichlet form on $L^{2}(\bar{D},m)$. Furthermore, under some assumptions on $D$ (e.g. $\partial D$ is smooth), $(\mathcal{E},H^{1}(D))$ is a regular Dirichlet form on $L^{2}(\bar{D},m)$.
Let $(\mathcal{E},H^{1}(D))$ is a regular Dirichlet form on $L^{2}(\bar{D},m)$ and $G$ be an open subset of $\bar{D}$ (not $D$!). Then, We can define the following space $$\mathcal{F}_{G}=\{f \in H^{1}(D) \mid \tilde{f}=0 \text{ q.e. on }\bar{D} \setminus G \},$$ where $\tilde{f}$ is a quasi continuous version of $f \in H^{1}(D)$.
$\mathcal{F}_{G}$ is a subspace of $\{f \in L^{2}(\bar{D},m)\mid f=0 \,m\text{-a.e. on }\bar{D}\setminus G\}$.
We regard $\mathcal{F}_{G}$ as a subspace of $L^{2}(G,m|_{G})$ under the natural identification of $L^{2}(G,m|_{G})$ with the closed linear subspace $\{f \in L^{2}(\bar{D},m)\mid f=0 \,m\text{-a.e. on }\bar{D}\setminus G\}$ of $L^{2}(\bar{D},m)$. It is known that $(\mathcal{E}|_{\mathcal{F}_{G} \times \mathcal{F}_{G}}, \mathcal{F}_G)$ is a regular Dirichlet form on $L^{2}(G,m|_{G})$.
My question
Let $(\mathcal{A},\mathcal{D}(\mathcal{A}))$ be a regular Dirichlet form on $L^{2}(G,m|_{G})$. To show two Dirichlet forms $(\mathcal{E}|_{\mathcal{F}_{G} \times \mathcal{F}_{G}}, \mathcal{F}_G)$ and $(\mathcal{A},\mathcal{D}(\mathcal{A}))$ agree, what should I prove? At least, we should prove $$\mathcal{D}(\mathcal{A}) \subset \mathcal{F}_{G}.$$
Suppose that I could prove the following claims:
- For any $f \in \mathcal{D}(\mathcal{A})$, there exists $F \in H^{1}(D)$ such that its quasi continuous version $\tilde{F}$ satisfies $\tilde{F}=0$ q.e. on $\bar{D} \setminus G$ (such $F$ is regarded as an element of $L^{2}(G,m)$) and $f=F$ $m|_{G}$-a.e.
Then, this claims shows $\mathcal{D}(\mathcal{A}) \subset \mathcal{F}_{G}$?
