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 $\mathcal{D}(\mathcal{A}) \subset \mathcal{F}\_{G}$?** 



Identifying Dirichlet forms of part processes, how to prove
If $f$, $F$, and $\tilde F$ are as in your claim, then $F\in\mathcal F\_G$. Because $f=F$, $m|\_G$-a.e., so too $f=\tilde F$, $m|\_G$-a.e., and $f\in\mathcal F\_G$.


