I want to ask a question that arises from reading [this paper](http://www.sciencedirect.com/science/article/pii/0022123676900173).



> 
> Let $X$ be a locally compact space which is countable at infinity and let $\xi$ be a Radon measure on $X$. Suppose $V$ is a Hilbert space and there is a continuous and coercive bilinear form $a:V \times V \to \mathbb{R}$ such that
> 
> 
> 1. $V$ is continuously embedded in $L^2(X,\xi)$
> 2. If $C\_0(X)$ is the space of continuous functions with compact support in $X$ then $V \cap C\_0(X)$ is dense in $C\_0(X)$
> 3. For all $v \in V$, $|v| \in V$ and $a(v^+,v^-) \leq 0$.
> 
> 
> We call $(V,a)$ a Dirichlet space. 
> 
> 
> 


(The $v^+$, $v^-$ makes sense if we consider $V$ as a sublattice of $L^2(X,\xi)$ so that $v\_1 \geq v\_2$ means $v\_1 \geq v\_2$ $\xi$-a.e. in $X$.)


The author then gives two examples, both on bounded regular domain $\Omega$ in $\mathbb{R}^n$ with $\xi$ chosen to be the Lebesgue measure.



> 
> * (Ex 1) $X=\Omega$, $V=H^1\_0(\Omega)$, $$a(u,v) = \int\_\Omega \nabla u\cdot \nabla v + uv$$
>  So far so ok.
> * (Ex 2) $X=\bar \Omega$ (closure of $\Omega$), $V=H^1(\Omega)$, $a(u,v)$ same as above.
> 
> 
> 


I have two questions:


1. Given a bilinear form $a$ and a space $V$ (let's say some Sobolev space on $\Omega$) and given the conditions that $X$ must satisfy, is this information enough to determine uniquely what $X$ should be?
2. Consider the two examples: why would I think to choose $X=\bar\Omega$ in the second example? I don't see what goes wrong if I choose plain $\Omega$ for $X$.


The author cites a thesis of Ancona for related reading, but this thesis is impossible to find (for me).



Two questions related to Dirichlet spaces and Sobolev spaces
For question 2, if you had $\Omega$ in place of $\bar{\Omega}$, the conditions you state would still be satisfied, but there are other interesting conditions that would not be. It would fail to be a *regular* Dirichlet space. In particular, you could not find a nice Markov process with state space $X = \Omega$ whose Dirichlet form was $a$. (Morally speaking, this process should be a Brownian motion in $\Omega$ that is killed at rate 1 inside $\Omega$, and reflected at the boundary. But in order to be reflected at the boundary, the process has to be able to *reach* the boundary, which you can't do with a process whose state space is only $\Omega$.)


(Incidentally, the author probably meant to say $V = H^1(\bar{\Omega})$.)


The standard modern reference on Dirichlet forms is Fukushima, Oshima and Takeda, *Dirichlet Forms and Symmetric Markov Processes*.


