Dirichlet energy with domain $W^{1,2}(M)$ or $W^{1,2}_{loc}(M)$ can be a specific Dirichlet form?

Question

M is a Riemannian manifold, $\varepsilon(f,g)=\int_M \langle {\nabla f,\nabla g}\rangle dvol$. Then with which domain is $\varepsilon$ a strongly local, regular and tight Dirichlet form? $W^{1,2}(M)$ or $W^{1,2}_{loc}(M)$(locally) or $W^{1,2}_0(M)$(compact support)?

Answer 1

@Nate Eldredge gives a good necessary and sufficient condition for tightness in his answer to this MO question.

The domain of a Dirichlet form is taken to be a subset of $L^2(M,\mu)$ for some measure $\mu$, assuming $\mu = vol$ and $M$ is not compact, $W^{1,2}_{loc}$ won't be a subset of $L^2(M,\mu)$.

Also, the domain of a Dirichlet form is usually taken to be closed with respect to the $W^{1,2}$-norm, so unless $M$ is compact and $W^{1,2} = W^{1,2}_0$, then $W_0^{1,2}$ is not the domain of a Dirichlet form...it is the domain of a closable form, which is often good enough.

The definitions strongly local and regular apply to $\varepsilon$ with $W^{1,2}_0$ as a domain (even though this isn't a Dirichlet form), but have to be adapted when dealing with $W^{1,2}_{loc}$.