Let M be an Riemannian manifold with the Dirichlet form $$\varepsilon (u,v) =-\int_M \langle \nabla u,\nabla v \rangle$$ for $u,v \in W^{1,2}_0(M)$. Let $\Delta^M:D(\Delta^M) \to L^2(M)$ denote the generator induced from $(\varepsilon, W^{1,2}_0(M)$. It's hard for me to describe $D(\Delta^M)$.

Let $f \in L^2(M)$ and $u \in W_0^{1,2}(M)$. If $$L_u(\phi)=\varepsilon (u,\phi)= \int_\Omega {f\phi dvol} $$ for all nonnegative $\phi \in W^{1,2}_0(M)$. In this case, we know the functional $L_u$ is a signed radon measure without sigular part w.r.t vol. And the absolute part is $fvol$. Then can we get $u \in D(\Delta^M)$ and $\Delta^M u=f$?

If this is right, can we extend this result to general metric measure space?