  We show the $L^r(\mathbb{R}^d, \mu)$-uniqueness for any $r \in (1, 2]$ and
the essential self-adjointness of a Dirichlet operator $Lf = \Delta f +\langle
\frac{1}{\rho}\nabla \rho , \nabla f \rangle$, $f \in
C_0^{\infty}(\mathbb{R}^d)$ with $d \geq 3$ and $\mu=\rho dx$. In particular,
$\nabla \rho$ is allowed to be in $L^d_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ or in
$L^{2+\varepsilon}_{loc}(\mathbb{R}^d, \mathbb{R}^d)$ for some $\varepsilon>0$,
while $\rho$ is required to be locally bounded below and above by strictly
positive constants. The main tools in this paper are elliptic regularity
results for divergence and non-divergence type operators and basic properties
of Dirichlet forms and their resolvents.
