  Let $\Omega$ be a domain in $\Ri^d$ with boundary $\Gamma$${\!,}$ $d_\Gamma$
the Euclidean distance to the boundary and $H=-\divv(C\,\nabla)$ an elliptic
operator with $C=(\,c_{kl}\,)>0$ where $c_{kl}=c_{lk}$ are real, bounded,
Lipschitz functions. We assume that $C\sim c\,d_\Gamma^{\,\delta}$ as
$d_\Gamma\to0$ in the sense of asymptotic analysis where $c$ is a strictly
positive, bounded, Lipschitz function and $\delta\geq0$. We also assume that
there is an $r>0$ and a $ b_{\delta,r}>0$ such that the weighted Hardy
inequality \[ \int_{\Gamma_{\!\!r}} d_\Gamma^{\,\delta}\,|\nabla \psi|^2\geq
b_{\delta,r}^{\,2}\int_{\Gamma_{\!\!r}} d_\Gamma^{\,\delta-2}\,| \psi|^2 \] is
valid for all $\psi\in C_c^\infty(\Gamma_{\!\!r})$ where
$\Gamma_{\!\!r}=\{x\in\Omega: d_\Gamma(x)<r\}$. We then prove that the
condition $(2-\delta)/2<b_\delta$ is sufficient for the essential
self-adjointness of $H$ on $C_c^\infty(\Omega)$ with $b_\delta$ the supremum
over $r$ of all possible $b_{\delta,r}$ in the Hardy inequality. This result
extends all known results for domains with smooth boundaries and also gives
information on self-adjointness for a large family of domains with rough, e.g.\
fractal, boundaries.
