  Let $\Omega$ be a domain in $\mathbf R^d$ and
$h(\varphi)=\sum^d_{k,l=1}(\partial_k\varphi, c_{kl}\partial_l\varphi)$ a
quadratic form on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the
$c_{kl}$ are real symmetric $L_\infty(\Omega)$-functions with
$C(x)=(c_{kl}(x))>0$ for almost all $x\in \Omega$. Further assume there are $a,
\delta>0$ such that $a^{-1}d_\Gamma^{\delta}\,I\le C\le
a\,d_\Gamma^{\delta}\,I$ for $d_\Gamma\le 1$ where $d_\Gamma$ is the Euclidean
distance to the boundary $\Gamma$ of $\Omega$.
  We assume that $\Gamma$ is Ahlfors $s$-regular and if $s$, the Hausdorff
dimension of $\Gamma$, is larger or equal to $d-1$ we also assume a mild
uniformity property for $\Omega$ in the neighbourhood of one $z\in\Gamma$. Then
we establish that $h$ is Markov unique, i.e. it has a unique Dirichlet form
extension, if and only if $\delta\ge 1+(s-(d-1))$. The result applies to forms
on Lipschitz domains or on a wide class of domains with $\Gamma$ a self-similar
fractal. In particular it applies to the interior or exterior of the von Koch
snowflake curve in $\mathbf R^2$ or the complement of a uniformly disconnected
set in $\mathbf R^d$.
