Let $��$ be an open subset of $\Ri^d$ with $0\in ��$. Further let $H_��=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j$ be a second-order partial differential operator with domain $C_c^\infty(��)$ where the coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\bar��)$ are real, $c_{ij}=c_{ji}$ and the coefficient matrix $C=(c_{ij})$ satisfies bounds $00$ where $��(s)=\int^s_0dt\,c(t)^{-1/2}$ then we establish that $H_��$ is $L_1$-unique, i.e.\ it has a unique $L_1$-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e.\ it has a unique $L_2$-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent with the capacity of the boundary of $��$, measured with respect to $H_��$, being zero. We also demonstrate that the capacity depends on two gross features, the Hausdorff dimension of subsets $A$ of the boundary the set and the order of degeneracy of $H_��$ at $A$.

21 pages