Markov uniqueness of degenerate elliptic operators

Let $\Omega$ be an open subset of $\Ri^d$ and $H_\Omega=-\sum^d_{i,j=1}\partial_i c_{ij} \partial_j$ a second-order partial differential operator on $L_2(\Omega)$ with domain $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}(\Omega)$ are real symmetric and $C=(c_{ij})$ is a strictly positive-definite matrix over $\Omega$. In particular, $H_\Omega$ is locally strongly elliptic. We analyze the submarkovian extensions of $H_\Omega$, i.e. the self-adjoint extensions which generate submarkovian semigroups. Our main result establishes that $H_\Omega$ is Markov unique, i.e. it has a unique submarkovian extension, if and only if $\capp_\Omega(\partial\Omega)=0$ where $\capp_\Omega(\partial\Omega)$ is the capacity of the boundary of $\Omega$ measured with respect to $H_\Omega$. The second main result establishes that Markov uniqueness of $H_\Omega$ is equivalent to the semigroup generated by the Friedrichs extension of $H_\Omega$ being conservative.