Markov uniqueness of degenerate elliptic operators

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

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