Uniqueness of diffusion operators and capacity estimates

Let $\Omega$ be a connected open subset of $\Ri^d$. We analyze $L_1$-uniqueness of real second-order partial differential operators $H=-\sum^d_{k,l=1}\partial_k\,c_{kl}\,\partial_l$ and $K=H+\sum^d_{k=1}c_k\,\partial_k+c_0$ on $\Omega$ where $c_{kl}=c_{lk}\in W^{1,\infty}_{\rm loc}( \Omega), c_k\in L_{\infty,{\rm loc}}(\Omega)$, $c_0\in L_{2,{\rm loc}}(\Omega)$ and $C(x)=(c_{kl}(x))>0$ for all $x\in\Omega$. Boundedness properties of the coefficients are expressed indirectly in terms of the balls $B(r)$ associated with the Riemannian metric $C^{-1}$ and their Lebesgue measure $|B(r)|$. \noindent\hspace{10mm}First we establish that if the balls $B(r)$ are bounded, the T\"acklind condition $\int^\infty_Rdr\,r(\log|B(r)|)^{-1}=\infty$ is satisfied for all large $R$ and $H$ is Markov unique then $H$ is $L_1$-unique. If, in addition, $C(x)\geq \kappa\, (c^{T}\!\otimes\, c)(x)$ for some $\kappa>0$ and almost all $x\in\Omega$, $\divv c\in L_{\infty,{\rm loc}}(\Omega)$ is upper semi-bounded and $c_0$ is lower semi-bounded then $K$ is also $L_1$-unique. \noindent\hspace{10mm}Secondly, if the $c_{kl}$ extend continuously to functions which are locally bounded on $\partial\Omega$ and if the balls $B(r)$ are bounded we characterize Markov uniqueness of $H$ in terms of local capacity estimates and boundary capacity estimates. For example, $H$ is Markov unique if and only if for each bounded subset $A$ of $\overline\Omega$ there exist $\eta_n \in C_c^\infty(\Omega)$ satisfying $\lim_{n\to\infty} \|\one_A\Gamma(\eta_n)\|_1 = 0$, where $\Gamma(\eta_n)=\sum^d_{k,l=1}c_{kl}\,(\partial_k\eta_n)\,(\partial_l\eta_n)$, and $\lim_{n\to\infty}\|\one_A (\one_\Omega-\eta_n )\, \varphi\|_2 = 0$ for each $\varphi \in L_2(\Omega)$ or if and only if $\capp(\partial\Omega)=0$.