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 $\capp(\partial\Omega)=0$.
Categories: math.AP