Uniqueness of diffusion operators and capacity estimates

Abstract

Let $\Omega$ be a connected open subset of $\Ri^d$. We analyze $L_1$-uniqueness of real second-order partial differential operators $H=-{\sum }_{k,l=1}^d{\partial }_k{c}_{kl}{\partial }_l$ and $K=H+{\sum }_{k=1}^d{c}_k{\partial }_k+c_0$ on $\Omega$ where $c_{kl}=c_{lk}\in W_{\mathrm{loc}}^{1,\infty }(\Omega ),c_k\in L_{\infty ,\mathrm{loc}}(\Omega )$, $c_0\in L_{2,\mathrm{loc}}(\Omega )$ 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 Täcklind condition ${\int }_R^{\infty }drr(\log |B(r)|{)}^{-1}=\infty$ is satisfied for all large $R$ and $H$ is Markov unique then $H$ is $L_1$-unique. If, in addition, $C(x)\ge \kappa (c^T\otimes c)(x)$ for some $\kappa >0$ and almost all $x\in \Omega$, $\divv c\in L_{\infty,{\rm loc}}(Ω)$ 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 $\lim_{n\to\infty} \|\one_AΓ(η_n)\|_1 = 0$, where $\Gamma ({\eta }_n)={\sum }_{k,l=1}^d{c}_{kl}({\partial }_k{\eta }_n)({\partial }_l{\eta }_n)$, and $\lim_{n\to\infty}\|\one_A (\one_Ω-η_n )\, φ\|_2 = 0$ for each $\phi \in L_2(\Omega )$ or if and only if $\capp(\partialΩ)=0$.