Uniqueness of diffusion operators and capacity estimates

Abstract

Let Ω be a connected open subset of R d . We analyse 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 Ω 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)|. First, we establish that if the balls B(r) are bounded, the Täcklind condition $${\int^\infty_R dr r({\rm log}|B(r)|)^{-1} = \infty}$$ 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 $${x \in \Omega}$$ , $${{\rm div} c \in L_{\infty,{\rm loc}}(\Omega)}$$ is upper semi-bounded and c 0 is lower semi-bounded, then K is also L 1-unique. Secondly, if the c kl extend continuously to functions which are locally bounded on ∂Ω 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 , where $${\Gamma(\eta_n) = \sum^d_{k,l=1}c_{kl} (\partial_k \eta_n) (\partial_l \eta_n)}$$ , and for each $${\varphi \in L_2(\Omega)}$$ or if and only if cap(∂Ω) = 0.