Uniqueness of diffusion operators and capacity estimates

Let Ω be a connected open subset of R^d. We analyse L₁-uniqueness of real second-order partial differential operators, and, on Ω where, and C(x) = (c_kl(x)) > 0 for all x ∈ Ω. 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 {pipe}B(r){pipe}. First, we establish that if the balls B(r) are bounded, the Täcklind condition ∫^∞_R dr r(log {pipe}B(r){pipe})^-1 = ∞ is satisfied for all large R and H is Markov unique then H is L₁-unique. If, in addition, C(x) ≥ κ (c^T ⊗ c)(x)} for some κ > 0 and almost all x ∈ Ω div, is upper semi-bounded and c₀ is lower semi-bounded, then K is also L₁-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, there exist, satisfying, where, and, for each, or if and only if cap(∂Ω) = 0.