Let Ω be an open subset of Rd with 0 ε Ω. Furthermore, let HΩ = - Σd 1,j=1 δicijδj be a second-order partial differential operator with domain C∞ c(Ω) where the coefficients Cij ε W 1, infin; loc(Ω̄) are real, Cij = Cji and the coefficient matrix C = (cij) satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ε ω. If ∫∞ 0 ds sd/2e-λμ(s) 2< ∞ for some λ > 0 where μ(s) = ∫s 0dtc(t)-1/2 then we establish that Hω is L1-unique, i.e. it has a unique L2-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique 1,2-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent to the capacity of the boundary of Ω, measured with respect to H ω, being zero. We also demonstrate that the capacity depends on two gross features, the Hausdorff dimension of subsets A of the boundary of the set and the order of degeneracy of Hω at A.