L₁-uniqueness of degenerate elliptic operators

Let Ω be an open subset of R^d with 0 ε Ω. Furthermore, let H_Ω = - Σ^d_1,j=1 δ_ic_ijδ_j be a second-order partial differential operator with domain C^∞_c(Ω) where the coefficients C_ij ε W ^{1, infin;} _loc(Ω̄) are real, C_ij = C_ji and the coefficient matrix C = (c_ij) satisfies bounds 0 < C(x) ≤ c(|x|)I for all x ε ω. If ∫^∞₀ ds s_d/2e-λμ(s)²< ∞ for some λ > 0 where μ(s) = ∫^s₀dtc(t)^-1/2 then we establish that H_ω 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 H_ω at A.