L1-uniqueness of degenerate elliptic operators

Let Omega be an open subset of R-d with 0 is an element of Omega. Furthermore, let H-Omega = -Sigma(d)(i,j=1) partial derivative(i)c(ij)partial derivative(j) be a second-order partial differential operator with domain C-c(infinity)(Omega) where the coefficients c(ij) is an element of W-loc(1,infinity)((Omega) over bar) are real, c(ij) = c(ji) and the coefficient matrix C = (c(ij)) satisfies bounds 0 < C(x) <= c(vertical bar x vertical bar)I for all x is an element of Omega. If integral(infinity)(0) ds s(d/2)e(-lambda mu(s)2) < infinity for some lambda > 0 where mu(s) = integral(s)(0) dt c(t)(-1/2) then we establish that H-Omega is L-1-unique, i.e. it has a unique L-1-extension which generates a continuous semigroup, if and only if it is Markov unique, i.e. it has a unique L-2-extension which generates a submarkovian semigroup. Moreover these uniqueness conditions are equivalent to the capacity of the boundary of Omega, measured with respect to H-Omega, 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-Omega at A.