Let Omega be an open subset of R(d) and H(Omega) = -Sigma(d)(i,j=1) partial derivative(i) c(ij) partial derivative(j) be a second-order partial differential operator on L(2)(Omega) with domain C(c)(infinity) (Omega), where the coefficients c(ij) epsilon W(l,infinity) (Omega) are real symmetric and C = (c(ij)) is a strictly positive-definite matrix over Omega. In particular, H(Omega) is locally strongly elliptic. We analyze the submarkovian extensions of H(Omega), i.e., the self-adjoint extensions that generate submarkovian semigroups. Our main result states that H(Omega) is Markov unique, i.e., it has a unique submarkovian extension, if and only if cap(Omega) (partial derivative Omega) = 0 where cap(Omega)(partial derivative Omega) is the capacity of the boundary of Omega measured with respect to H(Omega). The second main result shows that Markov uniqueness of H(Omega) is equivalent to the semigroup generated by the Friedrichs extension of H(Omega) being conservative.