  Segregated direct boundary-domain integral equations (BDIEs) based on a
parametrix and associated with the Dirichlet and Neumann boundary value
problems for the linear stationary diffusion partial differential equation with
a variable coefficient are formulated. The PDE right hand sides belong to the
Sobolev space $H^{-1}(\Omega)$ or $\tilde H^{-1}(\Omega)$, when neither
classical nor canonical co-normal derivatives are well defined. Equivalence of
the BDIEs to the original BVP, BDIE solvability, solution
uniqueness/non-uniqueness, and as well as Fredholm property and invertibility
of the BDIE operators are analysed in Sobolev (Bessel potential) spaces. It is
shown that the BDIE operators for the Neumann BVP are not invertible, and
appropriate finite-dimensional perturbations are constructed leading to
invertibility of the perturbed operators.
