  A system of boundary-domain integral equations is derived from the
bidimensional Dirichlet problem for the diffusion equation with variable
coefficient using the novel parametrix from [22] different from the one in
[5,18]. Mapping properties of the surface and volume parametrix based
potential-type operators are analysed. Invertibility of the single layer
potential is also studied in detail in appropriate Sobolev spaces. We show that
the system of boundary-domain integral equations derived is equivalent to the
Dirichlet problem prescribed and we prove the existence and uniqueness of
solution in suitable Sobolev spaces of the system obtained by using arguments
of compactness and Fredholm Alternative theory.
