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The direct segregated boundary-domain integral equations (BDIEs) for the mixed boundary value problem for a second order elliptic partial differential equation with variable coefficient in 2D is considered in this paper. An appropriate parametrix (Levi function) is used to reduce this BVP to the BDIEs. Although the theory of BDIEs in 3D is well developed, the BDIEs in 2D need a special consideration due to their different equivalence properties. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to insure the invertibility of corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The properties of corresponding potential operators are investigated. The equivalence of the original BVP and the obtained BDIEs is analysed.

In this paper, the Dirichlet boundary value problem for the second order stationary diffusion elliptic partial differential equation with variable coefficient is considered in unbounded (exterior) two-dimensional domain. Using an appropriate parametrix (Levi function), this problem is reduced to some direct segregated boundary-domain integral equations (BDIEs). We investigate the properties of corresponding parametrix-based integral volume and layer potentials in some weighted Sobolev spaces, as well as the unique solvability of BDIEs and their equivalence to the original BVP.