The Dirichlet and Neumann boundary value problems (BVPs) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain are considered. The PDE right-hand side belongs to $H^{-1}(\Omega)$ or $\widetilde{H}^{-1}(\Omega),$ when neither classical nor canonical conormal derivatives of solutions are well defined. Using the two-operator approach and appropriate parametrix (Levi function) each problem is reduced to two different systems of two-operator boundary-domain integral equations (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 ensure the invertibility of corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The equivalence of the two-operator BDIE systems to the original problems, BDIE system solubility, solution uniquness/nonuniquness and invertibility BDIE system are analyzed in the appropriate 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.

The formulation and analysis of two operator Boundary-Domain Integral Equation systems for variable coefficient mixed BVP with in two dimensional domain is discussed. To analyse the two-operator approach, we applied one of its linear versions to the mixed (Dirichlet-Neumann) BVP for a linear second-order scalar elliptic variable-coefficient PDE and reduced it to four different BDIE systems. %The two-operator BDIE systems are nonstandard systems of equations containing integral operators defined on the domain under consideration and potential type and pseudo-differential operators defined on open sub-manifolds of the boundary. Using the results as in CMN09 andAM11, a rigorous analysis of the two-operator BDIE systems is given and it is shown that they are equivalent to the mixed BVP and thus are uniquely solvable, while the corresponding boundary domain integral operators are invertible in the appropriate Sobolev-Slobodetski (Bessel potential) spaces.

In this paper, the Boundary-Domain Integral Equations (BDIEs) for the mixed boundary value problem(BVP) for a compressible Stokes system of partial differential equation (PDE) with variable coefficient in 2D is considered . An appropriate parametrix 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 spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The properties of corresponding potential operators are investigated. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems in appropriate Sobolev spaces are proved.

In this paper, the Boundary-Domain Integral Equations (BDIEs) for the mixed boundary value problem(BVP) for a compressible Stokes system of partial differential equation(PDE) with variable coefficient in 2D is considered . An appropriate parametrix 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 spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The properties of corresponding potential operators are investigated. Equivalence of the BDIE systems to the mixed BVP and invertibility of the matrix operators associated with the BDIE systems in appropriate Sobolev spaces are proved.

In this paper, the Dirichlet and Neumann boundary value problems for the steady-state Stokes system of partial differential equations for a compressible viscous fluid with variable viscosity coefficient is considered in two-dimensional bounded domain. Using an appropriate parametrix, this problem is reduced to a system of direct segregated boundary-domain integral equations (BDIEs). The BDIEs in the two-dimensional case have special properties in comparison with the three dimension because of the logarithmic term in the parametrix for the associated partial differential equations. Consequently, we need to set conditions on the function spaces or on the domain to ensure the invertibility of corresponding parametrix-based hydrodaynamic single layer and hypersingular potentials and hence the unique solvability of BDIEs. Equivalence of the BDIE systems to the Dirichlet and Neumann BVPs and the invertibility of the corresponding boundary-domain integral operators in appropriate Sobolev spaces are shown.

The Neumann boundary value problem (BVP) for the second order “stationary heat transfer” elliptic partial differential equation with variable coefficient is considered in two-dimensional bounded domain. Using an appropriate parametrix (Levi function) and applying the two-operator approach, this problem is reduced to some systems of boundary-domain integral equations (BDIEs). The two-operator BDIEs in 2D have special consideration due to their different equivalence properties as compared to higher dimensional case due to the logarithmic term in the parametrix for the associated partial differential equation. Consequently, we need to set conditions on the domain or function spaces to insure the invertibility of the corresponding layer potentials, and hence the unique solvability of BDIEs. Equivalence of the two operator BDIE systems to the original Neumann BVP, BDIEs solvability, uniqueness/non uniqueness of the solution, as well as Fredholm property and invertibility of the BDIE operator are analysed. Moreover, the two operator boundary domain integral operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading to invertibility of the perturbed operators.\\ \noindent\textbf{Key words:} Partial differential equation, Two-operator Boundary-Domain Integral Equations, finite-dimensional perturbations , equivalence, invertibility.

The Dirichlet, Neumann and mixed boundary value problems for the linear second-order scalar elliptic differential equation with variable coefficient in a bounded three-dimensional domain are considered. The PDE right-hand side belongs to $H^{-1}(\Omega)$ or $\widetilde{H}^{-1}(\Omega),$ when neither classical nor canonical co-normal derivatives are well defined. Using the two-operator approach and appropriate parametrix (Levi function) each problem is reduced to different systems of boundary domain integral equations (BDIEs). Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and as well as invertibility of the BDIE operators are analysed in appropriate 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.

In this paper, the Dirichlet and Neumann boundary value problems for the steady state Stokes system of partial differential equations for a compressible viscous fluid with variable viscosity coefficient is considered in two-dimensional bounded domain. Using an appropriate parametrix, this problem is reduced to a system of direct segregated boundary-domain integral equations (BDIEs). The BDIEs in the two dimensional case have special properties in comparison with the three dimension because of the logarithmic term in the parametrix for the associated partial differential equations. Consequently, we need to set conditions on the function spaces or on the domain to ensure the invertibility of corresponding parametrix-based hydrodaynamic single layer and hypersingular potentials and hence the unique solvability of BDIEs. Equivalence of the BDIE systems to the Dirichlet and Neumann BVPs and the invertibility of the corresponding boundary-domain integral operators in appropriate Sobolev spaces are shown.