Analysis of two-operator boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems in 2D with general right-hand side

Abstract

The Dirichlet and Neumann boundary value problems for the linear second-orderscalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain are considered. The right-hand side the PDE belongs to H´1 pΩq or Hr ´1 pΩq, when neither classical nor canonical conormal derivatives of solutions are well defined. The two-operator approach and appropriate parametrix (Levi function) are used to reduce each of the problems 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 solvability, solution uniqueness/nonuniqueness, and invertibility BDIE system are analyzed in the appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading toinvertibility of the perturbed operators.