Two‐operator boundary‐domain integral equation systems for variable‐coefficient mixed Boundary Value Problem in 2D

Abstract

In this paper, the mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second‐order scalar elliptic differential equation with variable coefficients in a bounded two‐dimensional domain is considered. The two‐operator approach and appropriate parametrix (Levi function) are used to reduce this BVP to four systems of two‐operator boundary‐domain integral equations (BDIEs). Although the theory of two‐operator 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 BVP is shown. The invertibility of the associated operators is proved in the appropriate Sobolev spaces.