Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Neumann BVP in 2D

Abstract

In this paper, the Dirichlet boundary value problem for the second order “stationary heat transfer” elliptic partial differential equation with variable coefficient is considered in 2D. Using an appropriate parametrix (Levi function), this problem is reduced to some direct segregated systems of 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. Consequently, we need to set conditions on the domain for the invertibility of corresponding parametrix-based integral layer potentialsand hence the unique solvability of BDIEs. The properties of corresponding potential operators are investigated. The equivalence of the original BVP and the obtained BDIEs are analyzed and the invertibility of the BDIE operators is proved.