Analysis of a Boundary-Domain Integral Equation System for the Mixed Interior Diffusion BVP with Variable Coefficient Based on a New Family of Parametrices

Abstract: A mixed boundary value problem for the diffusion equation in non-homogeneous media partial differential equation is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one employed by Chkadua, Mikhailov and Natroshvili in the paper 'Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility'. Mapping properties of boundedness and compactness of parametrix-based surface and volume potentials are analysed in appropriate Sobolev spaces. Using these properties we prove the equivalence between the original BVP and the corresponding BDIE system. Furthermore, we prove uniqueness of solution of the BDIE system by applying the Fredholm Alternative.