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

TL;DR

This paper develops a new boundary-domain integral equation system for a mixed diffusion boundary value problem with variable coefficients, proving its equivalence and solution uniqueness using advanced potential theory.

Contribution

It introduces a novel family of parametrices for BDIEs, establishing their properties and demonstrating their equivalence to the original PDE with a proof of solution uniqueness.

Findings

New parametrix-based BDIE system for variable coefficient diffusion BVP

Proved equivalence between the BVP and BDIE system

Established uniqueness of solutions via Fredholm theory

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.