Analysis of Boundary-Domain Integral Equations based on a new parametrix for the mixed diffusion BVP with variable coefficient in an interior Lipschitz domain

TL;DR

This paper introduces a new parametrix-based boundary-domain integral equation approach for solving mixed diffusion boundary value problems in Lipschitz domains with variable coefficients, establishing equivalence and operator properties.

Contribution

It develops a novel parametrix for BDIEs in inhomogeneous diffusion problems, differing from previous methods, and proves their equivalence and operator properties.

Findings

Proves the equivalence between BVP and BDIE system.

Analyzes invertibility and Fredholm properties of integral operators.

Introduces a new parametrix different from previous works.

Abstract

A mixed boundary value problem for the partial differential equation of diffusion in an inhomogeneous medium in a Lipschitz domain is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one employed in the papers by Mikhailov (2002, 2006) and Chkadua, Mikhailov, Natroshvili (2009). We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.