# On the Existence of Solution of the Boundary-Domain Integral Equation System derived from the 2D Dirichlet Problem for the Diffusion Equation with Variable Coefficient

## TL;DR

This paper derives a boundary-domain integral equation system for the 2D Dirichlet problem of the diffusion equation with variable coefficients, establishing its equivalence to the original problem and proving solution existence and uniqueness.

## Contribution

It introduces a novel parametrix for the integral equation system and analyzes its mapping properties, leading to new existence and uniqueness results.

## Key findings

- System is equivalent to the Dirichlet problem
- Invertibility of the single layer potential established
- Existence and uniqueness of solutions proved

## Abstract

A system of boundary-domain integral equations is derived from the bidimensional Dirichlet problem for the diffusion equation with variable coefficient using the novel parametrix from [22] different from the one in [5,18]. Mapping properties of the surface and volume parametrix based potential-type operators are analysed. Invertibility of the single layer potential is also studied in detail in appropriate Sobolev spaces. We show that the system of boundary-domain integral equations derived is equivalent to the Dirichlet problem prescribed and we prove the existence and uniqueness of solution in suitable Sobolev spaces of the system obtained by using arguments of compactness and Fredholm Alternative theory.