# Two-Operator Boundary-Domain Integral Equations for Variable Coefficient Dirichlet Problem in 2D

## Abstract

The Dirichlet problem for the second order elliptic PDE with variable coefficient is considered in two-dimensional bounded domain. Using an appropriate parametrix (Levi function) and applying the two-operator approach, this problem is reduced to two 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. As a result, we need to set conditions on the domain or on the associated Sobolev spaces to ensure the invertibility of corresponding parametrix-based integral layer potentials and hence the unique solvability of BDIEs. The properties of corresponding potential operators are investigated. The equivalence of the original BVP and the obtained BDIEs is analysed and the invertibility of the BDIE operators is proved in appropriate Sobolev-Slobodecki (Bessel potential) spaces.

## Notes

### Acknowledgements

The work on this paper of the first author was supported by EMS-Simmons for Africa and ISP Sweden and of the second author was supported by ISP, Sweden. They also thank the Department of Mathematics at Brunel University for hosting their research visit.

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