# Two-Operator Boundary–Domain Integral Equations for a Variable-Coefficient BVP

## Abstract

Partial differential equations (PDEs) with variable coefficients often arise in mathematical modeling of inhomogeneous media (e.g., functionally graded materials or materials with damage-induced inhomogeneity) in solid mechanics, electromagnetics, heat conduction, fluid flows through porous media, and other areas of physics and engineering.

Generally, explicit fundamental solutions are not available if the PDE coefficients are not constant, preventing formulation of explicit boundary integral equations, which can then be effectively solved numerically. Nevertheless, for a rather wide class of variable–coefficient PDEs, it is possible to use instead an explicit parametrix (Levi function) taken as a fundamental solution of corresponding frozen–coefficient PDEs, and reduce boundary value problems (BVPs) for such PDEs to explicit systems of boundary–domain integral equations (BDIEs); see, e.g., [Mi02, CMN09, Mi06] and references therein. However this (one–operator) approach does not work when the fundamental solution of the frozen–coefficient PDE is not available explicitly (as, e.g., in the Lam’e system of anisotropic elasticity).

## Keywords

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## References

- [CMN09]Chakuda, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary–domain integral equations for a mixed BVP with variable coefficient. I: Equivalence and invertibility.
*J. Integral Equations Appl.*(to appear).Google Scholar - [Co88]Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results.
*SIAM J. Math. Anal.*,**19**, 613–626 (1988).MATHCrossRefMathSciNetGoogle Scholar - [Gr85]Grisvard, P.:
*Elliptic Problems in Nonsmooth Domains*, Pitman, Boston-London-Melbourne (1985).MATHGoogle Scholar - [LiMa72]Lions, J.-L., Magenes, E.:
*Non-homogeneous Boundary Value Problems and Applications, Vol. 1*, Springer, Berlin-Heidberg-New York (1972).Google Scholar - [McL00]McLean, W.:
*Strongly Elliptic Systems and Boundary Integral Equations*, Cambridge University Press, Cambridge (2000).MATHGoogle Scholar - [Mi02]Mikhailov, S.E.: Localized boundary–domain integral formulations for problems with variable coefficients,
*Internat. J. Engng. Anal. Boundary Elements*,**26**, 681–690 (2002).MATHCrossRefGoogle Scholar - [Mi05]Mikhailov, S.E.: Localized direct boundary–domain integro-differential formulations for scalar nonlinear BVPs with variable coefficients.
*J. Engng. Math.*,**51**, 283–3002 (2005).MATHCrossRefGoogle Scholar - [Mi06]Mikhailov, S.E.: Analysis of united boundary-domain integral and integro-differential equations for a mixed BVP with variable coefficients.
*Math. Methods Appl. Sci.*,**29**, 715–739 (2006).MATHCrossRefMathSciNetGoogle Scholar - [Mi07]Mikhailov, S.E.: About traces, extensions and co-normal derivative operators on Lipschitz domains, in
*Integral Methods in Science and Engineering: Techniques and Applications*, Constanda, C., Potapenko, S. (eds.), 149–160, Birkhäuser, Boston, MA (2007).Google Scholar - [Mir70]Miranda, C.:
*Partial Differential Equations of Elliptic Type*, 2nd ed., Springer, Berlin-Heidelberg-New York (1970).MATHGoogle Scholar