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Two-operator boundary – domain integral equations for a variable-coefficient BVP. (English) Zbl 1221.35122

Constanda, C. (ed.) et al., Integral methods in science and engineering. Volume 1: Analytic methods. Papers presented at the 10th international conference on integral methods in science and engineering (IMSE 2008), Santander, Spain, July 7–10, 2008. Basel: Birkhäuser (ISBN 978-0-8176-4898-5/hbk; 978-0-8176-4899-2/ebook). 29-39 (2010).
From the Introduction: “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). 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 Lamé system of anisotropic elasticity). To overcome this difficulty, one can apply the two-operator approach, formulated in [S. E. Mikhailov, J. Eng. Math. 51, No. 3, 283–302 (2005; Zbl 1073.65136)] for some nonlinear problems, that employs a parametrix of another (second) PDE, not related with the PDE in question, for reducing the BVP to a BDIE system. Since the second PDE is rather arbitrary, one can always choose it in such a way that its parametrix is available explicitly. The simplest choice for the second PDE is the one with a fundamental solution explicitly available. To analyze the two-operator approach, we apply in this paper one of its linear versions to the mixed (Dirichlet-Neumann) BVP for a linear second-order scalar elliptic variable-coefficient PDE, reducing it to four different BDIE systems. Although the considered BVP can also be reduced to (other) BDIE systems by the one-operator approach, it can be considered as a simple “toy” model showing the main features of the two-operator approach arising also in reducing more general BVPs to BDIEs. The two-operator BDIE systems are nonstandard systems of equations containing integral operators defined on the domain under consideration and potential type and pseudo-differential operators defined on open submanifolds of the boundary. Using the results of O. Chkadua, S. E. Mikhailov and D. Natroshvili [J. Integral Equations Appl. 21, No. 4, 499–543 (2009; Zbl 1204.65139)], we give a rigorous analysis of the two-operator BDIEs and show that the BDIE systems are equivalent to the mixed BVP and thus are uniquely solvable, while the corresponding boundary-domain integral operators are invertible in appropriate Sobolev-Slobodetski (Bessel-potential) spaces.”
For the entire collection see [Zbl 1182.00023].

MSC:

35J25 Boundary value problems for second-order elliptic equations
45P05 Integral operators
47G10 Integral operators
35J08 Green’s functions for elliptic equations
47N20 Applications of operator theory to differential and integral equations
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