Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side

Ayele, Tsegaye G.

doi:10.1216/jie.2021.33.403

Cited by 3 publications

(3 citation statements)

References 25 publications

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“…1 pΩq, see(Mikhailov, 2015, Appendix A). For u P H 1 pΩq, the PDE Au in (1) is understood in the sense of distributions,…”

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confidence: 99%

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Analysis of two-operator boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems in 2D with general right-hand side

F. Yimer,

G. Ayele

2023

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The Dirichlet and Neumann boundary value problems for the linear second-orderscalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain are considered. The right-hand side the PDE belongs to H´1 pΩq or Hr ´1 pΩq, when neither classical nor canonical conormal derivatives of solutions are well defined. The two-operator approach and appropriate parametrix (Levi function) are used to reduce each of the problems to two different systems of two-operator 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 equivalence of the two-operator BDIE systems to the original problems, BDIE system solvability, solution uniqueness/nonuniqueness, and invertibility BDIE system are analyzed in the appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading toinvertibility of the perturbed operators.

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“…1 pΩq, see(Mikhailov, 2015, Appendix A). For u P H 1 pΩq, the PDE Au in (1) is understood in the sense of distributions,…”

mentioning

confidence: 99%

“…H s´2 pΩq is continuous for 1 2 ă s ă 3 2 , while V ´1 ∆ exists (Ayele Tsegaye and Bekele Solomon, 2019, Theorem 2). Hence we always represent F 1 " P b f˚, with…”

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confidence: 99%

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Analysis of two-operator boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems in 2D with general right-hand side

F. Yimer,

G. Ayele

2023

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Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet Problem in 2D with General Right-Hand Side

Ayele,

Yimer

2023

Integral Methods in Science and Engineering

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BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT HELMHOLTZ BVPs IN 2D

Ayele,

Demissie,

Mikhailov

2024

J Math Sci

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In this paper, we construct boundary-domain integral equations (BDIEs) of the Dirichlet and mixed boundary value problems for a two-dimensional variable-coefficient Helmholtz equation. Using an appropriate parametrix, these problems are reduced to several BDIE systems. It is shown that the BVPs and the formulated BDIE systems are equivalent. Fredholm properties and unique solvability and invertibility of BDIE systems are investigated in appropriate Sobolev spaces.

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Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side

Cited by 3 publications

References 25 publications

Analysis of two-operator boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems in 2D with general right-hand side

Analysis of two-operator boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems in 2D with general right-hand side

Two-Operator Boundary-Domain Integral Equations for Variable-Coefficient Dirichlet Problem in 2D with General Right-Hand Side

BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT HELMHOLTZ BVPs IN 2D

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