2021 Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side
Tsegaye G. Ayele
J. Integral Equations Applications 33(4): 403-426 (2021). DOI: 10.1216/jie.2021.33.403

Abstract

The mixed (Dirichlet–Neumann) boundary value problem (BVP) for the linear second-order scalar elliptic differential equation with variable coefficients in a bounded two-dimensional domain is considered. The PDE on the right-hand side belongs to H1(Ω) or H˜1(Ω), 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 this BVP to four 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 equivalence of the BDIE systems to the original BVP is shown. The invertibility of the associated operators is proved in the corresponding Sobolev spaces.

Citation

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Tsegaye G. Ayele. "Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP in 2D with general right-hand side." J. Integral Equations Applications 33 (4) 403 - 426, 2021. https://doi.org/10.1216/jie.2021.33.403

Information

Received: 9 September 2020; Revised: 17 April 2021; Accepted: 20 April 2021; Published: 2021
First available in Project Euclid: 11 March 2022

MathSciNet: MR4393375
zbMATH: 1495.35090
Digital Object Identifier: 10.1216/jie.2021.33.403

Subjects:
Primary: 31A10 , 35J25 , 45A05 , ‎45P05‎ , 47G10
Secondary: 47G30 , 47G40

Keywords: boundary-domain integral equations , equivalence , Parametrix , partial differential equations , unique solvability and invertibility. , Variable coefficients

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium

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Vol.33 • No. 4 • 2021
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