Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains
Abstract Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-05-01
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| Series: | Boundary Value Problems |
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| Online Access: | http://link.springer.com/article/10.1186/s13661-018-0992-0 |
| _version_ | 1817983385234046976 |
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| author | Sergey E. Mikhailov |
| author_facet | Sergey E. Mikhailov |
| author_sort | Sergey E. Mikhailov |
| collection | DOAJ |
| description | Abstract Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space Hs−2(Ω) $H^{s-2}(\Omega)$ or H˜s−2(Ω) $\widetilde{H}^{s-2}( \Omega)$, 12<s<32 $\frac{1}{2}< s<\frac{3}{2}$, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators. |
| first_indexed | 2024-04-13T23:31:43Z |
| format | Article |
| id | doaj.art-7a5b96eaa1b44142b9cd3e3f32674e75 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-04-13T23:31:43Z |
| publishDate | 2018-05-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-7a5b96eaa1b44142b9cd3e3f32674e752022-12-22T02:24:52ZengSpringerOpenBoundary Value Problems1687-27702018-05-012018115210.1186/s13661-018-0992-0Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domainsSergey E. Mikhailov0Department of Mathematics, Brunel University LondonAbstract Segregated direct boundary-domain integral equations (BDIEs) based on a parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE right-hand sides belong to the Sobolev (Bessel potential) space Hs−2(Ω) $H^{s-2}(\Omega)$ or H˜s−2(Ω) $\widetilde{H}^{s-2}( \Omega)$, 12<s<32 $\frac{1}{2}< s<\frac{3}{2}$, when neither strong classical nor weak canonical co-normal derivatives are well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators.http://link.springer.com/article/10.1186/s13661-018-0992-0Partial differential equationsNon-smooth coefficientsSobolev spacesParametrixIntegral equationsEquivalence |
| spellingShingle | Sergey E. Mikhailov Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains Boundary Value Problems Partial differential equations Non-smooth coefficients Sobolev spaces Parametrix Integral equations Equivalence |
| title | Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains |
| title_full | Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains |
| title_fullStr | Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains |
| title_full_unstemmed | Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains |
| title_short | Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains |
| title_sort | analysis of segregated boundary domain integral equations for bvps with non smooth coefficients on lipschitz domains |
| topic | Partial differential equations Non-smooth coefficients Sobolev spaces Parametrix Integral equations Equivalence |
| url | http://link.springer.com/article/10.1186/s13661-018-0992-0 |
| work_keys_str_mv | AT sergeyemikhailov analysisofsegregatedboundarydomainintegralequationsforbvpswithnonsmoothcoefficientsonlipschitzdomains |