The Neumann sieve problem revisited
Let Ω⊂Rn be a domain, Γ be a hyperplane intersecting it. Let ε>0, and Ωε=Ω∖Σε‾, where Σε (“sieve”) is an ε-neighborhood of Γ punctured by many narrow passages. When ε→0, the number of passages tends to infinity, while the diameters of their cross-sections tend to zero. For the case of identical s...
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| Published in | Journal of mathematical analysis and applications Vol. 554; no. 1; p. 129933 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Inc
01.02.2026
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| Online Access | Get full text |
| ISSN | 0022-247X |
| DOI | 10.1016/j.jmaa.2025.129933 |
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| Abstract | Let Ω⊂Rn be a domain, Γ be a hyperplane intersecting it. Let ε>0, and Ωε=Ω∖Σε‾, where Σε (“sieve”) is an ε-neighborhood of Γ punctured by many narrow passages. When ε→0, the number of passages tends to infinity, while the diameters of their cross-sections tend to zero. For the case of identical straight periodically distributed and appropriately scaled passages T. Del Vecchio (1987) proved that the Neumann Laplacian on Ωε converges in a strong resolvent sense to the Laplacian on Ω∖Γ subject to the so-called δ′-conditions on Γ. We will refine this result by deriving estimates on the rate of convergence in terms of various operator norms, and providing the estimate for the distance between the spectra. The assumptions we impose on the passages are rather general. For n=2 the results of T. Del Vecchio are not complete, some cases remain as open problems, and in this work we will fill these gaps. |
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| AbstractList | Let Ω⊂Rn be a domain, Γ be a hyperplane intersecting it. Let ε>0, and Ωε=Ω∖Σε‾, where Σε (“sieve”) is an ε-neighborhood of Γ punctured by many narrow passages. When ε→0, the number of passages tends to infinity, while the diameters of their cross-sections tend to zero. For the case of identical straight periodically distributed and appropriately scaled passages T. Del Vecchio (1987) proved that the Neumann Laplacian on Ωε converges in a strong resolvent sense to the Laplacian on Ω∖Γ subject to the so-called δ′-conditions on Γ. We will refine this result by deriving estimates on the rate of convergence in terms of various operator norms, and providing the estimate for the distance between the spectra. The assumptions we impose on the passages are rather general. For n=2 the results of T. Del Vecchio are not complete, some cases remain as open problems, and in this work we will fill these gaps. |
| ArticleNumber | 129933 |
| Author | Khrabustovskyi, Andrii |
| Author_xml | – sequence: 1 givenname: Andrii orcidid: 0000-0001-6298-9684 surname: Khrabustovskyi fullname: Khrabustovskyi, Andrii email: andrii.khrabustovskyi@uhk.cz organization: Department of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czech Republic |
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| Keywords | Homogenization Neumann sieve Resolvent convergence Varying Hilbert spaces Operator estimates Spectrum |
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| SubjectTerms | Homogenization Neumann sieve Operator estimates Resolvent convergence Spectrum Varying Hilbert spaces |
| Title | The Neumann sieve problem revisited |
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