Homogenisation and spectral convergence of high-contrast convolution type operators
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912462582841344 |
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| author | Cherdantsev, Mikhail Piatnitski, Andrey Velcic, Igor |
| author_facet | Cherdantsev, Mikhail Piatnitski, Andrey Velcic, Igor |
| contents | The paper deals with homogenisation problems for high-contrast symmetric convolution-type operators with integrable kernels in media with a periodic microstructure. We adapt the two-scale convergence method to nonlocal convolution-type operators and obtain the homogenisation result both for problems stated in the whole space and in bounded domains with the homogeneous Dirichlet boundary condition.
Our main focus is on spectral analysis. We describe the spectrum of the limit two-scale operator and characterize the limit behaviour of the spectrum of the original problem as the microstructure period tends to zero. It is shown that the spectrum of the limit operator is a subset the limit of the spectrum of the original operator, and that they need not coincide. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02638 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homogenisation and spectral convergence of high-contrast convolution type operators Cherdantsev, Mikhail Piatnitski, Andrey Velcic, Igor Analysis of PDEs 35B27, 45H99, 45M05, 45M15, 45P05 The paper deals with homogenisation problems for high-contrast symmetric convolution-type operators with integrable kernels in media with a periodic microstructure. We adapt the two-scale convergence method to nonlocal convolution-type operators and obtain the homogenisation result both for problems stated in the whole space and in bounded domains with the homogeneous Dirichlet boundary condition. Our main focus is on spectral analysis. We describe the spectrum of the limit two-scale operator and characterize the limit behaviour of the spectrum of the original problem as the microstructure period tends to zero. It is shown that the spectrum of the limit operator is a subset the limit of the spectrum of the original operator, and that they need not coincide. |
| title | Homogenisation and spectral convergence of high-contrast convolution type operators |
| topic | Analysis of PDEs 35B27, 45H99, 45M05, 45M15, 45P05 |
| url | https://arxiv.org/abs/2507.02638 |