A coarse-graining theory for elliptic operators and homogenization in high contrast
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918150671433728 |
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| author | Armstrong, Scott Kuusi, Tuomo |
| author_facet | Armstrong, Scott Kuusi, Tuomo |
| contents | We review a coarse-graining theory for divergence-form elliptic operators. The construction centers on a pair of coarse-grained matrices defined on spatial blocks that encode a scale-dependent notion of ellipticity, transmit precise information from small to large scales, and yield coarse-grained counterparts of standard elliptic estimates. Under simplifying assumptions, we give a complete proof of the result of [arXiv:2405.10732] that homogenization is reached within at most $C\log^2(1+Θ)$ dyadic length scales in the high-contrast regime, where $Θ$ is the ellipticity contrast. We argue that this scale-local notion of ellipticity is genuinely iterable across arbitrarily many scales, providing a framework for a rigorous renormalization group analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24887 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A coarse-graining theory for elliptic operators and homogenization in high contrast Armstrong, Scott Kuusi, Tuomo Analysis of PDEs Mathematical Physics Probability We review a coarse-graining theory for divergence-form elliptic operators. The construction centers on a pair of coarse-grained matrices defined on spatial blocks that encode a scale-dependent notion of ellipticity, transmit precise information from small to large scales, and yield coarse-grained counterparts of standard elliptic estimates. Under simplifying assumptions, we give a complete proof of the result of [arXiv:2405.10732] that homogenization is reached within at most $C\log^2(1+Θ)$ dyadic length scales in the high-contrast regime, where $Θ$ is the ellipticity contrast. We argue that this scale-local notion of ellipticity is genuinely iterable across arbitrarily many scales, providing a framework for a rigorous renormalization group analysis. |
| title | A coarse-graining theory for elliptic operators and homogenization in high contrast |
| topic | Analysis of PDEs Mathematical Physics Probability |
| url | https://arxiv.org/abs/2509.24887 |