Renormalization group and elliptic homogenization in high contrast
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916936396308480 |
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| author | Armstrong, Scott Kuusi, Tuomo |
| author_facet | Armstrong, Scott Kuusi, Tuomo |
| contents | We prove a quantitative estimate for the homogenization length scale in terms of the ellipticity ratio $Λ/λ$ of the coefficient field. This upper bound applies to high-contrast elliptic equations exhibiting near-critical behavior. Specifically, we show, assuming a suitable decay of correlations, the length scale at which homogenization occurs is at most $\exp(C \log^2(1+Λ/λ))$. The proof introduces the new concept of coarse-grained ellipticity, which measures the effective ellipticity ratio of the equation--and thus the strength of the disorder--after integrating out smaller scales. By a direct analytic argument, we derive an approximate differential inequality for this coarse-grained ellipticity as a function of the length scale. This approach may be viewed as a rigorous renormalization group argument and provides a quantitative framework for homogenization that can be iteratively applied across an arbitrary number of length scales. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10732 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Renormalization group and elliptic homogenization in high contrast Armstrong, Scott Kuusi, Tuomo Probability Mathematical Physics Analysis of PDEs We prove a quantitative estimate for the homogenization length scale in terms of the ellipticity ratio $Λ/λ$ of the coefficient field. This upper bound applies to high-contrast elliptic equations exhibiting near-critical behavior. Specifically, we show, assuming a suitable decay of correlations, the length scale at which homogenization occurs is at most $\exp(C \log^2(1+Λ/λ))$. The proof introduces the new concept of coarse-grained ellipticity, which measures the effective ellipticity ratio of the equation--and thus the strength of the disorder--after integrating out smaller scales. By a direct analytic argument, we derive an approximate differential inequality for this coarse-grained ellipticity as a function of the length scale. This approach may be viewed as a rigorous renormalization group argument and provides a quantitative framework for homogenization that can be iteratively applied across an arbitrary number of length scales. |
| title | Renormalization group and elliptic homogenization in high contrast |
| topic | Probability Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2405.10732 |