Diffuse Interface Energies with Microscopic Heterogeneities: Homogenization and Rare Events
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912002864054272 |
|---|---|
| author | Morfe, Peter S. Wagner, Christian |
| author_facet | Morfe, Peter S. Wagner, Christian |
| contents | We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $δ$ of the heterogeneities is much smaller (microscopic) than the interface width $ε$ (mesoscopic). In the main result, we prove that if the ratio $δε^{-1}$ decays fast enough compared to $ε$, then homogenization effects dominate, and the $Γ$-limit of the energy is the same as if the coefficients had been replaced by their homogenized values. As a byproduct of the proof, this implies that homogenization holds in the periodic setting whenever $δε^{-1}$ vanishes with $ε$, no matter how slowly. Via explicit examples, we prove this is sharp: if $δε^{-1}$ decays too slowly, then improbable or atypical local configurations of the medium begin to play a role, and the $Γ$-limit may be smaller than the one predicted by homogenization theory. We refer to this as the rare events regime, and we prove that it can occur in both random and almost periodic media. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14914 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diffuse Interface Energies with Microscopic Heterogeneities: Homogenization and Rare Events Morfe, Peter S. Wagner, Christian Analysis of PDEs We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $δ$ of the heterogeneities is much smaller (microscopic) than the interface width $ε$ (mesoscopic). In the main result, we prove that if the ratio $δε^{-1}$ decays fast enough compared to $ε$, then homogenization effects dominate, and the $Γ$-limit of the energy is the same as if the coefficients had been replaced by their homogenized values. As a byproduct of the proof, this implies that homogenization holds in the periodic setting whenever $δε^{-1}$ vanishes with $ε$, no matter how slowly. Via explicit examples, we prove this is sharp: if $δε^{-1}$ decays too slowly, then improbable or atypical local configurations of the medium begin to play a role, and the $Γ$-limit may be smaller than the one predicted by homogenization theory. We refer to this as the rare events regime, and we prove that it can occur in both random and almost periodic media. |
| title | Diffuse Interface Energies with Microscopic Heterogeneities: Homogenization and Rare Events |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.14914 |