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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2410.22726 |
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| _version_ | 1866913567925600256 |
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| author | Yang, Man |
| author_facet | Yang, Man |
| contents | In this study, we investigate the convergence rates for the homogenization of elliptic equations with lower-order terms under the spectral gap assumption, in both bounded domains and the entire space. Our analysis demonstrates that lower-order terms significantly affect the convergence rate, particularly in the full space, where the rate changes from \(O(ε)\) (observed without lower-order terms) to \(O(ε^{d/({d+2})})\) due to their influence. In contrast, in bounded domains, the convergence rate remains \(O(ε^{1/2})\), as boundary conditions exert a stronger influence than the lower-order terms. To manage the complexities introduced by lower-order terms, we developed a novel technique that localizes the analysis within small grids, enabling the application of the Poincaré inequality for effective estimates. This work builds upon existing frameworks, offering a refined approach to quantitative homogenization with lower-order terms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22726 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Influence of Lower-Order Terms on the Convergence Rates in Stochastic Homogenization of Elliptic Equations Yang, Man Analysis of PDEs In this study, we investigate the convergence rates for the homogenization of elliptic equations with lower-order terms under the spectral gap assumption, in both bounded domains and the entire space. Our analysis demonstrates that lower-order terms significantly affect the convergence rate, particularly in the full space, where the rate changes from \(O(ε)\) (observed without lower-order terms) to \(O(ε^{d/({d+2})})\) due to their influence. In contrast, in bounded domains, the convergence rate remains \(O(ε^{1/2})\), as boundary conditions exert a stronger influence than the lower-order terms. To manage the complexities introduced by lower-order terms, we developed a novel technique that localizes the analysis within small grids, enabling the application of the Poincaré inequality for effective estimates. This work builds upon existing frameworks, offering a refined approach to quantitative homogenization with lower-order terms. |
| title | Influence of Lower-Order Terms on the Convergence Rates in Stochastic Homogenization of Elliptic Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.22726 |