An operator-asymptotic approach to periodic homogenization for equations of linearized elasticity
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913604894195712 |
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| author | Lim, Yi-Sheng Žubrinić, Josip |
| author_facet | Lim, Yi-Sheng Žubrinić, Josip |
| contents | We present an operator-asymptotic approach to the problem of homogenization of periodic composite media in the setting of three-dimensional linearized elasticity. This is based on a uniform approximation with respect to the inverse wavelength $|χ|$ for the solution to the resolvent problem when written as a superposition of elementary plane waves with wave vector (``quasimomentum") $χ$. We develop an asymptotic procedure in powers of $|χ|$, combined with a new uniform version of the classical Korn inequality. As a consequence, we obtain $L^2\to L^2$, $L^2\to H^1$, and higher-order $L^2\to L^2$ norm-resolvent estimates in $\mathbb{R}^3$. The $L^2 \to H^1$ and higher-order $L^2 \to L^2$ correctors emerge naturally from the asymptotic procedure, and the former is shown to coincide with the classical formulae. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_00594 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An operator-asymptotic approach to periodic homogenization for equations of linearized elasticity Lim, Yi-Sheng Žubrinić, Josip Analysis of PDEs We present an operator-asymptotic approach to the problem of homogenization of periodic composite media in the setting of three-dimensional linearized elasticity. This is based on a uniform approximation with respect to the inverse wavelength $|χ|$ for the solution to the resolvent problem when written as a superposition of elementary plane waves with wave vector (``quasimomentum") $χ$. We develop an asymptotic procedure in powers of $|χ|$, combined with a new uniform version of the classical Korn inequality. As a consequence, we obtain $L^2\to L^2$, $L^2\to H^1$, and higher-order $L^2\to L^2$ norm-resolvent estimates in $\mathbb{R}^3$. The $L^2 \to H^1$ and higher-order $L^2 \to L^2$ correctors emerge naturally from the asymptotic procedure, and the former is shown to coincide with the classical formulae. |
| title | An operator-asymptotic approach to periodic homogenization for equations of linearized elasticity |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2308.00594 |