Homogenization of the first initial-boundary value problem for periodic hyperbolic systems. Principal term of approximation
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| Format: | Preprint |
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2023
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| _version_ | 1866929195496505344 |
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| author | Meshkova, Yulia |
| author_facet | Meshkova, Yulia |
| contents | Let $\mathcal{O}\subset \mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $ L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $A_{D,\varepsilon}$ with the Dirichlet boundary condition. Here $\varepsilon >0$ is a small parameter. The coefficients of the operator $A_{D,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. The principal terms of approximations for the operator cosine and sine functions are given in the $(H^2\rightarrow L_2)$- and $(H^1\rightarrow L_2)$-operator norms, respectively. The error estimates are of the precise order $O(\varepsilon)$ for a fixed time. The results in operator terms are derived from the quantitative homogenization estimate for approximation of the solution of the initial-boundary value problem for the equation $(\partial _t^2+A_{D,\varepsilon})\mathbf{u}_\varepsilon =\mathbf{F}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_15887 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Homogenization of the first initial-boundary value problem for periodic hyperbolic systems. Principal term of approximation Meshkova, Yulia Analysis of PDEs 35B27 (Primary), 35L53 (Secondary) Let $\mathcal{O}\subset \mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $ L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $A_{D,\varepsilon}$ with the Dirichlet boundary condition. Here $\varepsilon >0$ is a small parameter. The coefficients of the operator $A_{D,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. The principal terms of approximations for the operator cosine and sine functions are given in the $(H^2\rightarrow L_2)$- and $(H^1\rightarrow L_2)$-operator norms, respectively. The error estimates are of the precise order $O(\varepsilon)$ for a fixed time. The results in operator terms are derived from the quantitative homogenization estimate for approximation of the solution of the initial-boundary value problem for the equation $(\partial _t^2+A_{D,\varepsilon})\mathbf{u}_\varepsilon =\mathbf{F}$. |
| title | Homogenization of the first initial-boundary value problem for periodic hyperbolic systems. Principal term of approximation |
| topic | Analysis of PDEs 35B27 (Primary), 35L53 (Secondary) |
| url | https://arxiv.org/abs/2312.15887 |