Inverse homogenization problem for the Drichlet problem for Poisson equation for $W^{-1,\infty}$ potential
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909110233989120 |
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| author | Ishida, Hiroto |
| author_facet | Ishida, Hiroto |
| contents | We consider Poisson problems $-Δu^\varepsilon=f$ on perforated domains, and characterize the limit of $u^\varepsilon$ as the solution to $(-Δ+μ)u=f$ on domain $Ω\subset\mathbb{R}^d$ with some potential $μ\in W^{-1,\infty}(Ω).$ It is known that $μ$ is related to the capacity of holes when $μ\in L^\infty(Ω).$ In this paper, we characterize $μ$ as the limit of the density of the capacity of holes also for many $μ\in W^{-1,\infty}(Ω).$ We apply the result for the inverse homogenization problem, i.e. we construct holes corresponding to the given potential $μ\in L^d(Ω)+L^\infty(δ_S)$ where $δ_S$ is a surface measure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08911 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Inverse homogenization problem for the Drichlet problem for Poisson equation for $W^{-1,\infty}$ potential Ishida, Hiroto Analysis of PDEs 35B27 We consider Poisson problems $-Δu^\varepsilon=f$ on perforated domains, and characterize the limit of $u^\varepsilon$ as the solution to $(-Δ+μ)u=f$ on domain $Ω\subset\mathbb{R}^d$ with some potential $μ\in W^{-1,\infty}(Ω).$ It is known that $μ$ is related to the capacity of holes when $μ\in L^\infty(Ω).$ In this paper, we characterize $μ$ as the limit of the density of the capacity of holes also for many $μ\in W^{-1,\infty}(Ω).$ We apply the result for the inverse homogenization problem, i.e. we construct holes corresponding to the given potential $μ\in L^d(Ω)+L^\infty(δ_S)$ where $δ_S$ is a surface measure. |
| title | Inverse homogenization problem for the Drichlet problem for Poisson equation for $W^{-1,\infty}$ potential |
| topic | Analysis of PDEs 35B27 |
| url | https://arxiv.org/abs/2310.08911 |