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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.13169 |
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| _version_ | 1866909501890756608 |
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| author | Recke, Lutz |
| author_facet | Recke, Lutz |
| contents | We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } Ω$$ with Dirichlet boundary conditions. For small $\varepsilon>0$ we show existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized problem. Moreover, we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ for $\varepsilon \to 0$, and we estimate the corresponding rate of convergence. Our assumptions are, roughly speaking, as follows: $Ω$ is a bounded Lipschitz domain, $A$, $B$, $c(\cdot,u)$ and $d(\cdot,u)$ are bounded and measurable, $c(x,\cdot)$ and $d(x,\cdot)$ are $C^1$-smooth, $A$ is periodic, and $B$ is a localized defect. Neither global uniqueness is supposed nor growth restriction for $c(x,\cdot)$ or $d(x,\cdot)$.
The main tool of the proofs is an abstract result of implicit function theorem type which permits a common approach to nonlinear singular perturbation and homogenization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_13169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A common approach to singular perturbation and homogenization III: Nonlinear periodic homogenization with localized defects Recke, Lutz Analysis of PDEs 35B27 35D30 35J57 35J61 47J07 58C15 We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type $$ \nabla\cdot\Big(\Big(A(x/\varepsilon)+B(x/\varepsilon)\Big)\nabla u(x)+c(x,u(x)\Big)=d(x,u(x)) \mbox{ in } Ω$$ with Dirichlet boundary conditions. For small $\varepsilon>0$ we show existence of weak solutions $u=u_\varepsilon$ as well as their local uniqueness for $\|u-u_0\|_\infty \approx 0$, where $u_0$ is a given non-degenerate weak solution to the homogenized problem. Moreover, we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ for $\varepsilon \to 0$, and we estimate the corresponding rate of convergence. Our assumptions are, roughly speaking, as follows: $Ω$ is a bounded Lipschitz domain, $A$, $B$, $c(\cdot,u)$ and $d(\cdot,u)$ are bounded and measurable, $c(x,\cdot)$ and $d(x,\cdot)$ are $C^1$-smooth, $A$ is periodic, and $B$ is a localized defect. Neither global uniqueness is supposed nor growth restriction for $c(x,\cdot)$ or $d(x,\cdot)$. The main tool of the proofs is an abstract result of implicit function theorem type which permits a common approach to nonlinear singular perturbation and homogenization. |
| title | A common approach to singular perturbation and homogenization III: Nonlinear periodic homogenization with localized defects |
| topic | Analysis of PDEs 35B27 35D30 35J57 35J61 47J07 58C15 |
| url | https://arxiv.org/abs/2502.13169 |