Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates
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| Format: | Preprint |
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2024
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| _version_ | 1866915660401999872 |
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| author | Recke, Lutz |
| author_facet | Recke, Lutz |
| contents | We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$ \Big((A(x/\varepsilon)+B(x/\varepsilon))u'(x)+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0. $$ 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 $u=u_0$ is a given solution to the homogenized problem $$ \Big(A_0u'+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0,\; A_0:=\left(\int_0^1A(y)^{-1}dy\right)^{-1} $$ such that the linearized problem $$ \Big(A_0u'+\partial_uc(x,u_0(x))u(x)\Big)'= \partial_ud(x,u_0(x))u(x) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0 $$ does not have weak solutions $u\not=0$. Further, we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ and, if $c(\cdot,u)\in W^{1,\infty}((0,1);\mathbb{R}^n)$, that $\|u_\varepsilon-u_0\|_\infty=O(\varepsilon)$ for $\varepsilon \to 0$. Moreover, all these statements are true, roughly speaking, uniformly with respect to the localized defects $B$. We assume that $A \in L^\infty(\mathbb{R};\mathbb{M}_n)$ is 1-periodic, $B \in L^\infty(\mathbb{R};\mathbb{M}_n)\cap L^1(\mathbb{R};\mathbb{M}_n)$, $A(y)$ and $A(y)+B(y)$ are positive definite uniformly with respect to $y$, $c(x,\cdot),d(x,\cdot)\in C^1(\mathbb{R}^n;\mathbb{R}^n)$ and $c(\cdot,u),d(\cdot,u) \in L^\infty((0,1);\mathbb{R}^n)$. The main tool of the proofs is an abstract result of implicit function theorem type which has been tailored for applications to nonlinear singular perturbation and homogenization problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_06705 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates Recke, Lutz Classical Analysis and ODEs 34B15 34C29 35B27 47J07 58C15 We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type $$ \Big((A(x/\varepsilon)+B(x/\varepsilon))u'(x)+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0. $$ 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 $u=u_0$ is a given solution to the homogenized problem $$ \Big(A_0u'+c(x,u(x))\Big)'= d(x,u(x)) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0,\; A_0:=\left(\int_0^1A(y)^{-1}dy\right)^{-1} $$ such that the linearized problem $$ \Big(A_0u'+\partial_uc(x,u_0(x))u(x)\Big)'= \partial_ud(x,u_0(x))u(x) \mbox{ for } x \in (0,1),\; u(0)=u(1)=0 $$ does not have weak solutions $u\not=0$. Further, we prove that $\|u_\varepsilon-u_0\|_\infty\to 0$ and, if $c(\cdot,u)\in W^{1,\infty}((0,1);\mathbb{R}^n)$, that $\|u_\varepsilon-u_0\|_\infty=O(\varepsilon)$ for $\varepsilon \to 0$. Moreover, all these statements are true, roughly speaking, uniformly with respect to the localized defects $B$. We assume that $A \in L^\infty(\mathbb{R};\mathbb{M}_n)$ is 1-periodic, $B \in L^\infty(\mathbb{R};\mathbb{M}_n)\cap L^1(\mathbb{R};\mathbb{M}_n)$, $A(y)$ and $A(y)+B(y)$ are positive definite uniformly with respect to $y$, $c(x,\cdot),d(x,\cdot)\in C^1(\mathbb{R}^n;\mathbb{R}^n)$ and $c(\cdot,u),d(\cdot,u) \in L^\infty((0,1);\mathbb{R}^n)$. The main tool of the proofs is an abstract result of implicit function theorem type which has been tailored for applications to nonlinear singular perturbation and homogenization problems. |
| title | Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates |
| topic | Classical Analysis and ODEs 34B15 34C29 35B27 47J07 58C15 |
| url | https://arxiv.org/abs/2408.06705 |