Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates

[2408.06705] Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates Skip to main content Search Submit Donate Log in Mathematics > Classical Analysis and ODEs arXiv:2408.06705 (math) [Submitted on 13 Aug 2024 ( v1 ), last revised 3 Jul 2025 (this version, v5)] Title: Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates Authors: Lutz Recke View PDF HTML (experimental) Abstract: 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. Comments: arXiv admin note: substantial text overlap with arXiv:2309.15611 Subjects: Classical Analysis and ODEs (math.CA) MSC classes: 34B15 34C29 35B27 47J07 58C15 Cite as: arXiv:2408.06705 [math.CA] (or arXiv:2408.06705v5 [math.CA] for this version) https://doi.org/10.48550/arXiv.2408.06705 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Applicable Analysis 105, No.3, 473-488 (2026) Related DOI : https://doi.org/10.1080/00036811.2025.2527785 Focus to learn more DOI(s) linking to related resources Submission history From: Lutz Recke [ view email ] [v1] Tue, 13 Aug 2024 08:02:15 UTC (13 KB) [v2] Wed, 18 Sep 2024 07:42:55 UTC (13 KB) [v3] Mon, 21 Oct 2024 10:46:07 UTC (13 KB) [v4] Mon, 10 Mar 2025 11:33:43 UTC (14 KB) [v5] Thu, 3 Jul 2025 11:18:06 UTC (14 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CA < prev | next > new | recent | 2024-08 Change to browse by: math References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

Related documents

• Nonlinear non-periodic homogenization: Existence, local uniqueness and estimates#670290
• Singular Sets and Doubling Properties in Parabolic Homogenization#1143919
• Basic Reproduction Ratios for Almost Periodic Non-Densely Defined Cauchy Problems#1062364
• Quasi-periodicity and almost equality classes#659155
• Quasi-periodicity and almost equality classes#657019