2024/08/13 by Lutz Recke, Recke, Lutz
Computer Science · Engineering · #34B15 34C29 35B27 47J07 58C15 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #Composite Material Mechanics #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2408.06705
openalex publication_date 2024/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
We consider periodic homogenization with localized defects of boundary value problems for semilinear ODE systems of the type ((A(x/ε)+B(x/ε))u'(x)+c(x,u(x)))'= d(x,u(x)) for x ∈ (0,1), u(0)=u(1)=0. For small ε>0 we show existence of weak solutions u=uε as well as their local uniqueness for ‖u-u0‖_∞ ≈ 0, where u=u0 is a given solution to the homogenized problem (A0u'+c(x,u(x)))'= d(x,u(x)) for x ∈ (0,1), u(0)=u(1)=0, A0:=(∫01A(y)-1dy)-1 such that the linearized problem (A0u'+∂uc(x,u0(x))u(x))'= ∂ud(x,u0(x))u(x) for x ∈ (0,1), u(0)=u(1)=0 does not have weak solutions u\not=0. Further, we prove that ‖uε-u0‖_∞→ 0 and, if c(⋅,u)∈ W1,∞((0,1);ℝn), that ‖uε-u0‖_∞=O(ε) for ε → 0. Moreover, all these statements are true, roughly speaking, uniformly with respect to the localized defects B. We assume that A ∈ L^∞(ℝ;\mathbbMn) is 1-periodic, B ∈ L^∞(ℝ;\mathbbMn)∩ L1(ℝ;\mathbbMn), A(y) and A(y)+B(y) are positive definite uniformly with respect to y, c(x,⋅),d(x,⋅)∈ C1(ℝn;ℝn) and c(⋅,u),d(⋅,u) ∈ L^∞((0,1);ℝn). 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.