2025/02/17 by Lutz Recke, Recke, Lutz
Computer Science · #35B27 35D30 35J57 35J61 47J07 58C15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2502.13169
openalex publication_date 2025/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider periodic homogenization with localized defects for semilinear elliptic equations and systems of the type ∇⋅((A(x/ε)+B(x/ε))∇ u(x)+c(x,u(x))=d(x,u(x)) in Ω with Dirichlet boundary conditions. For small ε>0 we show existence of weak solutions u=uε as well as their local uniqueness for ‖u-u0‖_∞ ≈ 0, where u0 is a given non-degenerate weak solution to the homogenized problem. Moreover, we prove that ‖uε-u0‖_∞→ 0 for ε → 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(⋅,u) and d(⋅,u) are bounded and measurable, c(x,⋅) and d(x,⋅) are C1-smooth, A is periodic, and B is a localized defect. Neither global uniqueness is supposed nor growth restriction for c(x,⋅) or d(x,⋅). 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.