Minimax Theory and its Applications 08 (2023), No. 1, 085--108
Copyright Heldermann Verlag 2023
Convergence Rate for the Homogenization of Diffusions in Dilutely Perforated Domains with Reflecting Boundaries
Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
and: Yanqi Lake, Beijing Institute of Mathematical Sciences and Applications, Beijing, China
We revisit the homogenization problem for the Poisson equation in periodically perforated domains with zero Neumann data at the boundary of the holes and prescribed Dirichlet data at the outer boundary. It is known that, if the periodicity of the holes goes to zero but their volume fraction remains fixed and positive, the limit problem is a Dirichlet boundary value problem posed in the domain without the holes, and the effective diffusion coefficients are non-trivially modified; if that volume fraction goes to zero instead, i.e. the holes are dilute, the effective operator remains the Laplacian (that is, unmodified). Our main results contain the study of a "continuity" in those effective models with respect to the volume fraction of the holes and some new convergence rates for homogenization in the dilute setting. Our method explores the classical two-scale expansion ansatz and relies on asymptotic analysis of the rescaled cell problems using layer potential theory.
Keywords: Periodic homogenization, perforated domain, periodic layer potentials, Neumann boundary value problems, dilute suspensions.
MSC: 35B27, 35J08.
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