Transmission conditions obtained by homogenisation

Given a bounded open set in Rn, n≥2, and a sequence (Kj) of compact sets converging to an (n−1)-dimensional manifold M, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on Ω∖Kj, with Neumann boundary conditions on ∂(Ω∖Kj). We prove that the limit o...

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Bibliographic Details
Published in:Nonlinear analysis 2018-12, Vol.177, p.361-386
Main Authors: Dal Maso, Gianni, Franzina, Giovanni, Zucco, Davide
Format: Article
Language:English
Subjects:
[formula omitted]-convergence
Asymptotic properties
Boundary conditions
Capacitary measures
Integrals
Neumann sieve
Nonlinear systems
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Summary:Given a bounded open set in Rn, n≥2, and a sequence (Kj) of compact sets converging to an (n−1)-dimensional manifold M, we study the asymptotic behaviour of the solutions to some minimum problems for integral functionals on Ω∖Kj, with Neumann boundary conditions on ∂(Ω∖Kj). We prove that the limit of these solutions is a minimiser of the same functional on Ω∖M subjected to a transmission condition on M, which can be expressed through a measure μ supported on M. The class of all measures that can be obtained in this way is characterised, and the link between the measure μ and the sequence (Kj) is expressed by means of suitable local minimum problems.
ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2018.04.015