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Asymptotic Behaviour of the Energy Integral of a Two-Parameter Homogenization Problem with Nonlinear Periodic Robin Boundary Conditions

Published online by Cambridge University Press:  22 March 2019

Massimo Lanza de Cristoforis
Affiliation:
Dipartimento di Matematica ‘Tullio Levi-Civita’, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy (mldc@math.unipd.it; musolino@math.unipd.it)
Paolo Musolino
Affiliation:
Dipartimento di Matematica ‘Tullio Levi-Civita’, Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy (mldc@math.unipd.it; musolino@math.unipd.it)

Abstract

We consider a nonlinear Robin problem for the Poisson equation in an unbounded periodically perforated domain. The domain has a periodic structure, and the size of each cell is determined by a positive parameter δ. The relative size of each periodic perforation is determined by a positive parameter ε. Under suitable assumptions, such a problem admits a family of solutions which depends on ε and δ. We analyse the behaviour the energy integral of such a family as (ε, δ) tends to (0, 0) by an approach that represents an alternative to asymptotic expansions and classical homogenization theory.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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