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

Part of: Higher-dimensional theory Elliptic equations and systems Linear integral equations Nonlinear operators and their properties

Published online by Cambridge University Press:  22 March 2019

Massimo Lanza de Cristoforis  and
Paolo Musolino
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)
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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.

Keywords

Nonlinear Robin problemsingularly perturbed domainPoisson equationperiodically perforated domainhomogenizationenergy integralreal analytic continuation in Banach space

MSC classification

Primary: 35J25: Boundary value problems for second-order elliptic equations
Secondary: 31B10: Integral representations, integral operators, integral equations methods 45A05: Linear integral equations 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 4 , November 2019 , pp. 985 - 1016
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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