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Asymptotic behavior of integral functionals for a two‐parameter singularly perturbed nonlinear traction problem
Author(s) -
Falconi Riccardo,
Luzzini Paolo,
Musolino Paolo
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6920
Subject(s) - mathematics , nonlinear system , degenerate energy levels , singular perturbation , mathematical analysis , traction (geology) , perturbation (astronomy) , boundary value problem , domain (mathematical analysis) , method of matched asymptotic expansions , physics , quantum mechanics , geomorphology , geology
We consider a nonlinear traction boundary value problem for the Lamé equations in an unbounded periodically perforated domain. The edges lengths of the periodicity cell are proportional to a positive parameter δ , whereas the relative size of the holes is determined by a second positive parameter ε . Under suitable assumptions on the nonlinearity, there exists a family of solutions{ u ( ε , δ , · ) } ( ε , δ ) ∈ ] 0 , ε ′ [ × ] 0 , δ ′ [ . We analyze the asymptotic behavior of two integral functionals associated to such a family of solutions when the perturbation parameter pair ( ε , δ ) is close to the degenerate value (0, 0) .