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Asymptotic behavior of integral functionals for a two‐parameter singularly perturbed nonlinear traction problem

: Falconi, R.; Luzzini, P.; Musolino, P.


Mathematical Methods in the Applied Sciences 44 (2020), Nr.2, S.2111-2129
ISSN: 0170-4214
ISSN: 1099-1476
Fraunhofer ITWM ()

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).