Geodesic
Homogenization of a boundary-value problem in a domain perforated by cavities of arbitrary shape with a general nonlinear boundary condition on their boundaries: the case of critical values of the parameters
Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 32 (2019) no. 32, pp. 191-219

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A homogenized model is constructed (with rigorous justification) for a boundary-value problem for the Poisson equation in a periodically perforated domain with a nonlinear Robin condition on the boundary of the cavities. This condition contains a parameter depending on the period of the structure and a function $\sigma(x, u)$ responsible for the nonlinearity. The cavities can have an arbitrary shape and the parameters of the problem have “critical values”, which results in a homogenized problem with a different type of nonlinearity. 
@article{TSP_2019_32_32_a8,
     author = {M. N. Zubova and T. A. Shaposhnikova},
     title = {Homogenization of a boundary-value problem in a domain perforated by cavities of arbitrary shape with a general nonlinear boundary condition on their boundaries: the case of critical values of the parameters},
     journal = {Trudy Seminara im. I.G. Petrovskogo},
     pages = {191--219},
     publisher = {mathdoc},
     volume = {32},
     number = {32},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TSP_2019_32_32_a8/}
}
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M. N. Zubova; T. A. Shaposhnikova. Homogenization of a boundary-value problem in a domain perforated by cavities of arbitrary shape with a general nonlinear boundary condition on their boundaries: the case of critical values of the parameters. Trudy Seminara im. I.G. Petrovskogo, Trudy Seminara imeni I. G. Petrovskogo, Tome 32 (2019) no. 32, pp. 191-219. http://geodesic.mathdoc.fr/item/TSP_2019_32_32_a8/