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This investigation develops basic methods for the multi-scale analysis forproblems in thin porous layers. More precisely, we provide tools for thehomogenization in case of "tangentially" periodic structures and dimensionalreduction letting the layer thickness tend to zero proportional to the scaleparameter $\epsilon$. A crucial point is the identification of scale limits offunctions $v_{\epsilon}$ in subsets of function spaces characterized by uniforma priori estimates with respect to $\epsilon$, arising for solutions ofdifferential equations in heterogeneous media with thin layers, e.g., of aNavier-Stokes system, models in linear elasticity, or problems withfluid-structure interaction. Often in problems from continuum mechanics, in afirst step, the symmetric gradients of arising vector fields can be controlledand Korn's inequality in porous layers is required to estimate the gradients,such that crucial constants do not depend on $\epsilon$. Controllablepore-filling extension are constructed and, thus, the analysis is reduced to afixed basic domain. The proof of the required Korn-inequalities for porous thinlayers, formulated with respect to $L^p$-spaces, is based on theseconstructions. Also, the investigation of compactness with respect to two-scaleconvergence and the characterization of the scale limits is strongly based onthe extension theorem and the Korn-inequalities. To illustrate the range ofapplications of the developed analytic multiscale method a semi-linear elasticwave equation in a thin periodically perforated layer with an inhomogeneousNeumann boundary condition on the surface of the elastic substructure istreated and an homogenized, reduced system is derived.

Two-scale tools for homogenization and dimension reduction of perforated  thin layers: Extensions, Korn-inequalities, and two-scale compactness of  scale-dependent sets in Sobolev spaces