Two-Scale Tools for Homogenization and Dimension Reduction of Perforated Thin Layers: Extensions, Korn-Inequalities, and Compactness of Scale-Dependent Sets in Sobolev Spaces

In this investigation, we develop basic methods for the multi-scale analysis of problems in thin porous layers. More precisely, we provide tools for the homogenization of “tangentially” periodic structures, and dimensional reduction letting the layer thickness tend to zero prop ortional to the scale parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>ϵ</mml:mi> </mml:math> . A crucial point is the identification of scale limits of sequences <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>ϵ</mml:mi> </mml:msub> </mml:math> characterized by uniform a priori estimates with respect to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>ϵ</mml:mi> </mml:math> , arising as solutions of differential equations, like Navier–Stokes system, linear elasticity, or fluid-structure interaction problems, in media with thin layers. Often in such problems, in a first step, the symmetric gradients can be controlled, and Korn’s inequality in porous layers is required to estimate the gradients. We construct controllable pore-filling extensions and use them for the proof of the required Korn-inequalities in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mspace width="0.25em"/> <mml:mi>p</mml:mi> </mml:mrow> </mml:msup> </mml:math> -spaces. These results are the basis for the derivation of compactness results with respect to two-scale convergence and the characterization of the scale limits. To illustrate the range of application of the developed multi-scale methods, a semi-linear elastic wave equation in a thin periodically perforated layer with an inhomogeneous Neumann boundary condition on the surface of the elastic substructure is treated and a homogenized, reduced system is derived.