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

TL;DR

This paper develops multi-scale analysis tools for homogenization and dimension reduction in thin perforated layers, focusing on Korn-inequalities, two-scale compactness, and applications to elastic wave equations in heterogeneous media.

Contribution

It introduces new Korn-inequalities and extension methods for porous thin layers, enabling homogenization and dimension reduction in complex multi-scale problems.

Findings

Derived homogenized models for thin perforated elastic layers.

Established Korn-inequalities with constants independent of layer thickness.

Proved two-scale compactness results for functions in heterogeneous thin domains.

Abstract

This investigation develops basic methods for the multi-scale analysis for problems in thin porous layers. More precisely, we provide tools for the homogenization in case of "tangentially" periodic structures and dimensional reduction letting the layer thickness tend to zero proportional to the scale parameter $\epsilon$. A crucial point is the identification of scale limits of functions $v_{\epsilon}$ in subsets of function spaces characterized by uniform a priori estimates with respect to $\epsilon$, arising for solutions of differential equations in heterogeneous media with thin layers, e.g., of a Navier-Stokes system, models in linear elasticity, or problems with fluid-structure interaction. Often in problems from continuum mechanics, in a first step, the symmetric gradients of arising vector fields can be controlled and Korn's inequality in porous layers is required to estimate the gradients, such that crucial constants do not depend on $\epsilon$. Controllable pore-filling extension are constructed and, thus, the analysis is reduced to a fixed basic domain. The proof of the required Korn-inequalities for porous thin layers, formulated with respect to $L^p$-spaces, is based on these constructions. Also, the investigation of compactness with respect to two-scale convergence and the characterization of the scale limits is strongly based on the extension theorem and the Korn-inequalities. To illustrate the range of applications of the developed analytic multiscale method 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 an homogenized, reduced system is derived.