Asymptotic analysis of stratified elastic media in the space of functions with bounded deformation

TL;DR

This paper analyzes the asymptotic behavior of stratified elastic media with heterogeneous properties, deriving limit problems under measure convergence, and extends methods to elliptic systems and anisotropic heat equations.

Contribution

It introduces a novel asymptotic analysis framework for stratified elastic structures with measure-valued coefficients, addressing open problems in anisotropic heat equations.

Findings

Derived limit problems for stratified elastic media.

Extended analysis to elliptic PDE systems.

Addressed open problem for anisotropic heat equation.

Abstract

We consider a heterogeneous elastic structure which is stratified in some direction. We derive the limit problem under the assumption that the Lam\'e coefficients and their inverses weakly* converge to Radon measures. Our method applies also to linear second-order elliptic systems of partial differential equations and in particular, for the case $d=1$, this addresses the previously open problem of determining the asymptotic behaviour in this context for the general anisotropic heat equation.