Asymptotic analysis of boundary layer correctors in periodic homogenization

TL;DR

This paper analyzes the asymptotic behavior of boundary layer correctors in periodic homogenization, revealing convergence properties, asymptotic expansions, and the impact of microstructure boundary interactions on convergence speed.

Contribution

It extends previous results by studying arbitrary irrational directions using ergodicity and provides asymptotic expansions of Poisson's kernel.

Findings

Boundary layer correctors converge to a constant vector field.

Asymptotic expansion of Poisson's kernel for large distances.

Convergence can be arbitrarily slow in the general case.

Abstract

This paper is devoted to the asymptotic analysis of boundary layers in periodic homogenization. We investigate the behaviour of the boundary layer corrector, defined in the half-space $\Omega_{n,a}:=\{y\cdot n-a>0\}$, far away from the boundary and prove the convergence towards a constant vector field, the boundary layer tail. This problem happens to depend strongly on the way the boundary $\partial\Omega_{n,a}$ intersects the underlying microstructure. Our study complements the previous results obtained on the one hand for $n\in\mathbb R\mathbb Q^d$, and on the other hand for $n\notin\mathbb R\mathbb Q^d$ satisfying a small divisors assumption. We tackle the case of arbitrary $n\notin\mathbb R\mathbb Q^d$ using ergodicity of the boundary layer along $\partial\Omega_{n,a}$. Moreover, we get an asymptotic expansion of Poisson's kernel $P=P(y,\tilde{y})$, associated to the elliptic operator $-\nabla\cdot A(y)\nabla\cdot$ and $\Omega_{n,a}$, for $|y-\tilde{y}|\rightarrow\infty$. Finally, we show that, in general, convergence towards the boundary layer tail can be arbitrarily slow, which makes the general case very different from the rational or the small divisors one.