Asymptotic Analysis of Boundary Layer Correctors in Periodic Homogenization

Source abstract

This paper is devoted to the asymptotic analysis of boundary layers in periodic homogenization. We investigate the behavior 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 toward 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 toward the boundary layer tail can be arbitrarily slow, which makes the general case very different from the rational or the small divisors one.