Asymptotic Analysis of Boundary Layer Correctors in Periodic Homogenization
Christophe Prange · SIAM Journal on Mathematical Analysis · 2013
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 $n otin\mathbb R\mathbb Q^d$ satisfying a small divisors assumption. We tackle the case of arbitrary $n otin\mathbb R\mathbb Q^d$ 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 $- abla\cdot A(y) abla\cdot$ 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.