## Correctors for the homogenization limit: Applications to elliptic systems

In the terminology of homogenization method, “corrector” or “corrector estimate” is ascribed to be the error estimate between the approximate solution governed by the asymptotic procedure in two-scale analysis, and the exact solution. This word(s) typically originates from the French homogenization community since 20th century.

This note would follow up on those which we have had a glance before: “note 1” and “note 2“. To be more precise, it would present an extension of the homogenization method towards the elliptic systems posed in perforated media.

The semi-linear system we address here is likely to be a steady state-type of thermo-diffusion systems which describe the interplay between diffusion fluxes of a fixed number of colloidal populations and a heat flux incorporating the Dufour and Soret effects. Particularly, it is given by

$\nabla\cdot(-d_{i}^{\varepsilon}\nabla u_{i}^\varepsilon)=R_{i}(u_{1}^{\varepsilon},...,u_{N}^{\varepsilon}),$

associated with the boundary conditions

$d_{i}^{\varepsilon}\nabla u_{i}^\varepsilon\cdot n=\varepsilon(a_{i}^{\varepsilon}u_{i}^{\varepsilon}-b_{i}^{\varepsilon}F_{i}(u_{i}^{\varepsilon}))\;\mbox{on}\;\Gamma^{\varepsilon},$

$u_{i}^{\varepsilon}=0\;\mbox{on}\;\Gamma,$

for $i\in \left\{1,...,N\right\}$ the number of concentrations $u_{i}^{\varepsilon}$. Some notations could be elucidated here: follow the real-world application mentioned above, $d_{i}^{\varepsilon}$ represents the molecular diffusion with $R_{i}$ the volume reaction rate and $a_{i}^{\varepsilon},b_{i}^{\varepsilon}$ are deposition coefficients while $F_{i}$ indicates a surface chemical reaction for the immobile species. Notice that the quantity $\varepsilon$ is called the homogenization parameter or the scale factor.

A little bit unlike the media that we considered in “note 2“, here we require that the media with microstructure must be connected and that the boundaries are Lipschitz. In other words, the admissible domain can be imagined that the holes do not intersect the exterior boundary. To illustrate, we depict it together with the basic geometry of the microstructure in the figure underneath the entire content.

Such requirement allows us to prove the $L^{\infty}$-type estimate of solution by the Moser-like techniques. Otherwise, dependence of Sobolev embeddings on the dimension of media would ruin the whole proof. In addition, the well-posedness including the existence and uniqueness by variational techniques from the original work of Brezis and Oswald since 1994 can impose very well (but need modifications) in this problem.

The position of the $L^{\infty}$-type estimate plays a very crucial role in this situation. In real-world applications, the reaction term $R_{i}$ is mostly locally Lipschitz and may hinder computations in the homogenization process while the global Lipschitzian performs very well. With that estimate at hands, we may further assume that:

For the asymptotic up to $M$-level of expansion ($M\geq 2$) given by

$u_{i}^{\varepsilon}(x)=\displaystyle{\sum_{m=1}^{M}}\varepsilon^{m}u_{i,m}(x,y)+\mathcal{O}(\varepsilon^{M+1}),$

where $u_{i,m}^{\varepsilon}(x,y)$ is $Y$-periodic with $y=x/{\varepsilon}$ the fast variable representing the microscopic geometry, there exists a globally Lipschitz function $\bar{R}_{i}$ such that

$R_{i}(u_{1}^{\varepsilon},...,u_{N}^{\varepsilon})=\displaystyle{\sum_{m=1}^{M}}\varepsilon^{m}\bar{R}_{i}(u_{1,m},...,u_{N,m})+\mathcal{O}(\varepsilon^{M+1}).$

Consequently, we obtain the corrector estimates in $H^1(\Omega^{\varepsilon})$ (which means $N=1$ for simplicity) up to $M$-level of expansion:

$\left\Vert u^{\varepsilon} - \displaystyle{\sum_{m=0}^{M}}\varepsilon^{m}u_{m} \right\Vert\leq C(\varepsilon^{M-1}+\varepsilon^{M-3/2}),$

and for $m^{\varepsilon}$ the cut-off function defined in the book of Cioranescu and Saint Jean Paulin (1998), which allows us to have the corrector in a neighborhood of the boundary of the media $\Omega^{\varepsilon}$, that

$\left\Vert u^{\varepsilon} - u_{0}-m^{\varepsilon}\displaystyle{\sum_{m=1}^{M}}\varepsilon^{m}u_{m} \right\Vert\leq C\displaystyle{\sum_{m=1}^{M}}\varepsilon^{m-1/2},$

where $C$ is a generic positive constant independent of the choice of $\varepsilon$.

All of the above-mentioned results could be found in the preprint attached here.

Fig. 1: Admissible 2-D perforated domain (left) and basic geometry of the microstructure (right). (By courtesy of MSc. Mai Thanh Nhat Truong, Hankyong National University, Republic of Korea.)

## One thought on “Correctors for the homogenization limit: Applications to elliptic systems”

1. […] the previous note, we proved new results on the high-order correctors for the semi-linear elliptic […]