Consider the homogenization problem of the elliptic system
\begin{equation}
- \nabla \cdot A \left( \frac{x}{\varepsilon} \right) \nabla u (x) = 0, \ \ x \in D, \tag{1}
\end{equation}
in a domain $D\subset \mathbb{R}^d$, ($d\geq 2$), and with oscillating Dirichlet boundary data
\begin{equation}
u(x) = g \left(x , \frac{x}{\varepsilon} \right), \ \ x \in \partial D. \tag{2}
\end{equation}
Here $\varepsilon> 0$ is a small parameter, and $A= A^{\alpha \beta } (x) \in M_N(\mathbb{R})$, $x\in \mathbb{R}^d$ is a family of functions indexed by $1\leq \alpha, \beta \leq d$ and with values in the set of matrices $M_N( \mathbb{R})$. For each $\varepsilon>0$ let $\mathcal{L}_\varepsilon$ be the differential operator in question, i.e. the $i$-th component of its action on a vector function $u=(u_1,...,u_N)$ is defined as
$$
(\mathcal{L}_\varepsilon u)_i (x)= - \left( \nabla \cdot A \left( \frac{\cdot}{\varepsilon} \right) \nabla u \right)_{i} (x) =
-\partial_{x_\alpha} \left[ A^{\alpha \beta }_{ij} \left( \frac{\cdot}{\varepsilon} \right) \partial_{x_\beta} u_j \right],
$$
where $1\leq i \leq N$ (the case $N=1$ corresponds to scalar equations). We impose the following conditions on the problem $(1)$-$(2)$:
(Ellipticity) there exists a constant $\lambda>0$ such that for any $x\in \mathbb{R}^d$, and any $\xi=(\xi^\alpha_i)\in \mathbb{R}^{dN}$ one has
$$
\lambda \xi^\alpha_i \xi^\alpha_i \leq A^{\alpha \beta}_{ij}(x) \xi^\alpha_i \xi^\beta_j \leq \frac{1}{\lambda} \xi^\alpha_i \xi^\alpha_i.
$$
(Periodicity) $A$ and, $g$ in its second variable, are both $\mathbb{Z}^d$-periodic, i.e. $A(y+h) = A(y)$, and $g(x, y + h) = g(y)$ for all $x\in \partial D$, $y\in \mathbb{R}^d$, and $h\in \mathbb{Z}^d$.
We skip further assumptions on the smoothness and geometry of the domain, these are not relevant for introducing the general problem.
For each $\varepsilon > 0$ let $u_\varepsilon$ be the unique solution to $(1)$-$(2)$.
The main question, in general terms, asks:
$$
\textbf{What can we say about the limit of } u_\varepsilon \textbf{ as } \varepsilon \to 0 \ ?
$$
In other words does $(1)$-$(2)$ have a homogenized limit and if so what properties does it have?
Following is a brief discussion based on [1] (see also [2]-[4] for details) on how one may naturally appear at $(1)$-$(2)$.
Consider problem $(1)$ but with a fixed boundary data $g$, i.e.
$$
-\nabla \cdot A\left( \frac{x}{\varepsilon} \right) \nabla u (x) = 0 , \ x\in D \qquad \text{and} \qquad u= g(x), \ x\in \partial D.
$$
By Lax-Milgram for each $\varepsilon>0$ this problem has a unique weak solution $u_\varepsilon \in H^1(D; \mathbb{R}^N)$, which converges weakly in $H^1(D)$ to solution $u_0$ of the homogenized problem
$$
-\nabla \cdot A^0 \nabla u_0 (x) = 0 , \ x\in D \qquad \text{ and } \qquad u_0= g(x), \ x\in \partial D.
$$
Here $A^0$ is the homogenized coefficient tensor and is defined via the solutions of the cell-problem, namely
for $1\leq \gamma \leq d$ define $\chi = \chi^\gamma (y) \in M_N(\mathbb{R})$ to be the periodic solution of the problem
$$
-\partial_\alpha [ A^{\alpha \beta}(y) \partial_\beta \chi^\gamma(y) ] = \partial_\alpha A^{\alpha \gamma } (y) \text{ in } \mathbb{T}^d \text{ and } \int_{\mathbb{T}^d} \chi^\gamma(y) dy = 0 ,
$$
where $\mathbb{T}^d$ is the unit torus, and we adopted the summation convention of repeated indices. The homogenized coefficients are defined by
$$
A^{0, \alpha \beta} = \int_{ \mathbb{T}^d } A^{\alpha \beta} dy + \int_{\mathbb{T}^d} A^{\alpha \gamma} \partial_{\gamma } \chi^{\beta} dy.
$$
Setting $u_1 (x,y) = - \chi^{\alpha} (y) \partial_{\alpha} u_0(x)$ one obtains
$$
u_\varepsilon(x) = u_0(x) + \varepsilon u_1 \left( x, \frac{x}{\varepsilon} \right) + O( \varepsilon^{1/2} ) \text{ in } H^1(D).
$$
The latter is the justification that the first two terms of the formal two-scale expansion of $u_\varepsilon$ are correct.
Now observe that there is a mismatch of the boundary data in the left and right hand sides of the last expansion. The profile $u_1$ being periodic in its second variable oscillates near the boundary, giving rise to the so-called boundary layer phenomenon, which is responsible for the $\varepsilon^{1/2}$ loss in the approximation of $u_\varepsilon$. Quoting [1],
Of particular importance is the analysis of the behavior of solutions near boundaries and, possibly, any associated boundary layers. Relatively little seems to be known about this problem.
Indeed, correcting the boundary data by $u_{1,\varepsilon }^{bl}$ defined as
$$
\mathcal{L}_\varepsilon u_{1, \varepsilon }^{bl} = 0 \text{ in } D \qquad \text{and} \qquad u_{1, \varepsilon }^{bl} = -u_1 \left( x, \frac{x}{\varepsilon} \right) \text{ on } \partial D,
$$
we get
$$
u_\varepsilon(x) = u_0(x) + \varepsilon u_1 \left( x, \frac{x}{\varepsilon} \right) + \varepsilon u_{1, \varepsilon}^{bl} (x) + O( \varepsilon ) \text{ in } H^1(D).
$$
While this expansion shows that correcting the boundary data in the two-scale expansion of $u_\varepsilon$ gives a better approximation,
it is of little use as long as we do not understand the behavior of $u_{1, \varepsilon}^{bl}$ as $\varepsilon \to 0$.
The last question is precisely a partial case of $(1)$-$(2)$, with boundary data $g(x,y) = - \chi^{\alpha} (y) \partial_{\alpha} u_0(x) $.
Solution Description
In a bounded domain $D\subset \mathbb{R}^d$ ($d\geq 2$) consider homogenization of Dirichlet problem of the elliptic system
\begin{equation}
\begin{cases}
-\nabla \cdot A \left( \frac{x}{\varepsilon} \right) \nabla u(x) = 0, & x \in D, \\
u(x) = g \left(x , \frac{x}{\varepsilon} \right), & x \in \partial D
\end{cases} \tag{1}
\end{equation}
where $\varepsilon > 0$ is a small parameter and $A= A^{\alpha \beta } (x) \in M_N(\mathbb{R})$, $x\in \mathbb{R}^d$ is a family of functions indexed by $1\leq \alpha, \beta \leq d$ with values in the set of matrices $M_N( \mathbb{R})$.
Consider $(1)$ under the following conditions:
(Ellipticity) there exists a constant $\lambda>0$ such that $\forall x\in \mathbb{R}^d$, and $\forall \xi=(\xi^\alpha_i)\in \mathbb{R}^{d N}$ one has
$$
\lambda \xi^\alpha_i \xi^\alpha_i \leq A^{\alpha \beta}_{ij} (x) \xi^\alpha_i \xi^\beta_j \leq \frac{1}{\lambda} \xi^\alpha_i \xi^\alpha_i .
$$
(Periodicity) $A$ and, $g$ in its second variable, are both $\mathbb{Z}^d$-periodic, i.e. $A(y+h) = A(y)$, and $g(x, y + h) = g(y)$ for all $x\in \partial D$, $y\in \mathbb{R}^d$, and $h\in \mathbb{Z}^d$.
(Smoothness) The elements of $A$, the function $g$ in both variables, and the boundary of $D$ are $C^\infty$ smooth.
(Geometry) Domain $D$ is strictly convex.
For each $\varepsilon > 0$ let $u_\varepsilon$ be the unique (smooth) solution to $(1)$. The main result of [1] states:
Theorem: Under the conditions listed above, there exists an $L^\infty$ function $g_*:\partial D \to \mathbb{R}^N$, such that if $u_0$ is the solution to the Dirichlet problem with operator tensor $A^0$ (the classical homogenized coefficients), and boundary data $g_*$, then for any $0< \alpha < \frac{d-1}{3d + 5}$ one has
$$
|| u_\varepsilon - u_0 ||_{L^2(D)} \leq C_\alpha \varepsilon^\alpha,
$$
where the constant $C_\alpha = C(\alpha, D, A, g, d)$.
This result represents a major breakthrough in the study of the problem of homogenization of boundary layers introduced in [3] , and is the first result addressing the problem in domains with smooth geometry. Below we present an ultra-simplified sketch of the approach of [1].
A crucial mathematical difficulty in studying $(1)$ comes from the fact that oscillations of $u_\varepsilon$ near $\partial D$ are generally of no periodic character.
The rough summary of the approach which was introduced in [2] and developed further in [1], is to approximate the original domain by polygonal domains from outside having suitable normals for their bounding facets, and then, transfer the homogenization problem from the original domain to the approximating polygon, where only finitely many correctors (approximants) need be considered, namely one for each face. More precisely, fix $x_0\in \partial D$, and assume that $D$ lies locally on one side of its tangent plane, i.e. $D\subset \{x\in \mathbb{R}^d: \ (x-x_0)\cdot n>0 \}$ in a neighborhood of $x_0$. Then, in this neighborhood one tries to approximate the solution $u_\varepsilon$ to $(1)$ by a function of the form $v\left(x, \frac{x}{\varepsilon} \right)$, i.e. a function admitting separation of scales similar in the spirit to the classical two-scale expansion. Heuristically, plugging such a $v$ into $(1)$ and using an expansion of operator in $(1)$ coming from the two-scale ansatz,
one obtains that $v$ should solve
$$
\begin{cases}
-\nabla_y \cdot A(y) \nabla_y v(x_0, y) = 0 , & \qquad y\cdot n > \frac{x_0 \cdot n}{\varepsilon}, \\
v(x_0, y) = g(x_0, y) , & \qquad y \cdot n = \frac{x_0 \cdot n}{\varepsilon},
\end{cases}
$$
With a slight abuse of notation, dropping dependence on the position of the halfspace that comes through $x_0 /\varepsilon$, one gets a problem of the form
\begin{equation}
\begin{cases}
-\nabla_y \cdot A(y) \nabla_y v(y) = 0 , & \qquad y\cdot n > a, \\
v(y) = v_0(y) , & \qquad y \cdot n = a,
\end{cases} \tag{2}
\end{equation}
where $a\in \mathbb{R}$, and $v_0$ is smooth and $\mathbb{Z}^d$-periodic. As it turns out the behavior of $v$ depends on $n$ only when $n$ is not rational, and fixing $a$ is indeed a valid step. Problem $(2)$ is referred to as boundary layer system in [2] and plays a central role in the analysis of $(1)$.
It was proved in [2] that boundary layer systems are well-posed in the class of quasi-periodic functions. The behavior of solutions as $y\cdot n \to \infty$, which captures the homogenization effects for $u_\varepsilon$, was studied initially in [2], and with more detailed analysis in [1]. It is proved that under certain Diophantine condition on the normal $n$ the solution to $(2)$ converges as $y\cdot n \to \infty$, to a constant vector field, named as a boundary layer tail, which is independent of $a$.
Having this information at hand, one then tries to glue the all approximations by boundary layer systems to obtain an approximation in the vicinity of the boundary for the original problem $(1)$. The role of strict convexity of the domain is twofold: it first assures that almost all points of $\partial D$ have normals satisfying the Diophantine criteria
mentioned above, and secondly, convexity puts the domain on one side of its tangent planes making the approximation argument by boundary layer systems viable.
ArticleIs an originHomogenization and boundary layers
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