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 $1\leq i \leq N$.
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 that 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 breakthrough result in the analysis of homogenization of $(1)$-$(2)$ gives rise to the following natural question:
$$
\textbf{What is the regularity of the homogenized boundary condition } g_* \ ?
$$
The function $g_*$ in [1] is defined at all $x\in \partial D$ with Diophantine normal vector, where a unit vector $n \in \mathbb{R}^d$ is called Diophantine if there exist constants $\kappa,l>0$ such that $||P_{n^\perp} ( \xi ) || \geq \kappa ||\xi||^{-l}$ for all non-zero $\xi \in \mathbb{Z}^d$, where $P_{n^\perp}$ is the projection operator on the direction orthogonal to $n$. It is not hard to see that for any fixed $l>0$ satisfying $l(d-1)>1$ almost all points (with respect to the $\mathcal{H}^{d-1}$-measure on the sphere) are Diophantine with some constant $\kappa>0$ (the constant $\kappa>0$, however, is not bounded away from $0$). Thus, $g_*$ is defined almost everywhere on the boundary of $D$.
To outline how Diophantine condition comes into play, we next bring up the notion of boundary layer systems introduced in [2].
For a unit vector $n$, consider the following system
\begin{equation}\begin{cases}
-\nabla_y \cdot A(y) \nabla_y v(y) =0 , & \qquad y\cdot n > 0, \\
v(y)=v_0(y), & \qquad y \cdot n = 0
\end{cases} \tag{3}\end{equation}
where $v_0$ is smooth and $\mathbb{Z}^d$-periodic (and when applied to $(1)$-$(2)$ is defined via $g$ - the original boundary data). Systems of the form $(3)$ were introduced and studied in [2], and later in [1], and play a central role in the analysis of $(1)$-$(2)$. It was proved in [1] (see also [2]) that under the Diophantine condition on the normal $n$, the solution to $(3)$ converges as $y\cdot n \to \infty$ to a constant vector field named as a boundary layer tail. The homogenized boundary condition $g_*$ is defined via the function $x \mapsto v_\infty(n(x))$ where $x \in\partial D$ and has a Diophantine normal vector, and $v_\infty$ is the boundary layer tail corresponding to $n$. Hence the regularity of $g_*$ is boiled down to understanding the regularity of boundary layer tails with respect to the normal vector field of $\partial D$.
It is proved in $[1]$ that boundary layer tails are Lipschitz continuous, however, the Lipschitz constant blows up (as the Diophantine properties of the normal vectors deteriorate). From the (non-uniform) Lipschitz estimate it follows that $g_*$ is continuous at all points of $\partial D$ with Diophantine normal vector. But since the Lipschitz bounds on boundary layer tails are not uniform along $\partial D$, it is not clear, for example, if $g_*$ admits continuous extension to all points of $\partial D$ (recall that $g_*$ was defined only at points with Diophantine normals).
Understanding the regularity of $g_*$ presents a challenging mathematical question on its own right, and may lead to a better understanding of homogenization of $(1)$-$(2)$.
Solution Description
In [1], the authors considered the periodic homogenization of second-order elliptic systems in divergence form with oscillating Dirichlet data or Neumann data of first order. They proved that the homogenized boundary data belongs to $W^{1,p}$ for any $1< p< \infty$. In particular, this implies that the boundary layer tails are Hölder continuous of order $\alpha$ for any $\alpha \in (0, 1)$.
Precisely, we define the oscillating elliptic operator
\begin{equation*}
\mathcal{L}_\varepsilon = -\text{div} (A(x/\varepsilon) \nabla) = - \frac{\partial}{\partial x_i} \bigg\{ a^{\alpha\beta}_{ij} \Big( \frac{x}{\varepsilon}\Big) \frac{\partial}{\partial x_j}\bigg\},
\end{equation*}
We consider the Dirichlet problem
\begin{equation*}\tag{1}
\mathcal{L}_\varepsilon (u_\varepsilon) =0 \quad \text{ in } \Omega, \qquad \text{and} \qquad u_\varepsilon (x) = f(x, x/\varepsilon) \quad \text{ on } \partial\Omega,
\end{equation*}
where $f(x, y)$ is 1-periodic in $y$, and Neumnn problem
\begin{equation*}\tag{2}
\mathcal{L}_\varepsilon (v_\varepsilon) =0 \quad \text{ in } \Omega, \qquad \text{and} \qquad
\frac{\partial v_\varepsilon}{\partial \nu_\varepsilon} =
T_{ij} \cdot \nabla \big\{ g_{ij}(x, x/\varepsilon) \big\}
\quad \text{ on } \partial\Omega,
\end{equation*}
where $T_{ij}=n_i e_j -n_j e_i$ is a tangential vector field on $\partial\Omega$
and $\{ g_{ij} (x, y)\} $ are 1-periodic in $y$.
Under the assumptions that $A$ is smooth and 1-periodic , and $\Omega$ is a smooth and strictly convex domain in $\mathbb{R}^d$, it was proved in [2] that the homogenized problem for (1) is given by
\begin{equation*}\tag{3}
\mathcal{L}_0 (u_0) =0 \quad \text{ in } \Omega, \qquad \text{and} \qquad
u_0 = \overline{f} \quad \text{ on } \partial\Omega,
\end{equation*}
where $\mathcal{L}_0$ is the usual homogenized operator and $\overline{f} $ is a function whose value at $x\in \partial\Omega$ depends only on $A$, $f(x, \cdot)$ and the outward normal $n$ to $\partial\Omega$ at $x$. Similarly, it was proved in [3] that if $\Omega$ is smooth and strictly convex, the homogenized problem for (2) is given by
\begin{equation*}\tag{4}
\mathcal{L}_0 (v_0) =0 \quad \text{ in } \Omega, \qquad \text{and} \qquad
\frac{\partial v_0}{\partial \nu_0} =
T_{ij} \cdot \nabla \overline{g}_{ij}
\quad \text{ on } \partial\Omega,
\end{equation*}
where $\frac{\partial v_0}{\partial\nu_0}$ denotes the conormal derivative of $v_0$ associated with
$\mathcal{L}_0$, and $\{ \overline{g}_{ij} \}$
are functions on $\partial\Omega$ whose values at $x\in \partial\Omega$
depend only on $A$, $\{ g_{ij}(x, \cdot )\}$ and $n(x)$.
Then, it was proved in [1] that
Theorem 1 [Dirichlet Data]
Assume that $A$ is elliptic, smooth and 1-periodic. Let $\Omega$ be a smooth and strictly convex domain in $\mathbb{R}^d$. Let $\overline{f}$ denote the homogenized data in (3). Then
\begin{equation}
\| \overline{f}\|_{W^{1, p}(\partial\Omega)}
\le C_p \left(\int_{\mathbb{T}^d} \| f(\cdot, y)\|^2_{C^1(\partial\Omega)}\, dy\right)^{1/2}
\quad \text{ for any } 1< p< \infty,
\end{equation}
where $C_p$ depends only on $d$, $m$, $\lambda$, $p$, and $\| A\|_{C^k(\mathbb{T}^d)}$ for some $k=k(d, p)>1$.
Theorem 2 [Neumann Data]
Assume that $A$ is elliptic, smooth and 1-periodic. Let $\Omega$ be a smooth and strictly convex domain in $\mathbb{R}^d$. Let $\overline{g}=( \overline{g}_{ij}) $ denote the homogenized data in (4). Then
\begin{equation*}
\| \overline{g}\|_{W^{1, p}(\partial\Omega)}
\le C_p \left(\int_{\mathbb{T}^d} \| g(\cdot, y)\|^2_{C^1(\partial\Omega)}\, dy\right)^{1/2}
\quad \text{ for any } 1< p< \infty,
\end{equation*}
where $C_p$ depends only on $d$, $m$, $\lambda$, $p$, and $\| A\|_{C^k(\mathbb{T}^d)}$ for some $k=k(d, p)>1$.
The proofs for Dirichlet and Neumann are similar. The ingredients come from three parts: 1. Maximal principle for solutions in half-spaces; 2. Weighted estimates in half-spaces; 3. An interpolation argument that combines all these estimates. We also point that the results in Theorem 1 and 2 may be extended to domains of finite type considered in [4].
Article
Regularity of Homogenized Boundary Data in Periodic Homogenization of Elliptic Systems
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