Regularity of Homogenized Boundary Data in Periodic Homogenization of Elliptic Systems

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 $(1)$-$(2)$.