$L^p$ estimates in homogenization of Dirichlet problem for elliptic systems in divergence form

Partial SolutionYear of origin: 2015

Posted online: 2018-07-07 19:14:43Z by Hayk Aleksanyan 62

Cite as: S-180707.3

View pdf
Boundary layers in periodic homogenization of Dirichlet problem

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

For each $\varepsilon>0 $ let $ u_\varepsilon $ be the solution to the following Dirichlet problem $$ -\nabla \cdot A_\varepsilon(x) \nabla u_\varepsilon (x) =0 \text{ in } D \ \ \text{ and } \ \ u_\varepsilon(x) = g(x,x/ \varepsilon) \text{ on } \partial D, $$ where $D \subset \mathbb{R}^d $ $ (d\geq 2) $ is a bounded domain, $g(x, \cdot)$ is $ \mathbb{Z}^d $-periodic for any $ x\in \partial D$ and the operator is uniformly elliptic. We study the problem under strict convexity of the domain $D$ and $C^\infty$- smoothness of the data involved in the problem.

 

First, in the case when the coefficients $A_\varepsilon(x)$ do not depend on $ \varepsilon>0 $ (i.e. for some fixed $A$ one has $ A_\varepsilon \equiv A $; the operator is fixed, but not necessarily constant) for any $ 1\leq p < \infty $ we have the following convergence result \begin{equation} \tag{1} \|u_\varepsilon - u_0\|_{L^p(D)} \leq C_p \begin{cases} \varepsilon^{1/2p} ,& d=2 , \\ (\varepsilon |\ln \varepsilon |)^{1/ p}, & d=3 , \\ \varepsilon^{1/ p} , & d \geq 4, \end{cases} \end{equation} where $ u_0 $ satisfies the same elliptic system but with Dirichlet data set to $ \overline{g}(x):= \int_{\mathbb{T}^d} g(x, y) dy $, where $ \mathbb{T}^d $ is the unit torus of $\mathbb{R}^d $. By establishing a certain type of ergodic theorem for scaled surfaces (which essentially states that large scalings of a strictly convex smooth surface modulo $\mathbb{Z}^d$ become equidistributed in $\mathbb{T}^d$ when the size of the scaling tends to infinity) we also show that the convergence rates given in $(1)$ are (generically) sharp in dimensions higher than three, and sharp up to logarithmic factor in dimension three. This, in particular, answers a question, for dimensions higher than 2, posed in [3, page 139] concerning optimal rates of convergence in homogenization of this problem.

 

Second, when $ A_\varepsilon(x) \equiv A(x/ \varepsilon) $ for some $ \mathbb{Z}^d $-periodic tensor $A$, (the case of simultaneous oscillations in the operator and the boundary data) combining our method with a result on homogenization of Poisson kernel by Kenig, Lin, and Shen (see [4]) we prove in dimensions greater than two and for some special class of coefficients $A$ (namely, that the solutions to cell-problem for $A$ are trivial, this is equivalent to certain vector-fields build from components of $A$ being divergence-free), that \begin{equation} \tag{2} \|u_\varepsilon - u_0\|_{L^p(D)} \leq C_p [ \varepsilon (\ln(1/ \varepsilon))^2 ]^{1/p}, \end{equation} for any $1\leq p< \infty $. In this special case our result settles the homogenization in its optimal form. Here $u_0 $ solves the homogenized problem with constant coefficient system and boundary data (which in our special case can be computed explicitly) depending both on coefficients of the original as well as homogenized operators, the original boundary data, and the normal field of the boundary of the domain.

 

The point of departure in the approach leading to estimates $(1)$ and $(2)$ is the Poisson representation of the solution $u_\varepsilon$. Similar approach was used by the same authors in [2]; the motivation for revisiting the $L^p$-estimates obtained in [2] (as a corollary to pointwise bounds), comes from the following: for the problem with simultaneously oscillating operator and boundary data, as in $(2)$ above, the well-known paper [3] by Gérard-Varet and Masmoudi proves that for dimensions $d\geq 2$ one has homogenization of the $\varepsilon$-problem with convergence rate in $L^2$ of order $\varepsilon^{ (d-1)/(3d+5) -\delta } $ where $\delta>0$ is arbitrarily small. Comparing this result with the $L^2$-estimate of [2] in the case of constant coefficient operators when the settings of [3] and [2] coincide, we see that the convergence exponent $1/4$ established in [2], is better than $\frac{d-1}{3d+5} $ in dimensions up to $8$, and becomes worse for $d\geq 10$. This fact served as the motivating factor to revisit the problem and look for optimal exponents of the convergence.

 

The proof of $(1)$, as in [2], is based on the analysis of singular oscillatory integrals which enter the proof via Poisson kernel representation. The improvement of $L^p$-estimate over [2], is very roughly due to the fact that here one is dealing with domain integration, as opposed to surface integration studied in [2] for pointwise estimates. Here the singularity of type $ |x|^{-\alpha} $ (stemming from the Poisson kernel) near the origin has a better threshold for integration.

The refined analysis of [1] gives optimal bounds in $L^p$ however does not provide any pointwise estimates as we had in [2].

  1. Article Is an originApplications of Fourier analysis in homogenization of Dirichlet problem. $L^p$ estimates

    Archive for Rational Mechanics and Analysis 65 (1), 65-87, 2015fulltext

  2. Article Applications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise estimates

    Journal of Differential Equations 254 (6), 2626-2637, 2013arXivfulltext

  3. Article Homogenization and boundary layers

    Acta Mathematica 209 (1), 133-178, 2012

  4. Article Periodic Homogenization of Green and Neumann Functions

    Communications on Pure and Applied Mathematics 67 (8), 1219-1262, 2014arXiv

No remarks yet

  • Created at: 2018-07-07 19:14:43Z