Pointwise estimate in homogenization of Dirichlet problem for elliptic systems in divergence form

Partial SolutionYear of origin: 2013

Posted online: 2018-07-07 19:14:27Z by Hayk Aleksanyan 76

Cite as: S-180707.2

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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

We consider homogenization of Dirichlet problem for divergence type elliptic operator when the operator is fixed, and the boundary data is oscillating, which is a particular case of the setting introduced in [2].

Namely, for a small parameter $ \varepsilon>0 $ let $ u_\varepsilon $ be the solution of the following elliptic system with a Dirichlet boundary condition $$ -\nabla \cdot A(x) \nabla u_\varepsilon (x) =0 \text{ in } D \ \ \text{ and } \ \ u_\varepsilon(x) = g(x/ \varepsilon) \text{ on } \partial D, $$ where $ D \subset \mathbb{R}^d $ $ (d\geq 2) $ is a bounded domain, $g$ is $ \mathbb{Z}^d $-periodic, and the operator is uniformly elliptic. Let also $ u_0 $ be the solution to the same elliptic system in $D$ but with Dirichlet data equal to the average of $g$ over its unit cell of periodicity. The main result of [1] states that under strict convexity of the domain $D$, and $C^\infty$ smoothness of the data involved in the problem, for any $ \kappa>0 $ one has the following pointwise bound \begin{equation} | u_\varepsilon(x) - u_0(x) | \leq C_\kappa \min\left\{ 1, \frac{\varepsilon^{(d-1)/2}}{(d(x))^{d-1+\kappa} } \right\}, \qquad \forall x\in D, \tag{a} \end{equation} where $ d(x) $ denotes the distance of $ x $ from the boundary of $D$, and the constant $ C_\kappa =C(\kappa, d, A, D,g)>0 $. As a corollary to $(a)$, for any $ 1\leq p< \infty $ and any $ \kappa>0 $ we obtain \begin{equation} || u_\varepsilon - u_0 ||_{L^p(D)} \leq C_\kappa \varepsilon^{ \frac{1}{2p} - \kappa}. \tag{b} \end{equation}

 

The starting point of the proofs is representation of solutions via Poisson kernel; then the proof proceeds by an analysis of singular oscillatory integrals. There are two competing quantities in the Poisson integral representation; namely, the singularity of the Poisson kernel, and the oscillation of the boundary data. The smoothness of $A$ and $ \partial D $ allows one to obtain a nice (quantitative) control over singularities of the representation kernel and its derivatives. Next, the periodicity and smoothness of the boundary data $ g $, along with the strict convexity of the domain (ensuring non-zero Gauss curvature everywhere on the boundary) provide a lot of cancellations in the integral representing the solutions. With a careful trade-off between singularity of the integration kernel, and decay of the integral of the boundary data one obtains the desired estimate.

 

Remark 1. With some minor modifications, the proposed approach leads to homogenization when instead of strict convexity of the domain, one requires that at each point of the boundary at least $1\leq m \leq d-1$ of the principal curvatures are non-vanishing (strict convexity corresponds to the case when $ m=d-1 $). The pointwise estimate in this case wll be similar, with $ d-1 $ in the exponent replaced by $m$, while the $L^p$-estimates will remain the same.

 

Remark 2. The settings of [1] and [2] become the same for constant coefficient operators. Comparing bound $(b)$ with the exponent of $L^2$ estimate of [2] which equals $\frac{d-1}{3d+5} - $, we see that $(b)$ gives a better estimate for constant coefficient operators in dimensions up to $8$ and for dimensions at least $10$ the estimate of [2] takes over.

  1. Article Is an originApplications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise estimates

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

  2. Article Homogenization and boundary layers

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

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  • Created at: 2018-07-07 19:14:27Z