Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A brush problem. Homogenization involving thin domains and PDEs in graphs

José M. Arrieta, Joaquín Domínguez-de-Tena

TL;DR

The paper develops a nonperiodic homogenization theory for a Neumann elliptic problem in a brush domain consisting of a fixed base and many vanishing-width teeth. By adapting the unfolding operator to a nonperiodic setting and employing weighted Bochner spaces, it derives a limit problem on a fixed unfolded domain, characterized by a density function $\theta$ that may degenerate. The main result proves convergence of the original solutions to a pair $(u^a,u^b)$, with $u^b$ solving a base-domain problem and $\theta u^a$ governing the tooth-region coupling, and shows energy convergence yielding strong convergence. Under suitable geometric assumptions, the limit problem admits a graph interpretation, where the teeth are edges with weighted transmission conditions, providing a graph-based reformulation of the homogenized limit. These results extend previous periodic-geometry analyses to nonperiodic tooth distributions and more general tooth shapes, with implications for effective modeling of flow and diffusion in lung-, membrane-, or porous-structured media.

Abstract

This work analyses the homogenization of a linear elliptic equation with Neumann boundary conditions in a comb/brush domain, composed of a fixed base and a family of thin teeth. The teeth are defined as rescalings of order less than or equal to $\varepsilon$ of a model tooth of arbitrary shape. Periodicity in their distribution is not assumed; instead, the existence of an asymptotic limit density $θ$, which may vanish in certain regions, is assumed. The convergence analysis is performed using an adaptation of the unfolding operator method to a non-periodic framework. Finally, it is shown that, under certain conditions on the geometry of the teeth, the resulting limit problem can be interpreted as a differential equation on a graph.

A brush problem. Homogenization involving thin domains and PDEs in graphs

TL;DR

The paper develops a nonperiodic homogenization theory for a Neumann elliptic problem in a brush domain consisting of a fixed base and many vanishing-width teeth. By adapting the unfolding operator to a nonperiodic setting and employing weighted Bochner spaces, it derives a limit problem on a fixed unfolded domain, characterized by a density function that may degenerate. The main result proves convergence of the original solutions to a pair , with solving a base-domain problem and governing the tooth-region coupling, and shows energy convergence yielding strong convergence. Under suitable geometric assumptions, the limit problem admits a graph interpretation, where the teeth are edges with weighted transmission conditions, providing a graph-based reformulation of the homogenized limit. These results extend previous periodic-geometry analyses to nonperiodic tooth distributions and more general tooth shapes, with implications for effective modeling of flow and diffusion in lung-, membrane-, or porous-structured media.

Abstract

This work analyses the homogenization of a linear elliptic equation with Neumann boundary conditions in a comb/brush domain, composed of a fixed base and a family of thin teeth. The teeth are defined as rescalings of order less than or equal to of a model tooth of arbitrary shape. Periodicity in their distribution is not assumed; instead, the existence of an asymptotic limit density , which may vanish in certain regions, is assumed. The convergence analysis is performed using an adaptation of the unfolding operator method to a non-periodic framework. Finally, it is shown that, under certain conditions on the geometry of the teeth, the resulting limit problem can be interpreted as a differential equation on a graph.
Paper Structure (24 sections, 37 theorems, 282 equations, 11 figures)

This paper contains 24 sections, 37 theorems, 282 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Statement of the problem
  3. Description of the problem
  4. Main result
  5. Overview of the Method
  6. Notation
  7. Tools and functional framework
  8. Unfolding operator
  9. The space of solutions and some functional framework
  10. Weighted Lebesgue-Bochner Spaces
  11. The unfolding operator and Bochner spaces
  12. Traces
  13. Partially Constant Sobolev Spaces
  14. The space of solutions for the limit unfolded problem
  15. Derivation of the results
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $u_\varepsilon$ be the solution of eqn:main and let $(u^a, u^b)\in H(\theta)$ be the solution of the unfolded problem described above. Redefine $u^a(x)=0$ when $x\in\Theta_0$. Then, $u^a$ belongs to $L^2(\Omega';H^1(Y))$ and, defining for $\varepsilon>0$ we have $\bar{u}^a_\varepsilon \in H^1(\Omega_\varepsilon)$ and

Figures (11)

  • Figure 1: Examples of domains with thin structures. From left to right: three examples of thin domains, including one of the form $0 < y < \varepsilon g(x)$HR, an oscillating thin domain $0 < y < \varepsilon g(x/\varepsilon^\alpha)$VP1, and a thin domain of arbitrary shape with holes $\Omega_\varepsilon = \{(x,y) : (x, y/\varepsilon) \in \Omega\}$prizzi2001effect. The last two are domains with thin structures: a dumbbell domain, where two bodies are joined by a thin channel dumbell, and a comb domain, where a body has several teeth attached to its top gaudiello.
  • Figure 2: An example of a domain $\Omega_\varepsilon$ in 2 dimensions is shown in the figure. In blue, at the top, the domain in question is displayed. Below, their components $\Omega^b$ and $\Omega_\varepsilon^a$ are depicted in red at the bottom left and green at the bottom center, respectively. In pink, at the bottom right, the unit cell $Y$ and an example of a rescaled cell $Y_\varepsilon^n$ are shown. Observe that the set $\Omega^b$ is contained in the ball $B(0, R_0)$ and contains the set $\Omega' \times (-1,0)$. The set $\Omega_\varepsilon^a$ is composed of the cells $Y_\varepsilon^n$ for $n = 1, \ldots, N_\varepsilon$ which have different horizontal size. Note that $Y$ contains the set $\omega \times \delta_0$ and observe that the unit cell $Y$ has an arbitrary shape which may not be simply connected.
  • Figure 3: In the figure, two examples of 2-dimensional domains $\Omega_\varepsilon$ are shown for different values of $\varepsilon$. The first three figures at the left, in blue, represent a domain with periodically distributed teeth. In this case, the limit density $\theta$ takes a constant value. The other three figures at the right, in red, a domain with non-periodically distributed teeth is shown. In this case, the space between teeth increases linearly (the separation between the n-th and (n+1)-th teeth is n times the separation of the first and second teeth). In this case, the limit density has the form $\theta(x) = (1 - x)/2$.
  • Figure 4: On the left, in blue, a particular domain $\Omega_\varepsilon$ is shown and on the right, in yellow, its corresponding unfolded domain $W$.
  • Figure 5: Example of an extended cell.
  • ...and 6 more figures

Theorems & Definitions (100)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 90 more