Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic Analysis of Laplacian Operator in Thin Domains on the Sphere with Highly Oscillatory Boundary

Naísa C. Garcia, Raquel Lehrer, Marcus A. M. Marrocos

Abstract

In this work we analyse the convergence of solutions of the Poisson equation with Neumann boundary conditions in a thin domain with highly oscillatory behavior $\mathcal{U}^\varepsilon$ contained in the sphere $\mathbb{S}^2$. Using the Multiple Scales method, we obtain the homogenized limit problem and analyse the convergence of solutions, as $\varepsilon$ tends to $0$. Introducing appropriate correctors, we show strong convergence and give error estimates.

Asymptotic Analysis of Laplacian Operator in Thin Domains on the Sphere with Highly Oscillatory Boundary

Abstract

In this work we analyse the convergence of solutions of the Poisson equation with Neumann boundary conditions in a thin domain with highly oscillatory behavior contained in the sphere . Using the Multiple Scales method, we obtain the homogenized limit problem and analyse the convergence of solutions, as tends to . Introducing appropriate correctors, we show strong convergence and give error estimates.
Paper Structure (15 sections, 14 theorems, 231 equations)

This paper contains 15 sections, 14 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. The multiple scales method and the homogenized equation
  3. Convergence to the homogenized limit
  4. Construction of Tartar's test functions $\tau^\varepsilon$
  5. Limit of $\chi^{\varepsilon}$.
  6. Limit of functions extended by zero $\frac{\widetilde{\partial u^{\varepsilon}}}{\partial \varphi_1}$, $\frac{\widetilde{\partial u^{\varepsilon}}}{\partial \theta_1}$ and $\widetilde{f^\varepsilon}$
  7. Limit of the extended functions $P_{\varepsilon}u^{\varepsilon}$ and $\frac{\partial P_{\varepsilon}u^{\varepsilon} }{\partial \theta_1}$
  8. Limit in $\tau^{\varepsilon}$.
  9. Limit of $\widetilde{\frac{\partial \tau^\varepsilon}{\partial \varphi_1}}$.
  10. Proof of the main theorem
  11. Higher Order Multiscale Expansion of solutions
  12. Variational formulation of the problem
  13. First-order corrector
  14. Second-order corrector
  15. Concluding Remarks

Key Result

Lemma 3.1

Let $\sigma_{\varepsilon}=\cos(\varepsilon x_2)$ be the weight defined in $\mathcal{O}^{\varepsilon}$, and $W^{1,p}_{per}(\mathcal{O}^{\varepsilon})$ the set of functions in $W^{1,p}(\mathcal{O}^{\varepsilon})$ which are $I$-periodic in the variable $x_1$, analogously $L_{per}^2(\mathcal{O}^{\vareps a constant $K$ independent of $\varepsilon$ and $p$ such that for all $\varphi \in W^{1,p}(\mathca

Theorems & Definitions (35)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • proof
  • Proposition 3.6
  • ...and 25 more