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Influence of Lower-Order Terms on the Convergence Rates in Stochastic Homogenization of Elliptic Equations

Man Yang

Abstract

In this study, we investigate the convergence rates for the homogenization of elliptic equations with lower-order terms under the spectral gap assumption, in both bounded domains and the entire space. Our analysis demonstrates that lower-order terms significantly affect the convergence rate, particularly in the full space, where the rate changes from \(O(ε)\) (observed without lower-order terms) to \(O(ε^{d/({d+2})})\) due to their influence. In contrast, in bounded domains, the convergence rate remains \(O(ε^{1/2})\), as boundary conditions exert a stronger influence than the lower-order terms. To manage the complexities introduced by lower-order terms, we developed a novel technique that localizes the analysis within small grids, enabling the application of the Poincaré inequality for effective estimates. This work builds upon existing frameworks, offering a refined approach to quantitative homogenization with lower-order terms.

Influence of Lower-Order Terms on the Convergence Rates in Stochastic Homogenization of Elliptic Equations

Abstract

In this study, we investigate the convergence rates for the homogenization of elliptic equations with lower-order terms under the spectral gap assumption, in both bounded domains and the entire space. Our analysis demonstrates that lower-order terms significantly affect the convergence rate, particularly in the full space, where the rate changes from \(O(ε)\) (observed without lower-order terms) to \(O(ε^{d/({d+2})})\) due to their influence. In contrast, in bounded domains, the convergence rate remains \(O(ε^{1/2})\), as boundary conditions exert a stronger influence than the lower-order terms. To manage the complexities introduced by lower-order terms, we developed a novel technique that localizes the analysis within small grids, enabling the application of the Poincaré inequality for effective estimates. This work builds upon existing frameworks, offering a refined approach to quantitative homogenization with lower-order terms.

Paper Structure

This paper contains 10 sections, 13 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Setting and Assumptions
  4. Main Theorems
  5. Preliminary
  6. Flux Corrector and Localized Correctors
  7. Estimates for Correctors
  8. Main Estimates
  9. Proofs of Main Theorems
  10. Examples

Key Result

Theorem 1

Let $d\geq 3$. Let $\Lambda$, $\mathbf{a}^\epsilon$ and $\mathbf{b}^\epsilon$ be coefficients that satisfy the assumptions A1-A5, P1 and P2. Let $\bar{\mathbf{a}}$ and $\bar{\mathbf{b}}$ be as in Definition homogenized a. Let $u_{0} \in H^2(\mathbb{R}^d; \mathbb{R})\cap W^{1,\infty}(\mathbb{R}^d; \m in a distributional sense. Then we have where $\mathcal{C}(\omega_\epsilon)$ is a random constant

Figures (2)

  • Figure 1: Comparison of Gaussian random fields with short-range (left) and long-range (right) correlations. The covariance functions are indicated above each field.
  • Figure 2: Comparison of Gaussian Random Fields with Different Correlation Lengths.

Theorems & Definitions (26)

  • Definition 1: Spectral Gap Inequality
  • Definition 2: First-Order Corrector
  • Definition 3: Homogenized Coefficients
  • Theorem 1
  • Theorem 2
  • Definition 4: Flux Corrector
  • Lemma 1: Localized Correctors
  • Lemma 2: Localized Flux Correctors
  • Lemma 3
  • Lemma 4
  • ...and 16 more