[Paper Review] Weighted $L^2$ Estimates for Elliptic Homogenization in Lipschitz Domains

Zhongwei Shen|arXiv (Cornell University)|Apr 7, 2020

Advanced Mathematical Modeling in Engineering20 references4 citations

TL;DR

This paper establishes a necessary and sufficient condition for weighted $L^2$ estimates in Lipschitz domains for solutions to elliptic systems with rapidly oscillating, periodic, and VMO coefficients. Using a novel real-variable method, it reduces the weighted $W^{1,2}$ estimate to a reverse Hölder-type inequality involving the weight function, enabling uniform estimates across homogenization scales.

ABSTRACT

We develop a new real-variable method for weighted $L^p$ estimates. The method is applied to the study of weighted $W^{1, 2}$ estimates in Lipschitz domains for weak solutions of second-order elliptic systems in divergence form with bounded measurable coefficients. It produces a necessary and sufficient condition, which depends on the weight function, for the weighted $W^{1,2}$ estimate to hold in a fixed Lipschitz domain with a given weight. Using this condition, for elliptic systems in Lipschitz domains with rapidly oscillating, periodic and VMO coefficients, we reduce the problem of weighted estimates to the case of constant coefficients.

Motivation & Objective

• To develop a real-variable method for weighted $L^p$ estimates in elliptic homogenization.
• To identify a necessary and sufficient condition, dependent on the weight function, for $W^{1,2}$ estimates in bounded Lipschitz domains.
• To reduce the study of weighted estimates for oscillating coefficient systems to the case of constant coefficients.
• To establish uniform estimates in the homogenization limit ($\varepsilon \to 0$) under the derived condition.
• To extend known results on $L^p$ estimates to the weighted $L^2$ setting with $A_1$ weights, particularly for VMO and periodic coefficients.

Proposed method

• Introduces a new real-variable method tailored for weighted $L^p$ estimates in elliptic PDEs.
• Derives a reverse Hölder inequality condition (1.8) as a necessary and sufficient criterion for the weighted $W^{1,2}$ estimate.
• Applies the method to elliptic systems with bounded measurable, periodic, and VMO coefficients.
• Uses duality and extrapolation techniques to connect $A_1$ weight conditions to $L^p$ boundedness results.
• Employs Hardy-type inequalities and truncation arguments involving the distance to the boundary to control singular weights.
• Reduces the problem for oscillating coefficients to the constant-coefficient case via the homogenized matrix $\overline{A}$.

Experimental results

Research questions

• RQ1 What is a necessary and sufficient condition for the weighted $W^{1,2}$ estimate to hold in a fixed Lipschitz domain with a given $A_1$ weight?
• RQ2 How can weighted $L^2$ estimates for elliptic systems with rapidly oscillating coefficients be reduced to the constant-coefficient case?
• RQ3 Under what conditions on the weight $\omega$ does the inequality $\int_\Omega |\nabla u_\varepsilon|^2 \omega \,dx \leq C \int_\Omega |f|^2 \omega \,dx$ hold uniformly in $\varepsilon$?
• RQ4 Can the $A_1$ weight condition be used to derive $L^p$ estimates for $p>2$ via extrapolation?
• RQ5 What role does the VMO regularity of the coefficient matrix play in ensuring weighted $L^2$ boundedness?

Key findings

• Theorem 1.1 establishes that the weighted $W^{1,2}$ estimate holds if and only if the reverse Hölder inequality (1.8) holds for local solutions vanishing on the boundary.
• Condition (1.8) explicitly depends on the weight $\omega$, the domain $\Omega$, and the coefficient matrix $A$, making it sensitive to geometric and analytic structure.
• Theorem 1.2 shows that for $A \in \text{VMO}(\mathbb{R}^d)$ and $A$ periodic, the weighted estimate for $\mathcal{L}_\varepsilon$ reduces to the same estimate for the constant-coefficient operator with matrix $\overline{A}$.
• For power weights $\omega_\sigma(x) = \text{dist}(x,\partial\Omega)^\sigma$, the estimate holds uniformly for $|\sigma| \leq 2\kappa$ with $\kappa \in (0,1/2)$ depending on dimension, ellipticity, and Lipschitz character.
• Theorem 7.2 proves a weighted $L^2$ estimate with $\omega_\sigma(x) = \text{dist}(x,\partial\Omega)^\sigma$ for $|\sigma| \leq 2\kappa$, using a Hardy-type inequality and truncation via $\psi_t(x) = \text{dist}(x,\partial\Omega) + t$.
• The method enables uniform estimates in the homogenization limit, ensuring that the constant $C_\omega$ in (1.5) is independent of $\varepsilon > 0$.