[Paper Review] Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains

Li Wang, Qiang Xu+1 more|arXiv (Cornell University)|Jan 16, 2020

Advanced Mathematical Modeling in Engineering44 references4 citations

TL;DR

This paper establishes sharp O(ε) convergence rates for nonlinear elliptic homogenization in periodically perforated domains, overcoming boundary layer effects that otherwise degrade convergence to O(ε^{1/2}). By developing new L^p extension theory and Poincaré-Sobolev inequalities on perforated domains, the authors derive large-scale interior and boundary Lipschitz estimates and prove quenched Calderón-Zygmund estimates via real-variable methods.

ABSTRACT

This paper was devoted to study the quantitative homogenization problems for nonlinear elliptic operators in perforated domains. We obtained a sharp error estimate $O(\varepsilon)$ when the problem was anchored in the reference domain $\varepsilonω$. If concerning a bounded perforated domain, one will see a bad influence from the boundary layers, which leads to the loss of the convergence rate by $O(\varepsilon^{1/2})$. Equipped with the error estimates, we developed both interior and boundary Lipschitz estimates at large-scales. As an application, we received the so-called quenched Calderón-Zygumund estimates by Shen's real arguments. To overcome some difficulties, we improved the extension theory from (\cite[Theorem 4.3]{OSY}) to $L^p$-versions with $\frac{2d}{d+1}-ε

Motivation & Objective

• To establish quantitative convergence rates for nonlinear elliptic operators in periodically perforated domains with mixed boundary conditions.
• To address the degradation of convergence rate from O(ε) to O(ε^{1/2}) due to boundary layer effects in bounded perforated domains.
• To develop large-scale interior and boundary Lipschitz estimates for solutions to homogenized problems.
• To prove quenched Calderón-Zygumund estimates using real-variable techniques and improved extension theory.
• To extend the L^p extension theory to the range $\frac{2d}{d+1}-\epsilon < p < \frac{2d}{d-1}+\epsilon$ for $0 < \epsilon \ll 1$, enabling new Poincaré-Sobolev inequalities on perforated domains.

Proposed method

• Derive error estimates via a refined analysis of correctors and flux correctors in the periodic setting.
• Establish $L^p$-bounded extension operators for functions on perforated domains, extending prior results to a wider range of $p$ near $2d/(d\pm1)$.
• Use John-Nirenberg inequality and BMO estimates on $\ln u$ to control oscillations and derive quenched $L^p$ bounds.
• Apply real-variable methods inspired by Shen to prove quenched Calderón-Zygumund estimates for the homogenized operator.
• Prove local Poincaré-Sobolev inequalities on perforated domains using the extended $L^p$ theory and weighted norm estimates.
• Leverage the structure of the monotone operator $A(y,\xi)$ satisfying coercivity, growth, and Hölder continuity in $y$ to ensure solvability and stability.

Experimental results

Research questions

• RQ1 What is the optimal convergence rate for nonlinear elliptic homogenization in perforated domains when the reference domain is unbounded?
• RQ2 How do boundary layers in bounded perforated domains affect the convergence rate, and can this loss be quantitatively controlled?
• RQ3 Can large-scale interior and boundary Lipschitz estimates be established for solutions to nonlinear elliptic equations in perforated domains?
• RQ4 What is the sharp range of $p$ for which $L^p$ extension operators exist on perforated domains, and how does this extend previous results?
• RQ5 Can quenched Calderón-Zygumund estimates be proven for nonlinear elliptic operators using real-variable techniques in the context of stochastic homogenization?

Key findings

• A sharp $O(\varepsilon)$ convergence rate is obtained for the homogenization of nonlinear elliptic operators in the reference domain $\varepsilon\omega$, which is unbounded and 1-periodic.
• In bounded perforated domains, boundary layer effects reduce the convergence rate to $O(\varepsilon^{1/2})$, which is shown to be optimal under the given assumptions.
• The authors establish $L^p$ extension theory for $\frac{2d}{d+1}-\epsilon < p < \frac{2d}{d-1}+\epsilon$ with $0 < \epsilon \ll 1$, extending previous results to a critical range near the critical exponent for Sobolev embedding.
• Using the extended $L^p$ theory, the authors derive local Poincaré-Sobolev inequalities on perforated domains, which are essential for large-scale regularity estimates.
• Large-scale interior and boundary Lipschitz estimates are proven, providing control on the gradient of solutions at scales larger than $\varepsilon$.
• Quenched Calderón-Zygumund estimates are established via real-variable arguments, extending classical results to the nonlinear and stochastic homogenization setting.