Characterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form

TL;DR

This paper characterizes specific diffusion matrices that achieve an optimal $L^{ abla}infty$ convergence rate of order $\\varepsilon^2$ in periodic homogenization of elliptic equations in nondivergence form, advancing understanding of homogenization efficiency.

Contribution

It introduces a new class of diffusion matrices with optimal convergence rate and fully characterizes diagonal matrices in 2D, extending the theory to higher dimensions.

Findings

Identified a new class of $\\varepsilon^2$ diffusion matrices confirming a prior conjecture.

Provided a complete characterization of diagonal matrices in two dimensions.

Conducted systematic analysis for higher-dimensional cases.

Abstract

We characterize diffusion matrices that yield a $L^{\infty}$ convergence rate of $\mathcal{O}(\varepsilon^2)$ in the theory of periodic homogenization of linear elliptic equations in nondivergence-form. Such type-$\varepsilon^2$ diffusion matrices are of particular interest as the optimal rate of convergence in the generic case is only $\mathcal{O}(\varepsilon)$. First, we provide a new class of type-$\varepsilon^2$ diffusion matrices, confirming a conjecture posed in [15]. Then, we give a complete characterization of diagonal diffusion matrices in two dimensions and a systematic study in higher dimensions.