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

We consider periodic homogenization of the linear elliptic equation \(-A(x/\varepsilon ):D^2 u^{\varepsilon } = f\) $-A(x/\epsilon ):D^2{u}^{\epsilon }=f$ posed in a smooth domain \(\Omega \) $\mathrm{\Omega }$ subject to the Dirichlet boundary condition \(u^{\varepsilon } = g\) $u^{\epsilon }=g$ on \(\partial \Omega \) $\partial \mathrm{\Omega }$ . It is known that, while the optimal \(L^{\infty }\) $L^{\infty }$ -rate for the convergence of \((u^{\varepsilon })_{\varepsilon > 0}\) ${(u^{\epsilon })}_{\epsilon >0}$ to the solution of the homogenized problem is generically only \(\mathcal {O}(\varepsilon )\) $\mathcal{O}(\epsilon )$ , those diffusion matrices A for which the symmetric part of the third-order homogenized tensor vanishes lead to a \(L^{\infty }\) $L^{\infty }$ -rate of \(\mathcal {O}(\varepsilon ^2)\) $\mathcal{O}({\epsilon }^2)$ for all sufficiently regular f and g. In this work, we seek more explicit characterizations of such “type- \(\varepsilon ^2\) ${\epsilon }^2$ diffusion matrices" than the one via the third-order homogenized tensor. We start by providing a new class of type- \(\varepsilon ^2\) ${\epsilon }^2$ diffusion matrices, confirming an open conjecture. Thereafter, we give a complete characterization of diagonal type- \(\varepsilon ^2\) ${\epsilon }^2$ diffusion matrices in two dimensions and a systematic study in higher dimensions.