发表机构
SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院; 中国科学院大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对非散度型椭圆方程，在临界椭圆矩条件下建立定量均匀化估计，构造二次校正子，证明大尺度正则性与Liouville定理，并识别有效矩阵。
AI 中文摘要
我们证明了具有平稳、对称系数、有限依赖范围以及确定性上椭圆界的非散度型线性椭圆方程的定量均匀化估计。我们不施加确定性的正下界。我们假设单位球上最小特征值下确界的倒数具有阶为$d$的有限矩，并满足Armstrong和Smart的常见连续性条件。在这些假设下，我们获得了有限胞误差以及具有非零源的Dirichlet均匀化误差的代数概率界。我们在全空间上构造了模仿射函数的二次校正子，并证明了第一阶和第二阶大尺度正则性以及相应的Liouville定理。我们还通过不变密度识别了有效矩阵，并量化了密度和加权系数的空间平均的光滑性。证明将单位迹扩散与其物理时钟分离。对停止粗过程的格林函数的逆Hölder估计提供了在临界时刻控制时钟所需的可积性增益。应用包括加权梯度收敛和有效矩阵的有限域近似。
英文摘要
We prove quantitative homogenization estimates for linear elliptic equations in nondivergence form with stationary, symmetric coefficients, finite range of dependence, and a deterministic upper ellipticity bound. No deterministic positive lower bound is imposed. We assume that the reciprocal of the infimum of the smallest eigenvalue on a unit ball has a finite moment of order $d$, together with the common continuity condition of Armstrong and Smart. Under these assumptions, we obtain algebraic probability bounds for finite-cell errors and for Dirichlet homogenization errors with nonzero sources. We construct quadratic correctors on the whole space, modulo affine functions, and prove first- and second-order large-scale regularity and the corresponding Liouville theorems. We also identify the effective matrix through the invariant density and quantify smooth spatial averages of the density and the weighted coefficients. The proof separates a unit-trace diffusion from its physical clock. A reverse Hölder estimate for the Green function of a stopped coarse process yields the integrability gain needed to control the clock at the critical moment. Applications include weighted gradient convergence and a finite-domain approximation of the effective matrix.
Comments76 pages
