论文标题
关于以非差异形式的线性椭圆方程定期均质化的最佳收敛速率的评论
Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form
论文作者
论文摘要
我们研究并表征了以非差异形式的线性椭圆方程定期均质化中收敛速率。我们获得最佳的收敛速率为$ O（\ varepsilon）$或$ o（\ varepsilon^2）$，具体取决于扩散矩阵$ a $，源项$ f $和边界数据$ g $。此外，我们表明，在$ c^{2，α} $周期性，对称和正确定的矩阵中，给出最佳速率$ o（\ varepsilon）$的扩散矩阵$ a $是开放的，并且在一套$ c^{2，α} $中，这意味着一般的最佳速率为$ o（\ varepsilon）$。
We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $O(\varepsilon)$ or $O(\varepsilon^2)$ depending on the diffusion matrix $A$, source term $f$, and boundary data $g$. Moreover, we show that the set of diffusion matrices $A$ that give optimal rate $O(\varepsilon)$ is open and dense in the set of $C^{2,α}$ periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $O(\varepsilon)$.
