论文标题
具有快速振荡系数的临界椭圆方程组的关键集
Critical Sets of Elliptic Equations with Rapidly Oscillating Coefficients in Two Dimensions
论文作者
论文摘要
在本文中，我们继续研究以差异形式的二阶椭圆方程的关键解决方案$ u_ \ e $，并具有迅速振荡和周期性系数。在\ cite {lin-shen-3d}中，通过控制近似切线的“转动”，我们表明，$（d-2）$ - 尺寸的Hausdorff测量值相对于周期$ \ e $均匀地界定了$ \ e $，前提是解决方案的双倍指数是界限的。在本文中，我们根据$ u_ \ e $的加倍索引的减少来研究二维情况。证据依赖于以下事实：二维级或更高尺寸的均匀谐波多项式的临界集仅包含一个点。
In this paper we continue the study of critical sets of solutions $u_\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. In \cite{Lin-Shen-3d}, by controling the "turning" of approximate tangent planes, we show that the $(d-2)$-dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period $\e$, provided that doubling indices for solutions are bounded. In this paper we use a different approach, based on the reduction of the doubling indices of $u_\e$, to study the two-dimensional case. The proof relies on the fact that the critical set of a homogeneous harmonic polynomial of degree two or higher in dimension two contains only one point.
