Doubling inequalities and nodal sets in periodic elliptic homogenization
Doubling inequalities and nodal sets in periodic elliptic homogenization
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DOI：
10.1080/03605302.2021.1989699
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发表时间：
2021-01
影响因子：
1.9
通讯作者：
C. Kenig;Jiuyi Zhu;Jinping Zhuge
中科院分区：
文献类型：
--
作者：
C. Kenig;Jiuyi Zhu;Jinping Zhuge
摘要证明了一类周期系数快速振荡的线性椭圆型方程的显式倍增不等式，得到了弱解节点集的一致上界（在维Hausdorff测度下）。在不同的尺度上，用收敛速率、一个“解析性”的三球不等式和一个频率函数的单调性公式的组合证明了明显依赖于倍增指数的倍增不等式。利用二次不等式、调和函数逼近和迭代论证，给出了节点集的上界。
Abstract We prove explicit doubling inequalities and obtain uniform upper bounds (under -dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubling inequalities, explicitly depending on the doubling index, are proved at different scales by a combination of convergence rates, a three-ball inequality from certain “analyticity,” and a monotonicity formula of a frequency function. The upper bounds of nodal sets are shown by using the doubling inequalities, approximations by harmonic functions and an iteration argument.