Doubling inequalities and nodal sets in periodic elliptic homogenization
Carlos E. Kenig, Jiuyi Zhu, Jinping Zhuge · Communications in Partial Differential Equations · 2021
We prove explicit doubling inequalities and obtain uniform upper bounds (under (d−1)-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.