Doubling inequalities and nodal sets in periodic elliptic homogenization

Carlos E. Kenig; Jiuyi Zhu; Jinping Zhuge

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arXiv · 2101.04841

Doubling inequalities and nodal sets in periodic elliptic homogenization

Carlos E. Kenig·Jiuyi Zhu·Jinping Zhuge

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Abstract

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.

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Carlos E. Kenig, Jiuyi Zhu, Jinping Zhuge. 2021-01-13. Doubling inequalities and nodal sets in periodic elliptic homogenization. https://arxiv.org/abs/2101.04841

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