Doubling inequalities and nodal sets in periodic elliptic homogenization

TL;DR

This paper establishes explicit doubling inequalities and uniform bounds on the size of nodal sets for solutions to periodic elliptic equations with rapidly oscillating coefficients, advancing understanding in elliptic homogenization.

Contribution

It introduces new explicit doubling inequalities and uniform bounds on nodal sets for elliptic equations with oscillating coefficients, using a novel combination of techniques.

Findings

Explicit doubling inequalities depending on the doubling index.

Uniform upper bounds on nodal set measures.

Methodology combining convergence rates, three-ball inequalities, and frequency monotonicity.

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.