Nodal Sets and Doubling Conditions in Elliptic Homogenization

TL;DR

This paper establishes uniform measure bounds for nodal sets of solutions to elliptic equations with oscillating periodic coefficients, using doubling conditions and homogenization techniques.

Contribution

It provides the first uniform estimates on the measure of nodal sets in elliptic homogenization, connecting oscillating coefficients with homogenized solutions.

Findings

Uniform bounds on Hausdorff measure of nodal sets independent of oscillation parameter

Demonstrates the effectiveness of doubling conditions in homogenization context

Shows approximation of solutions by homogenized solutions preserves nodal set measures

Abstract

This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators $\{ \mathcal{L}_\e\}$ in divergence form with rapidly oscillating and periodic coefficients. We show that the $(d-1)$-dimensional Hausdorff measures of the nodal sets of solutions to $\mathcal{L}_\e (u_\e)=0$ in a ball in $\R^d$ are bounded uniformly in $\e>0$. The proof relies on a uniform doubling condition and approximation of $u_\e$ by solutions of the homogenized equation.