Lower Bound of Nodal Sets in Elliptic Homogenization and Functions with Strong Maximum Principle

TL;DR

This paper establishes a constant lower bound for the volume of nodal sets in elliptic homogenization and more general functions satisfying the strong maximum principle, with specific results in two dimensions.

Contribution

It introduces a uniform lower bound for nodal volumes in elliptic homogenization and extends this to broader classes of functions with the strong maximum principle.

Findings

Constant lower bound for nodal volume in 2D elliptic homogenization

Extension of lower bound results to general functions with strong maximum principle

Applicable beyond solutions to elliptic PDEs

Abstract

In this note, we first try to prove a uniform lower bound of nodal volume in elliptic homogenization setting. This lower bound is far from optimal. But, we can prove a constant lower bound in dimension two. Motivated by the proof, we extend this results to more general settings. To be more specific, we prove that the nodal volume has a constant lower bound for all continuous functions with strong maximum principle. Our result works for general functions beyond solutions to elliptic PDEs.