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

Jiahuan Li, Zhichen Ying

TL;DR

This paper establishes quantitative lower bounds for nodal sets in two complementary settings: elliptic homogenization and general SMP functions. For n≥3, it proves a lower bound on the nodal (n−1)-dimensional measure of ε-perturbed elliptic equations via harmonic approximation and doubling-index techniques, yielding a bound that behaves like C/N0^{n−1}. In dimension two, it proves a constant lower bound independent of ε, and separately shows that any continuous SMP function with u(0)=0 has nodal length at least 2. Together, these results extend nodal-set lower bounds beyond classical PDE contexts and highlight the role of maximum principles in guaranteeing nontrivial nodal structure.

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.

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

TL;DR

This paper establishes quantitative lower bounds for nodal sets in two complementary settings: elliptic homogenization and general SMP functions. For n≥3, it proves a lower bound on the nodal (n−1)-dimensional measure of ε-perturbed elliptic equations via harmonic approximation and doubling-index techniques, yielding a bound that behaves like C/N0^{n−1}. In dimension two, it proves a constant lower bound independent of ε, and separately shows that any continuous SMP function with u(0)=0 has nodal length at least 2. Together, these results extend nodal-set lower bounds beyond classical PDE contexts and highlight the role of maximum principles in guaranteeing nontrivial nodal structure.

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.

Paper Structure

This paper contains 6 sections, 19 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Uniform Lower Bound for Elliptic Homogenization
  3. Properties from Homogenization Theory
  4. Approximation and Volume Estimate
  5. Special Case in Two Dimension
  6. Special Case in SMP function

Key Result

Theorem 1.2

Let $B\subset\mathbb{R}^{n}$ be a unit ball and $n\geq 3$. For every $\varepsilon>0$, there exists a constant $c$ depending on dimension $n$ and $\varepsilon$ such that for every harmonic function $u : 4B\rightarrow \mathbb{R}$ with $u(0)=0$, we have Here $N_u(x,r)$ is the doubling index introduced later.

Theorems & Definitions (34)

  • Conjecture 1.1: MR645728
  • Theorem 1.2
  • Conjecture 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8: Nadirashvili's conjecture
  • Theorem 1.9
  • Theorem 2.1
  • ...and 24 more