Almost sharp lower bound for the nodal volume of harmonic functions

Abstract

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let $u$ be a real-valued harmonic function in ${\mathbb{R}}^n$ with $u(0)=0$ and $n\ge 3$. We prove $${\mathcal{H}}^{n-1}(\{u=0\}\cap B(0,2)){\gtrsim }_{\epsilon }{N}^{1-\epsilon },$$ where the doubling index $N$ is a notion of growth defined by $$\underset{B(0,1)}{\sup }|u|=2^N\underset{B(0,\frac{1}{2})}{\sup }|u|.$$ This gives an almost sharp lower bound for the Hausdorff measure of the zero set of $u$, which is conjectured to be linear in $N$. The new ingredients of the article are the notion of stable growth, and a multi-scale induction technique for a lower bound for the distribution of the doubling index of harmonic functions. It gives a significant improvement over the previous best-known bound ${\mathcal{H}}^{n-1}\left(\{u=0\}\cap 2B\right)\ge \exp (c\log N/\log \log N)$, which implied Nadirashvili's conjecture.

60 pages, 7 pictures, comments are welcome!